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Qm Math Oscillator

1 · Prerequisites

2 · Summary

The oscillator layer of the non-relativistic quantum mechanics mathematical prerequisites, all of it mathematics: the abstract diagonal operator as the exact model of an eigenbasis with prescribed real eigenvalues, the Hermite family of the line with its ladder operators and their closed maximal domains, the scaling that realises the one-dimensional oscillator as the self-adjoint differential operator $- \frac{ℏ^{2}}{2 m} \frac{d^{2}}{d x^{2}} + \frac{m ω^{2}}{2} x^{2}$ , the coherent vectors and their exact evolution and variance, and the product-basis completeness of the anisotropic multidimensional oscillator. Every item carries domain: mathematics and depends only on mathematical suppliers, published or from qm-math-composites; nothing here asserts a physical interpretation, measurement model or empirical claim, and the companion page qm-math-oscillator-examples stays mathematics as well.

The operator layer begins with Diagonal adjoint and graph-core lemma: for a complete orthonormal family $(e_{n})$ and real $λ_{n}$ , the finite-support operator $S e_{n} = λ_{n} e_{n}$ has exactly one self-adjoint extension, namely the diagonal operator $A$ with domain ${x : \sum_{n} λ_{n}^{2} ∣ ⟨ x, e_{n} ⟩ ∣^{2} < \infty}$ , and the algebraic span is a core on which the finite truncations converge in graph norm. Diagonal spectrum computes its spectrum, $σ (A) = \overline{{λ_{n}}}$ , proves that every $λ_{n}$ is an eigenvalue and exhibits the resolvent $R (λ, A) x = \sum_{n} (λ - λ_{n})^{- 1} ⟨ x, e_{n} ⟩ e_{n}$ with norm $(\inf_{n} ∣ λ - λ_{n} ∣)^{- 1}$ outside the closure.

The Hermite layer is the analytic heart of the page. Hermite recurrence equips $L^{2} (R, d y)$ with its complex Hilbert-space pairing and defines $g_{0} = π^{- 1 / 4} e^{- y^{2} / 2}$ , the polynomial-Gaussian space $P$ , the operators $a = (Y + \partial_{y}) / \sqrt{2}$ , $a^{†} = (Y - \partial_{y}) / \sqrt{2}$ and the vectors $g_{n} = (a^{†})^{n} g_{0} / \sqrt{n!}$ ; it proves the adjoint pairing on $P$ , the commutator $[a, a^{†}] = I$ , the ladder relations $a g_{n} = \sqrt{n} g_{n - 1}$ , $a^{†} g_{n} = \sqrt{n + 1} g_{n + 1}$ (with the convention $g_{- 1} = 0$ at $n = 0$ ), the Rodrigues polynomials $H_{n}$ with $g_{n} = H_{n} g_{0} / \sqrt{2^{n} n!}$ and the recurrences $H_{n}^{'} = 2 n H_{n - 1}$ , $H_{n + 1} = 2 y H_{n} - 2 n H_{n - 1}$ . Orthonormality is Hermite orthogonality, which also records the eigenvalue equation $(- \partial_{y}^{2} + Y^{2}) g_{n} = (2 n + 1) g_{n}$ ; completeness is Hermite completeness, proved by the moment criterion and the uniqueness of the $L^{1}$ Fourier transform, and it makes $(g_{n})$ a Hilbert basis and $P$ dense.

Closed ladder domains extracts the maximal closed ladder operators from the basis alone: $D (a) = {\sum_{n} n ∣ c_{n} ∣^{2} < \infty}$ , $D (a^{†}) = {\sum_{n} (n + 1) ∣ c_{n} ∣^{2} < \infty}$ , closedness, $a^{*} = a^{†}$ , $(a^{†})^{*} = a$ , the common domain $D (N)$ of the products, $a^{†} a = N$ and $a a^{†} = N + I$ , the commutator identity on $D (N)$ , and the identification of $a, a^{†}$ with the closures of the polynomial-Gaussian ladder operators. Oscillator scaling and exact realization then scales with $ℓ = \sqrt{ℏ / (m ω)}$ : $D f (x) = ℓ^{- 1 / 2} f (x / ℓ)$ is unitary, $H_{o s c} = \frac{ℏ ω}{2} D A D^{- 1}$ is self-adjoint with spectrum ${ℏ ω (n + \frac{1}{2})}$ and orthonormal eigenbasis $ψ_{n} = D g_{n}$ , the span of the $ψ_{n}$ is a core, on it $H_{o s c} = - \frac{ℏ^{2}}{2 m} \frac{d^{2}}{d x^{2}} + \frac{m ω^{2}}{2} x^{2}$ , and the same identity holds against every test function for every element of the domain.

The coherent layer begins with Coherent-state convergence: for every $z \in C$ the series $ψ_{z} = e^{- ∣ z ∣^{2} / 2} \sum_{n} \frac{z^{n}}{\sqrt{n!}} g_{n}$ converges to a unit vector in every polynomial number domain, with $a ψ_{z} = z ψ_{z}$ , $∥ a^{†} ψ_{z} ∥^{2} = ∣ z ∣^{2} + 1$ , $⟨ N ψ_{z}, ψ_{z} ⟩ = ∣ z ∣^{2}$ and $⟨ ψ_{z}, ψ_{w} ⟩ = e^{- (∣ z ∣^{2} + ∣ w ∣^{2}) / 2 + z \overline{w}}$ . Coherent evolution and variance transports these coherent vectors by $D$ to $L^{2} (R, d x)$ , identifies $H_{o s c} = ℏ ω (N_{ψ} + \frac{1}{2} I)$ , constructs the unitary group $U (t)$ generated by $- \frac{i}{ℏ} H_{o s c}$ (with the exact generator-domain statement), proves $U (t) ψ_{z} = e^{- i ω t / 2} ψ_{z e^{- i ω t}}$ , and computes the position and momentum means $⟨ Q ⟩ = \sqrt{2} ℓ Re z$ , $⟨ P ⟩ = \frac{\sqrt{2} ℏ}{ℓ} Im z$ and variances $ℓ^{2} / 2$ , $ℏ^{2} / (2 ℓ^{2})$ with product $ℏ^{2} / 4$ , invariant under the evolution.

Finally Multidimensional oscillator completeness builds the product basis $g_{N} = \prod_{j} g_{n_{j}}^{(j)}$ of $L^{2} (R^{d})$ from qm-math-composites, proves its completeness, transports the anisotropic operator $\sum_{j} (- \frac{ℏ^{2}}{2 m_{j}} \partial_{x_{j}}^{2} + \frac{m_{j} ω_{j}^{2}}{2} x_{j}^{2})$ with spectrum the closure of ${ℏ \sum_{j} ω_{j} (n_{j} + \frac{1}{2})}$ , and computes the isotropic multiplicities $(\binom{M + d - 1}{d - 1})$ , which is $(M + 1) (M + 2) / 2$ for $d = 3$ . All items are draft mathematics under construction; the physical oscillator pages consume these results as stated mathematical premises and add their own postulates and interpretation.

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

LemmaStatement: AI-adaptedProof: AI-adaptedOpen item page →

Diagonal adjoint and graph-core lemma

Statement

Assume the Axiom of Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $H$ be a complex Hilbert space (Hilbert space) with inner product linear in the first argument, let $(e_{n})_{n \in N}$ , where $N = {0, 1, 2, \dots}$ , be a complete orthonormal family of $H$ (Orthonormal families, complete orthonormal systems and Hilbert bases), and let $(λ_{n})_{n \in N} \subseteq R$ . Put

$D_{0} : = {\sum_{n \in F} c_{n} e_{n} : F \subseteq N finite, c_{n} \in C}, S (\sum_{n \in F} c_{n} e_{n}) : = \sum_{n \in F} λ_{n} c_{n} e_{n},$

and

$D (A) : = {x \in H : \sum_{n \in N} λ_{n}^{2} ∣ ⟨ x, e_{n} ⟩ ∣^{2} < \infty}, A x : = \sum_{n \in N} λ_{n} ⟨ x, e_{n} ⟩ e_{n} .$

Then:

(a) $S$ is a well-defined, densely defined linear operator with domain $D_{0}$ (Unbounded linear operators: domain, graph and extension), and it is symmetric (Symmetric, self-adjoint and essentially self-adjoint operators);

(b) the series defining $A x$ converges in $H$ for every $x \in D (A)$ , with $∥ A x ∥^{2} = \sum_{n} λ_{n}^{2} ∣ ⟨ x, e_{n} ⟩ ∣^{2}$ , so $A$ is a linear operator with domain $D (A)$ and $S \subseteq A$ ;

(c) $S^{*} = A$ (Adjoint of a densely defined operator), hence $A$ is closed and $A^{*} = A$ : $A$ is self-adjoint and $\overline{S} = A$ , so $S$ is essentially self-adjoint and $A$ is its unique self-adjoint extension;

(d) $D_{0}$ is a core for $A$ (Densely defined, closed and closable operators, and cores): the finite truncations $x_{N} : = \sum_{n = 0}^{N} ⟨ x, e_{n} ⟩ e_{n} \in D_{0}$ of $x \in D (A)$ satisfy $x_{N} \to x$ and $A x_{N} \to A x$ in $H$ as $N \to \infty$ ;

(e) $⟨ A x, x ⟩ = \sum_{n} λ_{n} ∣ ⟨ x, e_{n} ⟩ ∣^{2}$ for $x \in D (A)$ , so $⟨ A x, x ⟩$ is real and $⟨ A x, x ⟩ \geq c ∥ x ∥^{2}$ whenever $λ_{n} \geq c$ for all $n$ ; and $A e_{n} = λ_{n} e_{n}$ for every $n$ , so each $λ_{n}$ is an eigenvalue of $A$ with eigenvector $e_{n}$ .

No assertion is made about a differential or multiplication realisation of $A$ : the lemma concerns only a specified complete orthonormal family and real eigenvalue sequence.

Facts & Assumptions

Given: a complex Hilbert space $H$ ; a complete orthonormal family $(e_{n})_{n \in N}$ ; real numbers $λ_{n}$ ; the linear span $D_{0}$ ; the operators $S$ and $A$ displayed above.

[F1]

$⟨ e_{i}, e_{j} ⟩ = δ_{i j}$ ; every finite subfamily of $(e_{n})$ is linearly independent, so coefficients in a finite expansion in $D_{0}$ are unique; and for every $x \in H$ the finite-subset net of partial sums $\sum_{n \in F} ⟨ x, e_{n} ⟩ e_{n}$ converges to $x$ in $H$ , with $\sum_{n} ∣ ⟨ x, e_{n} ⟩ ∣^{2} = ∥ x ∥^{2}$ (Orthonormal families, complete orthonormal systems and Hilbert bases, Parseval equivalences for an orthonormal family, clauses 3 and 4).

[F2]

For a densely defined $T$ , a vector $y \in H$ lies in $D (T^{*})$ exactly when $x \mapsto ⟨ T x, y ⟩$ is bounded on $(D (T), ∥ \cdot ∥)$ ; then $T^{*} y$ is the unique $w \in H$ with $⟨ T x, y ⟩ = ⟨ x, w ⟩$ for all $x \in D (T)$ . The adjoint $T^{*}$ is closed, and $T \subseteq T^{*}$ holds exactly when $⟨ T x, y ⟩ = ⟨ x, T y ⟩$ for all $x, y \in D (T)$ ; a symmetric operator is closable with symmetric closure, and a self-adjoint operator has no proper symmetric extension and is closed (Adjoint of a densely defined operator, Symmetric, self-adjoint and essentially self-adjoint operators, The adjoint is well defined, closed, and reverses inclusions).

[F3]

$\overline{T} = T^{* *}$ and $T^{*} = {\overline{T}}^{*}$ for a densely defined closable $T$ (Closability is equivalent to density of the adjoint domain).

[F4]

$∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥ ∥ y ∥$ and $∣ \sum_{n \in F} a_{n} \overline{b_{n}} ∣ \leq (\sum_{n \in F} ∣ a_{n} ∣^{2})^{1 / 2} (\sum_{n \in F} ∣ b_{n} ∣^{2})^{1 / 2}$ for finite $F$ (Cauchy–Schwarz: $∣ ⟨ u, v ⟩ ∣ \leq ∥ u ∥ ∥ v ∥$ , with equality exactly for linearly dependent vectors).

Proof

1.1givenF1

[Well-definedness and density] $D_{0}$ is the linear span of the family and is a linear subspace; by [F1] the coefficients in a finite expansion are unique, so $S$ is well defined, and it is linear and maps $D_{0}$ into $H$ . By [F1] the finite partial sums of the expansion of any $x \in H$ converge to $x$ , so the span $D_{0}$ is dense in $H$ .

1.2givenF1F4algebra

[Symmetry and the series for $A$ ] Let $x = \sum_{n \in F} c_{n} e_{n}$ and $y = \sum_{m \in G} d_{m} e_{m}$ lie in $D_{0}$ , with $c_{n} = ⟨ x, e_{n} ⟩$ and $d_{m} = ⟨ y, e_{m} ⟩$ by [F1]. Then $⟨ S x, y ⟩ = \sum_{n \in F} λ_{n} c_{n} \overline{d_{n}} = \sum_{n \in F} c_{n} \overline{λ_{n} d_{n}} = ⟨ x, S y ⟩$ because $λ_{n}$ is real: $S$ is symmetric. For $x \in D (A)$ the sequence $(λ_{n} ⟨ x, e_{n} ⟩)_{n}$ is square summable with $\sum_{n} λ_{n}^{2} ∣ ⟨ x, e_{n} ⟩ ∣^{2} < \infty$ , so its finite partial sums are Cauchy: $∥ \sum_{m \leq M < N} λ_{m} ⟨ x, e_{m} ⟩ e_{m} ∥^{2} = \sum_{M < m \leq N} λ_{m}^{2} ∣ ⟨ x, e_{m} ⟩ ∣^{2} \to 0$ by [F1]; by completeness of $H$ the series $\sum_{n} λ_{n} ⟨ x, e_{n} ⟩ e_{n}$ converges, its sum $A x$ satisfies $∥ A x ∥^{2} = \sum_{n} λ_{n}^{2} ∣ ⟨ x, e_{n} ⟩ ∣^{2}$ (the finite-subset net of squared norms has the same supremum), and $A$ is linear because coefficient extraction is linear. If moreover $x \in D_{0}$ , then $⟨ x, e_{n} ⟩ = 0$ for all but finitely many $n$ and $A x = S x$ ; hence $S \subseteq A$ .

1.3givenF1F2F4algebra

[ $S^{*} = A$ ] Let $y \in D (S^{*})$ and $w : = S^{*} y$ . For each $n$ , $λ_{n} \overline{⟨ y, e_{n} ⟩} = ⟨ S e_{n}, y ⟩ = ⟨ e_{n}, w ⟩ = \overline{⟨ w, e_{n} ⟩}$ , so $λ_{n} ⟨ y, e_{n} ⟩ = ⟨ w, e_{n} ⟩$ . By [F1] applied to $w$ , $\sum_{n} λ_{n}^{2} ∣ ⟨ y, e_{n} ⟩ ∣^{2} = \sum_{n} ∣ ⟨ w, e_{n} ⟩ ∣^{2} = ∥ w ∥^{2} < \infty$ , so $y \in D (A)$ ; and $\sum_{n} λ_{n} ⟨ y, e_{n} ⟩ e_{n} = \sum_{n} ⟨ w, e_{n} ⟩ e_{n} = w$ by [F1], so $S^{*} y = A y$ . Hence $S^{*} \subseteq A$ . Conversely let $y \in D (A)$ and $x = \sum_{n \in F} c_{n} e_{n} \in D_{0}$ ; then $⟨ S x, y ⟩ = \sum_{n \in F} λ_{n} c_{n} \overline{⟨ y, e_{n} ⟩} = ⟨ x, A y ⟩$ , so $x \mapsto ⟨ S x, y ⟩$ is the restriction to the dense subspace $D_{0}$ of the bounded functional $⟨ \cdot, A y ⟩$ and is therefore bounded on $(D_{0}, ∥ \cdot ∥)$ ; by [F2] $y \in D (S^{*})$ and $S^{*} y = A y$ . Thus $S^{*} = A$ .

2.1step 1.3F1F2F4algebra

[ $A$ is self-adjoint] Let $z \in D (A^{*})$ and $v : = A^{*} z$ . For $y \in D (A)$ the pairing $⟨ A y, z ⟩ = \sum_{n} λ_{n} ⟨ y, e_{n} ⟩ \overline{⟨ z, e_{n} ⟩}$ is absolutely convergent by [F4] and $∥ A y ∥ ∥ z ∥$ . Testing $y = e_{m}$ gives $λ_{m} ⟨ e_{m}, z ⟩ = ⟨ A e_{m}, z ⟩ = ⟨ e_{m}, v ⟩$ , that is $λ_{m} ⟨ z, e_{m} ⟩ = ⟨ v, e_{m} ⟩$ . By [F1], $\sum_{m} λ_{m}^{2} ∣ ⟨ z, e_{m} ⟩ ∣^{2} = \sum_{m} ∣ ⟨ v, e_{m} ⟩ ∣^{2} = ∥ v ∥^{2} < \infty$ , so $z \in D (A)$ ; and then $A z = \sum_{m} λ_{m} ⟨ z, e_{m} ⟩ e_{m} = \sum_{m} ⟨ v, e_{m} ⟩ e_{m} = v = A^{*} z$ by [F1]. Hence $D (A^{*}) \subseteq D (A)$ ; the reverse inclusion was proved in step 1.3 as $A \subseteq S^{*}$ with $S^{*} = A$ , and directly: for $z \in D (A)$ and $y \in D (A)$ , $⟨ A y, z ⟩ = \sum_{n} λ_{n} ⟨ y, e_{n} ⟩ \overline{⟨ z, e_{n} ⟩} = ⟨ y, A z ⟩$ by absolute convergence and realness of $λ_{n}$ , so $z \in D (A^{*})$ with $A^{*} z = A z$ . Therefore $A^{*} = A$ .

2.2step 1.2F1

[Graph core] Let $x \in D (A)$ and $x_{N} : = \sum_{n = 0}^{N} ⟨ x, e_{n} ⟩ e_{n} \in D_{0} \subseteq D (A)$ . By [F1], $∥ x - x_{N} ∥^{2} = \sum_{n > N} ∣ ⟨ x, e_{n} ⟩ ∣^{2} \to 0$ , and by step 1.2, $∥ A x - A x_{N} ∥^{2} = \sum_{n > N} λ_{n}^{2} ∣ ⟨ x, e_{n} ⟩ ∣^{2} \to 0$ because the series converges. Thus $(x_{N}, A x_{N}) \to (x, A x)$ in $H \oplus H$ ; since $x \in D (A)$ was arbitrary, $D_{0}$ is dense in $(D (A), ∥ \cdot ∥_{A})$ , that is, $D_{0}$ is a core for $A$ .

3.1step 1.2step 1.3step 2.1F2F3

[Closure, essential self-adjointness and uniqueness] Since $S$ is symmetric, [F2] makes $S$ closable with symmetric closure $\overline{S}$ , and [F3] gives $\overline{S} = S^{* *}$ . Since $A = S^{*}$ is the adjoint of a densely defined operator, $A$ is closed by [F2]; as $S \subseteq A$ , the least closed extension satisfies $\overline{S} \subseteq A$ . Conversely, for $x \in D (A)$ step 1.2 supplies $x_{N} \in D_{0}$ with $x_{N} \to x$ and $S x_{N} = A x_{N} \to A x$ , so $(x, A x) \in \overline{Γ (S)} = Γ (\overline{S})$ and $A \subseteq \overline{S}$ . Hence $\overline{S} = A$ , and $S$ is essentially self-adjoint because $\overline{S} = A$ is self-adjoint. If $T$ is any self-adjoint extension of $S$ , then $T$ is closed and contains $\overline{S} = A$ ; since $A$ is self-adjoint and $A \subseteq T$ with $T$ symmetric, [F2] forces $T = A$ . So $A$ is the unique self-adjoint extension of $S$ .

4.1step 1.2F1F4∎

[Positivity and eigenvalues] For $x \in D (A)$ the series $\sum_{n} λ_{n} ∣ ⟨ x, e_{n} ⟩ ∣^{2}$ is absolutely convergent by [F4] and step 1.2: $\sum_{n} ∣ λ_{n} ∣ ∣ ⟨ x, e_{n} ⟩ ∣^{2} \leq (\sum_{n} λ_{n}^{2} ∣ ⟨ x, e_{n} ⟩ ∣^{2})^{1 / 2} (\sum_{n} ∣ ⟨ x, e_{n} ⟩ ∣^{2})^{1 / 2} < \infty$ , and $⟨ A x, x ⟩ = \sum_{n} λ_{n} ∣ ⟨ x, e_{n} ⟩ ∣^{2}$ by continuity of the inner product in the first argument applied to the convergent series of step 1.2. Since the $λ_{n}$ are real the value is real, and $λ_{n} \geq c$ for all $n$ gives $⟨ A x, x ⟩ \geq c ∥ x ∥^{2}$ by [F1]. Finally $A e_{n} = λ_{n} e_{n}$ directly from the definition of $A$ and [F1], so each $λ_{n}$ is an eigenvalue of $A$ with eigenvector $e_{n} \neq 0$ .

TheoremStatement: AI-adaptedProof: AI-adaptedOpen item page →

Diagonal spectrum

Statement

Assume the Axiom of Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $H$ be a complex Hilbert space (Hilbert space), let $(e_{n})_{n \in N}$ be a complete orthonormal family of $H$ (Orthonormal families, complete orthonormal systems and Hilbert bases), let $(λ_{n})_{n \in N} \subseteq R$ , and let $A$ be the self-adjoint diagonal operator of Diagonal adjoint and graph-core lemma:

$D (A) = {x \in H : \sum_{n} λ_{n}^{2} ∣ ⟨ x, e_{n} ⟩ ∣^{2} < \infty}, A x = \sum_{n} λ_{n} ⟨ x, e_{n} ⟩ e_{n} .$

Then, with $ρ (A)$ , $σ (A)$ and $R (λ, A)$ as in Resolvent and spectrum of a closed operator on a Banach space:

(a) $A e_{n} = λ_{n} e_{n}$ for every $n$ ; every $λ_{n}$ is an eigenvalue of $A$ , hence belongs to $σ (A)$ ;

(b) $σ (A) = \overline{{λ_{n} : n \in N}}$ , the closure in $C$ ; equivalently $λ \in ρ (A)$ exactly when $d : = \inf_{n} ∣ λ - λ_{n} ∣ > 0$ , and then

$R (λ, A) x = \sum_{n} (λ - λ_{n})^{- 1} ⟨ x, e_{n} ⟩ e_{n}, ∥ R (λ, A) ∥ = d^{- 1} = (\inf_{n} ∣ λ - λ_{n} ∣)^{- 1};$

(c) if $λ \in σ (A) ∖ {λ_{n} : n \in N}$ , then $λ$ is an approximate eigenvalue: for some subsequence $(n_{k})$ , $∥ (λ I - A) e_{n_{k}} ∥ \to 0$ with $∥ e_{n_{k}} ∥ = 1$ . Moreover, $\ker (λ I - A) = {0}$ , so $λ$ is not an eigenvalue of $A$ ; no further identification of spectral types is made.

Facts & Assumptions

Given: the Hilbert space $H$ , the complete orthonormal family $(e_{n})$ , real numbers $λ_{n}$ , and the self-adjoint diagonal operator $A$ of Diagonal adjoint and graph-core lemma.

[F1]

$D (A) = {x : \sum_{n} λ_{n}^{2} ∣ ⟨ x, e_{n} ⟩ ∣^{2} < \infty}$ , $A x = \sum_{n} λ_{n} ⟨ x, e_{n} ⟩ e_{n}$ with $∥ A x ∥^{2} = \sum_{n} λ_{n}^{2} ∣ ⟨ x, e_{n} ⟩ ∣^{2}$ , $A e_{n} = λ_{n} e_{n}$ , and $A$ is self-adjoint (Diagonal adjoint and graph-core lemma(a),(b),(e)); for every $x \in H$ the expansion $x = \sum_{n} ⟨ x, e_{n} ⟩ e_{n}$ converges and $\sum_{n} ∣ ⟨ x, e_{n} ⟩ ∣^{2} = ∥ x ∥^{2}$ (Parseval equivalences for an orthonormal family).

[F2]

$λ \in ρ (A)$ exactly when $λ I - A : D (A) \to H$ is bijective with bounded inverse, and then that inverse is $R (λ, A)$ ; $σ (A) = C ∖ ρ (A)$ (Resolvent and spectrum of a closed operator on a Banach space).

[F3]

$∣ \sum_{n} a_{n} \overline{b_{n}} ∣ \leq (\sum_{n} ∣ a_{n} ∣^{2})^{1 / 2} (\sum_{n} ∣ b_{n} ∣^{2})^{1 / 2}$ and $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥ ∥ y ∥$ (Cauchy–Schwarz: $∣ ⟨ u, v ⟩ ∣ \leq ∥ u ∥ ∥ v ∥$ , with equality exactly for linearly dependent vectors).

Proof

1.1givenF1F2

[Eigenvalues are spectral] By [F1], $A e_{n} = λ_{n} e_{n}$ with $e_{n} \neq 0$ , so $λ_{n}$ is an eigenvalue of $A$ . If $λ_{n} \in ρ (A)$ , then $λ_{n} I - A$ is injective by [F2], contradicting $(λ_{n} I - A) e_{n} = 0$ . Hence ${λ_{n} : n \in N} \subseteq σ (A)$ .

1.2givenF1F2F3algebra

[The resolvent outside the closure] Let $λ \in C$ with $d = \inf_{n} ∣ λ - λ_{n} ∣ > 0$ , and define $R x : = \sum_{n} (λ - λ_{n})^{- 1} ⟨ x, e_{n} ⟩ e_{n}$ for $x \in H$ . The series converges and $∥ R x ∥^{2} = \sum_{n} ∣ λ - λ_{n} ∣^{- 2} ∣ ⟨ x, e_{n} ⟩ ∣^{2} \leq d^{- 2} ∥ x ∥^{2}$ by [F1], so $R \in B (H)$ with $∥ R ∥ \leq d^{- 1}$ ; testing on $e_{n}$ gives $∥ R ∥ \geq \sup_{n} ∣ λ - λ_{n} ∣^{- 1} = d^{- 1}$ , so $∥ R ∥ = d^{- 1}$ . For $x \in H$ , the coefficients of $R x$ satisfy $\sum_{n} λ_{n}^{2} ∣ (λ - λ_{n})^{- 1} ⟨ x, e_{n} ⟩ ∣^{2} \leq (1 + ∣ λ ∣ d^{- 1})^{2} \sum_{n} ∣ ⟨ x, e_{n} ⟩ ∣^{2} < \infty$ because $∣ λ_{n} ∣ \leq ∣ λ - λ_{n} ∣ + ∣ λ ∣ \leq ∣ λ - λ_{n} ∣ (1 + ∣ λ ∣ d^{- 1})$ for every $n$ ; hence $R x \in D (A)$ by [F1]. Computing coefficients with [F1] gives $(λ I - A) R x = \sum_{n} ⟨ x, e_{n} ⟩ e_{n} = x$ , and symmetrically $R (λ I - A) x = \sum_{n} (λ - λ_{n})^{- 1} (λ - λ_{n}) ⟨ x, e_{n} ⟩ e_{n} = x$ for $x \in D (A)$ . So $λ I - A$ is bijective with bounded inverse $R$ , that is, $λ \in ρ (A)$ and $R (λ, A) = R$ by [F2].

2.1step 1.1F1F2algebra

[Closure points are spectral] Let $λ \in \overline{{λ_{n}}}$ . If $λ = λ_{m}$ for some $m$ , then $λ \in σ (A)$ by step 1.1. Otherwise pick a subsequence with $λ_{n_{k}} \to λ$ and $λ_{n_{k}} \neq λ$ for all $k$ , and suppose for contradiction that $λ \in ρ (A)$ . Then $1 = ∥ e_{n_{k}} ∥ = ∥ R (λ, A) (λ I - A) e_{n_{k}} ∥ = ∥ R (λ, A) (λ - λ_{n_{k}}) e_{n_{k}} ∥ \leq ∥ R (λ, A) ∥ ∣ λ - λ_{n_{k}} ∣ \to 0$ , which is impossible. Hence $λ \in σ (A)$ .

3.1step 1.2step 2.1step 1.1F1∎

[Conclusion] Steps 1.1 and 2.1 give $\overline{{λ_{n}}} \subseteq σ (A)$ , and step 1.2 gives $C ∖ \overline{{λ_{n}}} \subseteq ρ (A)$ ; taking complements yields $σ (A) \subseteq \overline{{λ_{n}}}$ . Hence $σ (A) = \overline{{λ_{n}}}$ , and the equivalence " $λ \in ρ (A)$ exactly when $\inf_{n} ∣ λ - λ_{n} ∣ > 0$ " together with the resolvent formula and its norm are exactly the contents of step 1.2. For (c), if $λ \in σ (A) ∖ {λ_{n}}$ , the spectrum equality gives $λ \in \overline{{λ_{n}}}$ , so a subsequence with $λ_{n_{k}} \to λ$ satisfies $∥ (λ I - A) e_{n_{k}} ∥ = ∣ λ - λ_{n_{k}} ∣ \to 0$ and $∥ e_{n_{k}} ∥ = 1$ . If $x \in D (A)$ and $A x = λ x$ , then comparing coefficients in the complete orthonormal family gives $(λ_{n} - λ) ⟨ x, e_{n} ⟩ = 0$ for every $n$ ; because $λ \neq λ_{n}$ , every coefficient vanishes, and Parseval gives $x = 0$ . Thus $\ker (λ I - A) = {0}$ and $λ$ is not an eigenvalue. No further spectral-type classification is asserted.

TheoremStatement: AI-adaptedProof: AI-adaptedOpen item page →

Hermite recurrence

Statement

Assume the Axiom of Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $L^{2} = L^{2} (R, d y)$ be the complex Lebesgue space with inner product $⟨ φ, ψ ⟩ = \int_{R} φ (y) \overline{ψ (y)} d y$ linear in the first argument (The space $L^{p} (μ)$ as the quotient by null functions, Complex Lp classes and Euclidean test-function conventions); this pairing makes $L^{2}$ a complex Hilbert space by $L^{2}$ with the integral pairing is a Hilbert space. For complex-valued functions, smoothness and derivatives use the componentwise convention of Complex Lp classes and Euclidean test-function conventions. Write $g_{0}$ for the function $g_{0} (y) : = π^{- 1 / 4} e^{- y^{2} / 2}$ and also for its class in $L^{2}$ . Put

$P : = {p g_{0} : p \in C [y]} \subseteq L^{2},$

the set of classes of polynomial multiples of $g_{0}$ . Each class in $P$ has a unique representative of the form $p g_{0}$ : if $p g_{0} = q g_{0}$ almost everywhere, the continuous function $(p - q) g_{0}$ vanishes almost everywhere and therefore everywhere, and $g_{0} (y) > 0$ for every $y$ , so $p = q$ . Let $Y$ be multiplication by $y$ and $\partial_{y}$ the derivative, acting on this unique smooth representative. Set $g_{- 1} : = 0$ for use in the recurrence formulas at $n = 0$ . Define

$a : = \frac{Y + \partial_{y}}{\sqrt{2}}, a^{†} : = \frac{Y - \partial_{y}}{\sqrt{2}}, g_{n} : = \frac{(a^{†})^{n} g_{0}}{\sqrt{n!}} (n \in N) .$

Then:

(a) $g_{0} \in L^{2}$ with $∥ g_{0} ∥ = 1$ ; more generally $y^{k} g_{0} \in L^{2}$ for every $k \in N$ , so $P$ is a linear subspace of $L^{2}$ all of whose members are represented by $C^{\infty}$ functions;

(b) $Y, \partial_{y}, a, a^{†}$ map $P$ into $P$ , and $\partial_{y} (p g_{0}) = (p^{'} - y p) g_{0}$ for every polynomial $p$ ;

(d) $a^{†} a - a a^{†} = - I$ on $P$ , i.e. $[a, a^{†}] = a a^{†} - a^{†} a = I$ on $P$ , and $a g_{0} = 0$ ;

(e) for every $n \in N$ : $a g_{n} = \sqrt{n} g_{n - 1}$ (with $a g_{0} = 0$ ), $a^{†} g_{n} = \sqrt{n + 1} g_{n + 1}$ , $a^{†} a g_{n} = n g_{n}$ and $a a^{†} g_{n} = (n + 1) g_{n}$ ;

(f) the polynomials

$H_{n} (y) : = (- 1)^{n} e^{y^{2}} \frac{d^{n}}{d y^{n}} e^{- y^{2}}$

are polynomials of degree $n$ with leading coefficient $2^{n}$ , $H_{0} = 1$ , $H_{1} (y) = 2 y$ , $H_{n} (- y) = (- 1)^{n} H_{n} (y)$ , and they satisfy

$H_{n}^{'} = 2 n H_{n - 1}, H_{n + 1} = 2 y H_{n} - 2 n H_{n - 1} (n \geq 1);$

moreover

$g_{n} = \frac{H_{n} g_{0}}{\sqrt{2^{n} n!}};$

(g) consequently

$\sqrt{2} y g_{n} = \sqrt{n} g_{n - 1} + \sqrt{n + 1} g_{n + 1}, \sqrt{2} g_{n}^{'} = \sqrt{n} g_{n - 1} - \sqrt{n + 1} g_{n + 1} .$

Facts & Assumptions

Given: the space $L^{2} (R, d y)$ with its inner product; the function $g_{0} (y) = π^{- 1 / 4} e^{- y^{2} / 2}$ ; the space $P$ of polynomial multiples of $g_{0}$ ; the operators $Y, \partial_{y}, a, a^{†}$ and the vectors $g_{n}$ .

[F1]

$L^{2}$ is a complex Hilbert space for $⟨ f, g ⟩ = \int f \overline{g}$ , with $∥ f ∥_{2}^{2} = \int ∣ f ∣^{2}$ ; the pairing is well defined on a.e. classes, positive definite, and satisfies $∣ ⟨ f, g ⟩ ∣ \leq \int ∣ f g ∣ \leq ∥ f ∥_{2} ∥ g ∥_{2}$ (The space $L^{p} (μ)$ as the quotient by null functions, Complex Lp classes and Euclidean test-function conventions, $L^{2}$ with the integral pairing is a Hilbert space, Cauchy-Schwarz inequality for $L^{2}$ ).

[F2]

For $t > 0$ , the Lebesgue Gaussian transform at dimension $n = 1$ , frequency $ξ = 0$ and parameter $t / π$ gives $\int_{R} e^{- t y^{2}} d y = \sqrt{π / t}$ ; every polynomial times a positive real Gaussian is absolutely integrable (Euclidean Gaussian transform with the 2π normalization). In particular $\int e^{- y^{2}} d y = \sqrt{π}$ and $\int e^{- y^{2} / 2} d y = \sqrt{2 π}$ .

[F3]

For $t \geq 0$ and $k \in N$ one has $t^{k} \leq 2^{k} k! e^{t / 2}$ : by The complex exponential by its power series and The complex exponential series converges absolutely for every complex argument the exponential is the sum of the absolutely convergent series $\sum_{j} (t / 2)^{j} / j!$ with nonnegative terms, so $e^{t / 2} \geq (t / 2)^{k} / k!$ .

[F4]

The real exponential is $C^{\infty}$ with $(\exp)^{'} = \exp$ (The exponential function is smooth and $(\exp)^{'} = \exp$ ); the chain rule (The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g (c)$ , then $f \circ g$ is differentiable at $c$ with $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ ) and product rule (Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ ) hold for real-valued functions. For complex-valued functions apply those rules separately to the real and imaginary parts, as in the componentwise convention of Complex Lp classes and Euclidean test-function conventions. Consequently $(e^{- y^{2} / 2})^{'} = - y e^{- y^{2} / 2}$ and, for any complex polynomial $p$ , $(p g_{0})^{'} = (p^{'} - y p) g_{0}$ on $R$ .

[F5]

For every real $x$ , $e^{x} > 0$ and $e^{- x} = 1 / e^{x}$ (The exponential is positive and satisfies $\exp (- x) = 1 / \exp (x)$ ); the real exponential is strictly increasing (The exponential function is strictly increasing). The complex exponential extends the real one and satisfies $e^{z + w} = e^{z} e^{w}$ for all $z, w \in C$ ( $\exp (z + w) = \exp z \exp w$ , and the complex exponential extends the real exponential).

[F6]

If $F : [a, b] \to R$ is differentiable at every point of $[a, b]$ and $F^{'}$ is integrable there, then $\int_{a}^{b} F^{'} = F (b) - F (a)$ (The second fundamental theorem: if $G$ is differentiable on $[a, b]$ with $G^{'} = f$ and $f$ is integrable, then $\int_{a}^{b} f = G (b) - G (a)$ ); complex integration by parts on $R$ is valid when both derivative products are in $L^{1}$ and the product vanishes at both ends (Complex integration by parts on intervals and decaying lines). For $φ, ψ \in P$ one has $φ (y) \overline{ψ (y)} \to 0$ as $∣ y ∣ \to \infty$ , because its modulus is bounded by $C (1 + y^{2 k}) e^{- y^{2}}$ for some $C > 0$ and $k \in N$ ; [F3] bounds this by $C (1 + 2^{k} k!) e^{- y^{2} / 2}$ , which tends to zero since $e^{y^{2} / 2} \geq 1 + y^{2} / 2$ by the same nonnegative series and [F5].

Proof

1.1givenF1F2F3F5algebra

[The Gaussian moments and polynomial classes] If $p g_{0} = q g_{0}$ as $L^{2}$ classes, $(p - q) g_{0}$ is a continuous function that vanishes almost everywhere, hence vanishes everywhere; since $g_{0} > 0$ , $p = q$ . Thus the representative used to define $Y$ and $\partial_{y}$ is unique, so these operators are well defined on $P$ . By [F2], $∥ g_{0} ∥^{2} = π^{- 1 / 2} \int e^{- y^{2}} d λ = π^{- 1 / 2} \sqrt{π} = 1$ , so $g_{0} \in L^{2}$ and $∥ g_{0} ∥ = 1$ . For $k \in N$ and $y \in R$ , $∣ y^{k} g_{0} (y) ∣^{2} = π^{- 1 / 2} y^{2 k} e^{- y^{2}} \leq π^{- 1 / 2} 2^{k} k! e^{- y^{2} / 2}$ by [F3]; hence $∥ y^{k} g_{0} ∥^{2} \leq π^{- 1 / 2} 2^{k} k! \sqrt{2 π} < \infty$ by [F2], so $y^{k} g_{0} \in L^{2}$ . Every $p g_{0}$ with $\deg p = m$ is a finite linear combination of $y^{0} g_{0}, \dots, y^{m} g_{0}$ , so $p g_{0} \in L^{2}$ : $P \subseteq L^{2}$ is a linear subspace.

1.2givenF4F5algebra

[Invariance of $P$ and $a g_{0} = 0$ ] For a complex polynomial $p = p_{1} + i p_{2}$ with real polynomials $p_{1}, p_{2}$ , the real product and chain rules [F4] applied to $p_{1} g_{0}$ and $p_{2} g_{0}$ give $\partial_{y} (p g_{0}) = (p^{'} - y p) g_{0}$ ; $Y (p g_{0}) = (y p) g_{0}$ has polynomial factor $y p$ . Thus both $Y$ and $\partial_{y}$ , hence $a = (Y + \partial_{y}) / \sqrt{2}$ and $a^{†} = (Y - \partial_{y}) / \sqrt{2}$ , map $P$ into $P$ . Since $g_{0}^{'} = - y g_{0}$ by [F4], $(Y + \partial_{y}) g_{0} = y g_{0} + g_{0}^{'} = 0$ , i.e. $a g_{0} = 0$ . All members of $P$ are represented by componentwise $C^{\infty}$ functions by [F4].

1.3givenF1F2F4F5F6algebra

[Adjoint pairing] Let $φ, ψ \in P$ . They are componentwise $C^{\infty}$ by [F4], and $φ^{'} \overline{ψ}$ and $φ \overline{ψ^{'}}$ are polynomial multiples of $e^{- y^{2}}$ , hence absolutely integrable by [F2]. Their product $φ \overline{ψ}$ tends to zero at both ends by [F6]. Apply the whole-line complex integration-by-parts identity in [F6] to $u = φ$ and $v = \overline{ψ}$ to obtain $\int_{R} φ^{'} \overline{ψ} = - \int_{R} φ \overline{ψ^{'}}$ . Therefore $⟨ a^{†} φ, ψ ⟩ = 2^{- 1 / 2} \int (y φ - φ^{'}) \overline{ψ} = 2^{- 1 / 2} \int φ \overline{y ψ + ψ^{'}} = ⟨ φ, a ψ ⟩$ , since $y$ is real and $\overline{ψ^{'}}$ is the derivative of $\overline{ψ}$ ; the second identity follows by exchanging $φ$ and $ψ$ and conjugating.

1.4givenF4F5algebra

[Rodrigues polynomials] Define $H_{n} (y) : = (- 1)^{n} e^{y^{2}} \frac{d^{n}}{d y^{n}} e^{- y^{2}}$ ; each $H_{n}$ is a polynomial because differentiating $e^{- y^{2}}$ produces $e^{- y^{2}}$ times a polynomial, and $H_{0} = 1$ , $H_{1} = 2 y$ . Put $F_{n} : = e^{y^{2}} \frac{d^{n}}{d y^{n}} e^{- y^{2}}$ , so $H_{n} = (- 1)^{n} F_{n}$ . By the product rule, $F_{n + 1} = e^{y^{2}} \frac{d}{d y} (e^{- y^{2}} F_{n}) = - 2 y F_{n} + F_{n}^{'}$ , hence $H_{n + 1} = 2 y H_{n} - H_{n}^{'}$ . We prove $H_{n}^{'} = 2 n H_{n - 1}$ for all $n \geq 1$ by strong induction: for $n = 1$ this is $H_{1}^{'} = 2 = 2 H_{0}$ , and $H_{2} = 4 y^{2} - 2$ gives $H_{2}^{'} = 8 y = 4 H_{1}$ . For $n \geq 2$ , if it holds for all $1 \leq j \leq n$ , then differentiating the displayed relation at $n$ gives $H_{n + 1}^{'} = 2 H_{n} + 2 y H_{n}^{'} - H_{n}^{''}$ , while the induction hypothesis gives $H_{n}^{'} = 2 n H_{n - 1}$ and $H_{n}^{''} = 2 n H_{n - 1}^{'} = 4 n (n - 1) H_{n - 2}$ , and the same relation at $n - 1$ reads $H_{n} = 2 y H_{n - 1} - H_{n - 1}^{'}$ , so $2 y H_{n - 1} = H_{n} + 2 (n - 1) H_{n - 2}$ ; substituting, $H_{n + 1}^{'} = 2 H_{n} + 2 n (H_{n} + 2 (n - 1) H_{n - 2}) - 4 n (n - 1) H_{n - 2} = 2 (n + 1) H_{n}$ , which is the claim for $n + 1$ . Hence $H_{n}^{'} = 2 n H_{n - 1}$ for all $n \geq 1$ , and the displayed relation becomes the three-term recurrence $H_{n + 1} = 2 y H_{n} - 2 n H_{n - 1}$ for $n \geq 1$ . Induction on this recurrence gives that $H_{n}$ has degree $n$ with leading coefficient $2^{n}$ (the term $2 y H_{n}$ contributes $2 \cdot 2^{n} = 2^{n + 1}$ to $y^{n + 1}$ and $- 2 n H_{n - 1}$ only degree $n - 1$ ) and that $H_{n} (- y) = (- 1)^{n} H_{n} (y)$ , true for $n = 0, 1$ and inherited.

2.1step 1.2F4algebra

[Commutator] On any $C^{\infty}$ function $φ$ , $2 (a a^{†} - a^{†} a) = (Y + \partial_{y}) (Y - \partial_{y}) - (Y - \partial_{y}) (Y + \partial_{y})$ , and expanding with the product rule [F4] gives $(Y + \partial_{y}) (Y - \partial_{y}) = Y^{2} - Y \partial_{y} + \partial_{y} Y - \partial_{y}^{2}$ and $(Y - \partial_{y}) (Y + \partial_{y}) = Y^{2} + Y \partial_{y} - \partial_{y} Y - \partial_{y}^{2}$ ; subtracting gives $2 (\partial_{y} Y - Y \partial_{y}) = 2 I$ because $\partial_{y} (Y φ) - Y (\partial_{y} φ) = φ$ . Hence $[a, a^{†}] = I$ on $P$ by step 1.2.

2.2step 1.2step 1.4F4algebra

[Identification of $g_{n}$ ] We prove $g_{n} = H_{n} g_{0} / \sqrt{2^{n} n!}$ by induction on $n$ . For $n = 0$ this is $g_{0} = H_{0} g_{0}$ . Assume it holds for $n$ ; by step 1.2 and [F4], $(Y - \partial_{y}) (H_{n} g_{0}) = (2 y H_{n} - H_{n}^{'}) g_{0} = H_{n + 1} g_{0}$ by step 1.4, so $g_{n + 1} = a^{†} g_{n} / \sqrt{n + 1} = (Y - \partial_{y}) H_{n} g_{0} / (\sqrt{2} \sqrt{n + 1} \sqrt{2^{n} n!}) = H_{n + 1} g_{0} / \sqrt{2^{n + 1} (n + 1)!}$ , which completes the induction.

3.1step 2.1step 1.2algebra

[Ladder relations] From $[a, a^{†}] = I$ on $P$ we prove $[a, (a^{†})^{n}] = n (a^{†})^{n - 1}$ on $P$ for every $n \geq 1$ : for $n = 1$ this is step 2.1, and if it holds for $n$ then $[a, (a^{†})^{n + 1}] = [a, (a^{†})^{n}] a^{†} + (a^{†})^{n} [a, a^{†}] = n (a^{†})^{n - 1} a^{†} + (a^{†})^{n} = (n + 1) (a^{†})^{n}$ , all evaluated on $P$ , which $(a^{†})^{n}$ and $a$ preserve by step 1.2. Since $a g_{0} = 0$ , for $n \geq 1$ we get $a g_{n} = [a, (a^{†})^{n}] g_{0} / \sqrt{n!} = n (a^{†})^{n - 1} g_{0} / \sqrt{n!} = \sqrt{n} g_{n - 1}$ , while $a g_{0} = 0$ ; by definition $a^{†} g_{n} = (a^{†})^{n + 1} g_{0} / \sqrt{n!} = \sqrt{n + 1} g_{n + 1}$ for every $n$ , hence $a^{†} a g_{n} = \sqrt{n} a^{†} g_{n - 1} = n g_{n}$ and, using $a a^{†} = a^{†} a + I$ from step 2.1, $a a^{†} g_{n} = (n + 1) g_{n}$ .

4.1step 3.1step 1.4step 2.2∎

[The $y$ and $\partial_{y}$ actions] In the formulas containing $g_{n - 1}$ at $n = 0$ , set $g_{- 1} : = 0$ ; this is the zero vector, so both formulas at $n = 0$ are exactly the two ladder identities with $a g_{0} = 0$ . Since $Y = (a + a^{†}) / \sqrt{2}$ and $\partial_{y} = (a - a^{†}) / \sqrt{2}$ on $P$ by the definitions, applying step 3.1 gives $\sqrt{2} y g_{n} = a g_{n} + a^{†} g_{n} = \sqrt{n} g_{n - 1} + \sqrt{n + 1} g_{n + 1}$ and $\sqrt{2} g_{n}^{'} = a g_{n} - a^{†} g_{n} = \sqrt{n} g_{n - 1} - \sqrt{n + 1} g_{n + 1}$ . All identities are in $L^{2}$ , where equalities mean equality of classes, and the underlying functions agree pointwise because both sides are represented by continuous functions; the polynomial identities of step 1.4 are pointwise identities of polynomials.

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Hermite orthogonality

Statement

(a) $⟨ g_{n}, g_{m} ⟩ = δ_{n m}$ ; in particular $(g_{n})_{n \in N}$ is an orthonormal family in $L^{2}$ (Orthonormal families, complete orthonormal systems and Hilbert bases);

(b) the eigenvalue equation $(- \partial_{y}^{2} + Y^{2}) g_{n} = (2 n + 1) g_{n}$ holds as an identity of $L^{2}$ classes; more precisely $Y^{2} - \partial_{y}^{2} = 2 a^{†} a + I$ as operators on $P$ , so that on $P$

$- φ^{''} + y^{2} φ = 2 a^{†} a φ + φ (φ \in P);$

No completeness of the family $(g_{n})$ is asserted here.

Facts & Assumptions

Given: the family $(g_{n})$ of Hermite recurrence with $g_{0} (y) = π^{- 1 / 4} e^{- y^{2} / 2}$ , the ladder operators $a, a^{†}$ on $P$ , and the identities

[F1]

$∥ g_{0} ∥ = 1$ ; $g_{n} = (a^{†})^{n} g_{0} / \sqrt{n!}$ ; $a g_{n} = \sqrt{n} g_{n - 1}$ for $n \geq 1$ with $a g_{0} = 0$ ; $a^{†} g_{n} = \sqrt{n + 1} g_{n + 1}$ ; $a^{†} a g_{n} = n g_{n}$ ; $a a^{†} g_{n} = (n + 1) g_{n}$ ; $a^{†} a - a a^{†} = - I$ on $P$ (Hermite recurrence(a),(d),(e)).

[F2]

$⟨ a^{†} φ, ψ ⟩ = ⟨ φ, a ψ ⟩$ for all $φ, ψ \in P$ (Hermite recurrence(c)).

[F3]

$⟨ \cdot, \cdot ⟩$ is linear in the first argument and conjugate-linear in the second, and $∥ v ∥^{2} = ⟨ v, v ⟩ \geq 0$ with equality only for $v = 0$ (The space $L^{p} (μ)$ as the quotient by null functions, Complex Lp classes and Euclidean test-function conventions, Cauchy-Schwarz inequality for $L^{2}$ ).

[F4]

$Y^{2} - \partial_{y}^{2} = 2 a^{†} a + I$ on $P$ , where $Y$ is multiplication by $y$ and $\partial_{y}$ is the derivative on smooth representatives; this is the computation $(Y - \partial_{y}) (Y + \partial_{y}) = Y^{2} - \partial_{y}^{2} - [\partial_{y}, Y] = Y^{2} - \partial_{y}^{2} - I$ combined with $a^{†} a = \frac{1}{2} (Y - \partial_{y}) (Y + \partial_{y})$ (Hermite recurrence(b),(d)).

Proof

1.1givenF1F2algebra

[Orthogonality] Let $m, n \in N$ with $n \geq 1$ . By [F2] with $φ = a g_{n}$ and $ψ = g_{m}$ , $⟨ a^{†} a g_{n}, g_{m} ⟩ = ⟨ a g_{n}, a g_{m} ⟩$ ; by [F2] with $φ = a g_{m}$ and $ψ = g_{n}$ and then conjugating, $⟨ g_{n}, a^{†} a g_{m} ⟩ = ⟨ a g_{n}, a g_{m} ⟩$ as well. Using $a^{†} a g_{k} = k g_{k}$ from [F1] on both sides, $n ⟨ g_{n}, g_{m} ⟩ = m ⟨ g_{n}, g_{m} ⟩$ . If $n = m$ this gives no information; if $n \neq m$ and $n, m \geq 1$ it gives $⟨ g_{n}, g_{m} ⟩ = 0$ . If $m = 0$ use $a g_{0} = 0$ : then $⟨ a g_{n}, a g_{0} ⟩ = 0$ , and the first identity reads $n ⟨ g_{n}, g_{0} ⟩ = 0$ , so $⟨ g_{n}, g_{0} ⟩ = 0$ for $n \geq 1$ ; conjugate symmetry gives $⟨ g_{0}, g_{n} ⟩ = 0$ as well.

1.2givenF1F4algebra

[Eigenvalue equation] By [F4], $Y^{2} - \partial_{y}^{2} = 2 a^{†} a + I$ on $P$ . Evaluating at $g_{n}$ and using [F1] gives $(- \partial_{y}^{2} + Y^{2}) g_{n} = 2 n g_{n} + g_{n} = (2 n + 1) g_{n}$ as classes in $L^{2}$ ; the left side is computed on the smooth representative $g_{n}$ , and the equality of classes holds because the two sides are continuous functions agreeing in $L^{2}$ . The displayed form for $φ \in P$ with $Y^{2} φ = y^{2} φ$ is the same identity.

2.1step 1.1F1F2F3algebra

[Unit norms] $∥ g_{0} ∥ = 1$ by [F1]. For $n \geq 0$ , applying [F2] with $φ = g_{n}$ , $ψ = a^{†} g_{n}$ gives $∥ a^{†} g_{n} ∥^{2} = ⟨ a^{†} g_{n}, a^{†} g_{n} ⟩ = ⟨ g_{n}, a a^{†} g_{n} ⟩ = (n + 1) ∥ g_{n} ∥^{2}$ by $a a^{†} g_{n} = (n + 1) g_{n}$ from [F1]. Since $g_{n + 1} = a^{†} g_{n} / \sqrt{n + 1}$ by [F1], $∥ g_{n + 1} ∥^{2} = ∥ a^{†} g_{n} ∥^{2} / (n + 1) = ∥ g_{n} ∥^{2}$ . Induction gives $∥ g_{n} ∥ = 1$ for every $n$ . Together with step 1.1, $⟨ g_{n}, g_{m} ⟩ = δ_{n m}$ .

3.1step 2.1F1algebra∎

[Ladder pairings] For $n \neq m$ , $⟨ a g_{n}, a g_{m} ⟩ = \sqrt{n m} ⟨ g_{n - 1}, g_{m - 1} ⟩ = 0$ by step 2.1 when $n, m \geq 1$ , and $⟨ a g_{0}, \cdot ⟩ = 0$ ; for $n = m$ it equals $n ∥ g_{n - 1} ∥^{2} = n$ . Likewise $⟨ a^{†} g_{n}, a^{†} g_{m} ⟩ = \sqrt{(n + 1) (m + 1)} ⟨ g_{n + 1}, g_{m + 1} ⟩ = (n + 1) δ_{n m}$ .

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Hermite completeness

Statement

Assume the Axiom of Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $L^{2} = L^{2} (R, d y)$ be the complex Lebesgue space with inner product linear in the first argument, and let $g_{0}, P, (g_{n})_{n \in N}$ and $H_{n}$ be as in Hermite recurrence. Define $h : = f g_{0} \in L^{1} (R)$ for $f \in L^{2}$ and write $\hat{h} (ξ) = \int_{R} h (y) e^{- 2 π i ξ y} d y$ for its $L^{1}$ Fourier transform. Then:

(a) (moment criterion) if $\int_{R} f q g_{0} = 0$ for every polynomial $q \in C [y]$ , then $f = 0$ almost everywhere;

(b) the family $(g_{n})_{n \in N}$ is a complete orthonormal family, i.e. a Hilbert basis, of $L^{2}$ (Orthonormal families, complete orthonormal systems and Hilbert bases): the only $f \in L^{2}$ with $⟨ f, g_{n} ⟩ = 0$ for all $n$ is $f = 0$ , and consequently every $f \in L^{2}$ satisfies $f = \sum_{n} ⟨ f, g_{n} ⟩ g_{n}$ in $L^{2}$ with $∥ f ∥^{2} = \sum_{n} ∣ ⟨ f, g_{n} ⟩ ∣^{2}$ (Parseval equivalences for an orthonormal family);

Facts & Assumptions

Given: the orthonormal family $(g_{n})$ with $g_{n} = H_{n} g_{0} / \sqrt{2^{n} n!}$ and $H_{n}$ a real polynomial of degree $n$ with leading coefficient $2^{n}$ (Hermite recurrence(a),(f)); the measures and integrals are Lebesgue measure on $R$ .

[F1]

$(g_{n})$ is orthonormal in $L^{2}$ and $P$ is a linear subspace of $L^{2}$ containing every $g_{n}$ (Hermite orthogonality(a), Hermite recurrence(a)).

[F2]

For a complete orthonormal family the following are equivalent: the closed linear span is $H$ ; the only vector orthogonal to every member is $0$ ; Parseval holds; and the finite-subset net of partial sums converges to each vector (Parseval equivalences for an orthonormal family, Orthonormal families, complete orthonormal systems and Hilbert bases).

[F3]

$∥ f g ∥_{1} \leq ∥ f ∥_{2} ∥ g ∥_{2}$ for $f, g \in L^{2}$ , so $h = f g_{0} \in L^{1}$ with $∥ h ∥_{1} \leq ∥ f ∥_{2}$ because $∥ g_{0} ∥ = 1$ (Cauchy-Schwarz inequality for $L^{2}$ , Hermite recurrence(a)).

[F4]

The $L^{1}$ Fourier transform is $\hat{h} (ξ) = \int h (y) e^{- 2 π i ξ y} d y$ (Fourier transform on complex L1 classes), and $\hat{h} = 0$ implies $h = 0$ a.e. (Uniqueness of the L1 Fourier transform); if $h_{N} \to h$ pointwise a.e. with $∣ h_{N} ∣ \leq G$ a.e. for a fixed $G \in L^{1}$ , then $\int h_{N} \to \int h$ (Dominated convergence).

[F5]

For real $w$ , $e^{w} = \sum_{j \geq 0} w^{j} / j!$ is the sum of an absolutely convergent series (The complex exponential by its power series, The complex exponential series converges absolutely for every complex argument), and for nonnegative terms every partial sum is at most the full sum (A series of nonnegative terms converges iff its partial sums are bounded, and then the sum is their supremum).

[F6]

The Lebesgue Gaussian transform at $n = 1$ , $ξ = 0$ and parameter $1 / (2 π)$ gives $\int_{R} e^{- y^{2} / 2} d y = \sqrt{2 π} < \infty$ (Euclidean Gaussian transform with the 2π normalization).

Proof

1.1givenF1F2

[Reduction to the moment criterion] Since $(g_{n})$ is orthonormal by [F1], [F2] reduces completeness to the statement that no nonzero $f \in L^{2}$ is orthogonal to every $g_{n}$ ; the expansion and Parseval identities are part of the same equivalence. The moment criterion (a) is proved first, and then applied.

1.2givenF1F3algebra

[Vanishing moments] Let $f \in L^{2}$ satisfy the moment hypothesis of (a), and put $h : = f g_{0}$ . By [F3], $h \in L^{1}$ and $∥ h ∥_{1} \leq ∥ f ∥_{2}$ ; every polynomial moment is integrable by Cauchy–Schwarz because $q g_{0} \in L^{2}$ by [F1], and the hypothesis gives $\int h q = 0$ , in particular $\int h (y) y^{j} d y = 0$ for every $j \geq 0$ . If instead $f$ is orthogonal to every $g_{n}$ , then real-valuedness of $H_{n}$ and $g_{0}$ gives $0 = ⟨ f, g_{n} ⟩ = (2^{n} n!)^{- 1 / 2} \int f H_{n} g_{0}$ . The polynomials $H_{0}, \dots, H_{m}$ have respective degrees $0, \dots, m$ and nonzero leading coefficients, so successive subtraction of leading terms expresses every polynomial of degree at most $m$ as their linear combination. Thus orthogonality implies the moment hypothesis of (a). It remains to prove $h = 0$ under that hypothesis.

2.1step 1.2F3F4F5F6algebra

[The transform vanishes] Fix $ξ \in R$ , write $w : = 2 π ξ y$ and $S_{N} (y) : = \sum_{j = 0}^{N} (- 2 π i ξ y)^{j} / j!$ . Then $S_{N} (y) \to e^{- 2 π i ξ y}$ for every $y$ by [F5], and $∣ S_{N} (y) ∣ \leq \sum_{j = 0}^{N} ∣ 2 π ξ y ∣^{j} / j! \leq e^{2 π ∣ ξ ∣ ∣ y ∣} = : G (y)$ by [F5]. Moreover $h G \in L^{1}$ : by [F3], $\int ∣ h (y) ∣ G (y) d y \leq ∥ f ∥_{2} ∥ g_{0} G ∥_{2} = ∥ f ∥_{2} (\int G (y)^{2} g_{0} (y)^{2} d y)^{1 / 2}$ , and $G (y)^{2} g_{0} (y)^{2} = π^{- 1 / 2} e^{4 π ∣ ξ ∣ ∣ y ∣ - y^{2}} \leq π^{- 1 / 2} e^{8 π^{2} ξ^{2}} e^{- y^{2} / 2}$ because $4 π ∣ ξ ∣ ∣ y ∣ - y^{2} \leq 8 π^{2} ξ^{2} - ∣ y ∣^{2} / 2$ ; the last integral is finite by [F6]. Hence [F4] applies with $\hat{h} (ξ) = \lim_{N \to \infty} \int h S_{N} = \lim_{N \to \infty} \sum_{j = 0}^{N} \frac{(- 2 π i ξ)^{j}}{j!} \int h (y) y^{j} d y = 0$ , the first equality by dominated convergence, the interchange of the finite sum with the integral by linearity of the integral, and the final equality because all moments of $h$ vanish by step 1.2.

3.1step 1.1step 1.2step 2.1

[Conclusion of (a) and (b)] Since $\hat{h} = 0$ identically, [F4] gives $h = f g_{0} = 0$ a.e.; as $g_{0} (y) = π^{- 1 / 4} e^{- y^{2} / 2} > 0$ for every $y$ , $f = 0$ a.e. This proves (a), and with step 1.1 it proves that $(g_{n})$ is complete. By [F2] applied to the complete orthonormal family $(g_{n})$ , every $f \in L^{2}$ has the stated expansion and Parseval identity, which is (b).

4.1step 3.1F1∎

[Density of $P$ ] By [F2] and step 3.1 the closed linear span of $(g_{n})$ is $L^{2}$ . Since every $g_{n} \in P$ by [F1] and $P$ is a linear subspace, $P$ contains the span of the family; a subspace containing a dense subspace is dense, so $P$ is dense. This is (c).

TheoremStatement: AI-adaptedProof: AI-adaptedOpen item page →

Oscillator scaling and exact realization

Statement

Assume the Axiom of Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Fix real parameters $m > 0$ , $ω > 0$ , $ℏ > 0$ , put $ℓ : = \sqrt{ℏ / (m ω)} > 0$ , and let $g_{0}, P, (g_{n})_{n \in N}$ be the Hermite data of Hermite recurrence in $L^{2} (R, d y)$ , so that $(g_{n})$ is a complete orthonormal family by Hermite completeness. Define

$D : L^{2} (R, d y) \to L^{2} (R, d x), (D f) (x) : = ℓ^{- 1 / 2} f (x / ℓ),$

and let $A$ be the self-adjoint diagonal operator of Diagonal adjoint and graph-core lemma for the complete orthonormal family $(g_{n})$ and the eigenvalues $λ_{n} = 2 n + 1$ , with the core $P$ and spectrum ${2 n + 1 : n \in N}$ (Diagonal spectrum).

Then:

(a) $D$ is a linear isometry of $L^{2} (R, d y)$ onto $L^{2} (R, d x)$ with two-sided inverse $(D^{- 1} g) (y) = ℓ^{1 / 2} g (ℓ y)$ ; the functions $ψ_{n} : = D g_{n}$ , $ψ_{n} (x) = ℓ^{- 1 / 2} g_{n} (x / ℓ)$ , form a complete orthonormal family of $L^{2} (R, d x)$ ;

(b) the operator

$H_{o s c} : = \frac{ℏ ω}{2} D A D^{- 1}, D (H_{o s c}) : = {f \in L^{2} (R, d x) : D^{- 1} f \in D (A)},$

is self-adjoint, satisfies $H_{o s c} ψ_{n} = ℏ ω (n + \frac{1}{2}) ψ_{n}$ for every $n$ , and has

$D (H_{o s c}) = {f : \sum_{n} ℏ^{2} ω^{2} (n + \frac{1}{2})^{2} ∣ c_{n} ∣^{2} < \infty}, H_{o s c} f = \sum_{n} ℏ ω (n + \frac{1}{2}) c_{n} ψ_{n},$

where $c_{n} : = ⟨ D^{- 1} f, g_{n} ⟩ = ⟨ f, ψ_{n} ⟩$ ; its spectrum is $σ (H_{o s c}) = {ℏ ω (n + \frac{1}{2}) : n \in N}$ , its ground energy is $ℏ ω / 2$ with eigenvector $ψ_{0}$ , and $⟨ H_{o s c} f, f ⟩ \geq \frac{ℏ ω}{2} ∥ f ∥^{2}$ for every $f \in D (H_{o s c})$ ;

(c) $D (P) = span {ψ_{n}}$ is a core for $H_{o s c}$ , and for every polynomial $p$ the function $f : = D (p g_{0})$ is $C^{\infty}$ on $R$ with

$H_{o s c} f = - \frac{ℏ^{2}}{2 m} f^{''} + \frac{m ω^{2}}{2} x^{2} f$

pointwise on $R$ ;

(d) for every $f \in D (H_{o s c})$ and every test function $φ \in C_{c}^{\infty} (R)$ ,

$\int_{R} (H_{o s c} f) (x) φ (x) d x = \int_{R} f (x) (- \frac{ℏ^{2}}{2 m} φ^{''} (x) + \frac{m ω^{2}}{2} x^{2} φ (x)) d x,$

so $H_{o s c}$ is the self-adjoint realization of the differential expression $- \frac{ℏ^{2}}{2 m} \frac{d^{2}}{d x^{2}} + \frac{m ω^{2}}{2} x^{2}$ in the weak sense; no identification of $D (H_{o s c})$ with a separately defined maximal differential-operator domain is asserted.

All parameters enter as positive real numbers; no physical identification of $m, ω, ℏ$ with measured quantities is made.

Facts & Assumptions

Given: real $m, ω, ℏ > 0$ ; $ℓ = \sqrt{ℏ / (m ω)}$ ; the complete orthonormal family $(g_{n})$ in $L^{2} (R, d y)$ ; the diagonal operator $A$ with $A g_{n} = (2 n + 1) g_{n}$ ; the map $D$ and the vectors $ψ_{n} = D g_{n}$ .

[F1]

$(g_{n})$ is orthonormal and complete in $L^{2} (R, d y)$ , hence a Hilbert basis; coefficients in expansions are $⟨ f, g_{n} ⟩$ and $\sum_{n} ∣ ⟨ f, g_{n} ⟩ ∣^{2} = ∥ f ∥^{2}$ (Hermite orthogonality, Hermite completeness, Complex Lp classes and Euclidean test-function conventions).

[F2]

$A$ has domain $D (A) = {f : \sum_{n} (2 n + 1)^{2} ∣ ⟨ f, g_{n} ⟩ ∣^{2} < \infty}$ , action $A g_{n} = (2 n + 1) g_{n}$ , is self-adjoint with $σ (A) = {2 n + 1 : n \in N}$ , and its algebraic eigenvector span is a core for $A$ ; the Hermite polynomials $H_{n}$ have degree $n$ with nonzero leading coefficient, so triangular elimination gives $P = span {g_{n}}$ ; also $⟨ A f, f ⟩ \geq ∥ f ∥^{2}$ for $f \in D (A)$ (Diagonal adjoint and graph-core lemma(b),(d),(e), Diagonal spectrum, Hermite recurrence(f)).

[F3]

$(- \partial_{y}^{2} + Y^{2}) φ = A φ$ for $φ \in P$ ; explicitly $- φ^{''} + y^{2} φ = 2 a^{†} a φ + φ$ on $P$ and $A g_{n} = (2 n + 1) g_{n}$ (Hermite orthogonality(b), [F2]).

[F4]

For a $C^{1}$ diffeomorphism $T : U \to V$ of open subsets of $R$ and $h \in L^{1} (V)$ , $\int_{V} h (y) d y = \int_{U} h (T (x)) ∣ T^{'} (x) ∣ d x$ (A C^1 diffeomorphism satisfies the change-of-variables formula for L^1 functions).

[F5]

$∥ u v ∥_{1} \leq ∥ u ∥_{2} ∥ v ∥_{2}$ ; the real chain and product rules for derivatives, applied componentwise to complex-valued functions under the convention of Complex Lp classes and Euclidean test-function conventions (Cauchy-Schwarz inequality for $L^{2}$ , The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g (c)$ , then $f \circ g$ is differentiable at $c$ with $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ , Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ ); and for real $G \in C^{1} [a, b]$ with integrable derivative, $\int_{a}^{b} G^{'} = G (b) - G (a)$ (The second fundamental theorem: if $G$ is differentiable on $[a, b]$ with $G^{'} = f$ and $f$ is integrable, then $\int_{a}^{b} f = G (b) - G (a)$ ).

[F6]

$C_{c}^{\infty} (R) \subseteq L^{2} (R)$ and, for $g \in C_{c}^{\infty} (R)$ , $g^{''} \in L^{2}$ and $x \mapsto x^{2} g (x) \in L^{2}$ ; the integral pairing $(u, φ) \mapsto \int u φ$ satisfies $∣ \int u φ ∣ \leq ∥ u ∥_{2} ∥ φ ∥_{2}$ (Test function space d of an open set, Complex Lp classes and Euclidean test-function conventions, [F5]).

Proof

1.1givenF4algebra

[ $D$ is a bijective isometry] Define $D$ initially on a measurable representative $u$ of an $L^{2} (d y)$ class by the representative $ℓ^{- 1 / 2} u (x / ℓ)$ . With $T (x) = x / ℓ$ in [F4], $\int h (x / ℓ) d x = ℓ \int h (y) d y$ for every $h \in L^{1} (R)$ . If $u, v$ represent the same class, apply this identity to $h = ∣ u - v ∣^{2} \in L^{1}$ : then $∥ ℓ^{- 1 / 2} (u - v) (\cdot / ℓ) ∥_{2}^{2} = ∥ u - v ∥_{2}^{2} = 0$ , so $D$ is independent of the representative. Applying the same change of variables to $∣ u ∣^{2}$ gives $∥ D u ∥^{2} = ℓ^{- 1} \int ∣ u (x / ℓ) ∣^{2} d x = ∥ u ∥^{2}$ ; $D$ is linear, hence a linear isometry. The formula $D^{- 1} v (y) : = ℓ^{1 / 2} v (ℓ y)$ is also independent of representatives by the same change of variables, is linear, and satisfies $D (D^{- 1} v) = v$ and $D^{- 1} (D u) = u$ as $L^{2}$ classes. Thus $D$ is a bijective linear isometry.

2.1step 1.1F1

[Transport of the orthonormal basis] By step 1.1 and the definition $ψ_{n} = D g_{n}$ , for all $m, n$ we have $⟨ ψ_{n}, ψ_{m} ⟩_{L^{2} (d x)} = \int ℓ^{- 1} g_{n} (x / ℓ) \overline{g_{m} (x / ℓ)} d x = \int g_{n} (y) \overline{g_{m} (y)} d y = δ_{n m}$ by the substitution of step 1.1 applied to $g_{n} \overline{g_{m}} \in L^{1}$ and [F1]. If $h \in L^{2} (d x)$ satisfies $⟨ h, ψ_{n} ⟩ = 0$ for all $n$ , then $⟨ D^{- 1} h, g_{n} ⟩ = ⟨ h, ψ_{n} ⟩ = 0$ for all $n$ because $D$ is an isometry, so $D^{- 1} h = 0$ by completeness of $(g_{n})$ and $h = 0$ . Hence $(ψ_{n})$ is a complete orthonormal family.

3.1step 1.1step 2.1F1F2

[ $H_{o s c}$ is the diagonal operator in the transported basis] Let $B$ be the diagonal operator of Diagonal adjoint and graph-core lemma for the complete orthonormal family $(ψ_{n})$ and the eigenvalues $λ_{n}^{'} = ℏ ω (n + \frac{1}{2})$ , so $D (B) = {f : \sum_{n} (λ_{n}^{'})^{2} ∣ ⟨ f, ψ_{n} ⟩ ∣^{2} < \infty}$ and $B f = \sum_{n} λ_{n}^{'} ⟨ f, ψ_{n} ⟩ ψ_{n}$ . By that lemma and Diagonal spectrum, $B$ is self-adjoint with $σ (B) = \overline{{λ_{n}^{'}}} = {ℏ ω (n + \frac{1}{2})}$ (a closed discrete set) and $⟨ B f, f ⟩ \geq \frac{ℏ ω}{2} ∥ f ∥^{2}$ because $λ_{n}^{'} \geq \frac{ℏ ω}{2}$ . We claim $B = H_{o s c}$ as operators. First, $⟨ f, ψ_{n} ⟩ = ⟨ D^{- 1} f, g_{n} ⟩$ by the isometry of step 1.1 applied to $f = D (D^{- 1} f)$ and $ψ_{n} = D g_{n}$ ; hence $D (H_{o s c}) = {f : D^{- 1} f \in D (A)} = D (B)$ . Second, if $f \in D (H_{o s c})$ and $c_{n} = ⟨ D^{- 1} f, g_{n} ⟩$ , then $D^{- 1} f \in D (A)$ means $\sum_{n} (2 n + 1)^{2} ∣ c_{n} ∣^{2} < \infty$ , and the series $\sum_{n} (2 n + 1) c_{n} g_{n}$ converges in $L^{2}$ to $A D^{- 1} f$ by [F2]; since $D$ is a bijective isometry it preserves limits of series, so $H_{o s c} f = \frac{ℏ ω}{2} D (\sum_{n} (2 n + 1) c_{n} g_{n}) = \sum_{n} ℏ ω (n + \frac{1}{2}) c_{n} ψ_{n} = B f$ , and $\sum_{n} (λ_{n}^{'})^{2} ∣ c_{n} ∣^{2} = \frac{ℏ^{2} ω^{2}}{4} \sum_{n} (2 n + 1)^{2} ∣ c_{n} ∣^{2} < \infty$ . Hence $B = H_{o s c}$ , giving self-adjointness, the eigenvalue formulas $H_{o s c} ψ_{n} = ℏ ω (n + \frac{1}{2}) ψ_{n}$ , the spectrum, the domain formula, the ground energy $ℏ ω / 2$ and the lower bound.

4.1step 3.1F2F3F5algebra

[The differential expression on the core] For a polynomial $p$ put $F : = p g_{0} \in P$ and $f : = D F$ , so $f (x) = ℓ^{- 1 / 2} p (x / ℓ) g_{0} (x / ℓ)$ . By the chain rule and product rule [F5], $f$ is $C^{\infty}$ with $f^{'} (x) = ℓ^{- 3 / 2} F^{'} (x / ℓ)$ and $f^{''} (x) = ℓ^{- 5 / 2} F^{''} (x / ℓ)$ . Since $ℏ^{2} / (2 m ℓ^{2}) = ℏ ω / 2$ and $m ω^{2} ℓ^{2} / 2 = ℏ ω / 2$ by $ℓ^{2} = ℏ / (m ω)$ , for all $x$ one has $- \frac{ℏ^{2}}{2 m} f^{''} (x) + \frac{m ω^{2}}{2} x^{2} f (x) = ℓ^{- 1 / 2} (- \frac{ℏ^{2}}{2 m ℓ^{2}} F^{''} + \frac{m ω^{2} ℓ^{2}}{2} y^{2} F) ∣_{y = x / ℓ} = \frac{ℏ ω}{2} ℓ^{- 1 / 2} (- F^{''} + y^{2} F) (x / ℓ)$ . By [F3], $- F^{''} + y^{2} F = A F$ as functions on $R$ , because the operator identity on $P$ holds for the smooth representatives; hence the right side equals $\frac{ℏ ω}{2} (D A F) (x) = H_{o s c} f (x)$ by step 3.1. This proves the pointwise identity. Moreover, if $\deg p \leq m$ , then $H_{0}, \dots, H_{m}$ have respective degrees $0, \dots, m$ and nonzero leading coefficients by Hermite recurrence; triangular elimination shows that they span all polynomials of degree at most $m$ . Since $g_{n} = H_{n} g_{0} / \sqrt{2^{n} n!}$ , every $p g_{0} \in P$ is a finite linear combination of the $g_{n}$ , and the reverse inclusion follows from $g_{n} \in P$ . Thus $P = span {g_{n}}$ and $D (P) = span {ψ_{n}}$ . Since $P$ is a core for $A$ by [F2], its image under the unitary $D$ is a core for $H_{o s c} = \frac{ℏ ω}{2} D A D^{- 1}$ : for $f \in D (H_{o s c})$ with $D^{- 1} f \in D (A)$ choose $φ_{k} \in P$ with $φ_{k} \to D^{- 1} f$ and $A φ_{k} \to A D^{- 1} f$ ; then $D φ_{k} \to f$ and $H_{o s c} (D φ_{k}) = \frac{ℏ ω}{2} D A φ_{k} \to \frac{ℏ ω}{2} D A D^{- 1} f = H_{o s c} f$ .

5.1step 4.1F5F6

[Weak identity for general elements] Let $f \in D (H_{o s c})$ and $φ \in C_{c}^{\infty} (R)$ ; put $S φ : = - \frac{ℏ^{2}}{2 m} φ^{''} + \frac{m ω^{2}}{2} x^{2} φ$ , an element of $L^{2} (R)$ by [F6]. By step 4.1 with the core $D (P)$ , choose $f_{k} \in D (P)$ with $f_{k} \to f$ and $H_{o s c} f_{k} \to H_{o s c} f$ in $L^{2}$ . For each $k$ , write $f_{k} = D F$ with $F \in P$ ; by applying [F5] separately to the real and imaginary parts on $[- R, R]$ , and using the fact that $F$ is a polynomial times $g_{0}$ , whose products with $φ$ and their derivatives tend to $0$ at $\pm \infty$ , integrating by parts twice gives $\int_{R} (- \frac{ℏ^{2}}{2 m} f_{k}^{''} + \frac{m ω^{2}}{2} x^{2} f_{k}) φ d x = \int_{R} f_{k} (- \frac{ℏ^{2}}{2 m} φ^{''} + \frac{m ω^{2}}{2} x^{2} φ) d x$ . By step 4.1 the left side is $\int (H_{o s c} f_{k}) φ$ . Letting $k \to \infty$ , [F6] and [F5] give $∣ \int (H_{o s c} f_{k} - H_{o s c} f) φ ∣ \leq ∥ H_{o s c} f_{k} - H_{o s c} f ∥_{2} ∥ φ ∥_{2} \to 0$ and $∣ \int (f_{k} - f) S φ ∣ \leq ∥ f_{k} - f ∥_{2} ∥ S φ ∥_{2} \to 0$ , so the identity of the statement holds for $f$ .

6.1step 2.1step 3.1step 4.1step 5.1∎

[Summary of the hypotheses used] The construction uses only the completeness of $(g_{n})$ (step 2.1), the diagonal lemma and spectrum (step 3.1), the eigenvalue equation on $P$ (step 4.1) and the two integration-by-parts limits (step 5.1); no claim identifies $D (H_{o s c})$ with the domain of a maximal differential expression, and no physical meaning is attached to $m, ω, ℏ$ or to $f$ .

LemmaStatement: AI-adaptedProof: AI-adaptedOpen item page →

Closed ladder domains

Statement

Assume the Axiom of Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $H$ be a complex Hilbert space (Hilbert space) with inner product linear in the first argument, let $(e_{n})_{n \in N}$ , $N = {0, 1, 2, \dots}$ , be a complete orthonormal family of $H$ (Orthonormal families, complete orthonormal systems and Hilbert bases), and for $x \in H$ write $c_{n} : = ⟨ x, e_{n} ⟩$ , so that $x = \sum_{n} c_{n} e_{n}$ and $\sum_{n} ∣ c_{n} ∣^{2} = ∥ x ∥^{2}$ . Define

$D (a) : = {x : \sum_{n} n ∣ c_{n} ∣^{2} < \infty}, a x : = \sum_{m \geq 0} \sqrt{m + 1} c_{m + 1} e_{m};$ $D (a^{†}) : = {x : \sum_{n} (n + 1) ∣ c_{n} ∣^{2} < \infty}, a^{†} x : = \sum_{m \geq 1} \sqrt{m} c_{m - 1} e_{m};$ $D (N) : = {x : \sum_{n} n^{2} ∣ c_{n} ∣^{2} < \infty}, N x : = \sum_{n} n c_{n} e_{n} .$

Then:

(a) $a, a^{†}, N$ are well-defined linear operators with the displayed domains; these domains contain the algebraic span $D_{0} = span {e_{n}}$ and are dense in $H$ ; and $∥ a x ∥^{2} = \sum_{n} n ∣ c_{n} ∣^{2}$ , $∥ a^{†} x ∥^{2} = \sum_{n} (n + 1) ∣ c_{n} ∣^{2}$ , $∥ N x ∥^{2} = \sum_{n} n^{2} ∣ c_{n} ∣^{2}$ ;

(b) $a$ and $a^{†}$ are closed, densely defined operators with $a^{*} = a^{†}$ and $(a^{†})^{*} = a$ ;

(d) $D (a a^{†}) = D (a^{†} a) = D (N)$ ; on $D (N)$ one has $a^{†} a = N$ and $a a^{†} = N + I$ , hence $[a, a^{†}] x = a a^{†} x - a^{†} a x = x$ for every $x \in D (N)$ ; and $D (N)$ is exactly the set of $x \in H$ for which both $a (a^{†} x)$ and $a^{†} (a x)$ are defined;

(e) $N$ is self-adjoint with $N e_{n} = n e_{n}$ and spectrum $σ (N) = N$ (Diagonal spectrum), and for $x \in D (N)$ the partial sums $x_{M} : = \sum_{n \leq M} c_{n} e_{n}$ satisfy $x_{M} \to x$ and $N x_{M} \to N x$ ;

(f) if $H = L^{2} (R, d y)$ with the Hermite orthonormal family $e_{n} = g_{n}$ of Hermite recurrence, then the operators $a_{P}, a_{P}^{†} : P \to P$ of that theorem are the restrictions of $a, a^{†}$ to $P$ , and $a$ and $a^{†}$ are exactly the closures of $a_{P}$ and $a_{P}^{†}$ .

Facts & Assumptions

Given: a complex Hilbert space $H$ , a complete orthonormal family $(e_{n})$ , and the operators $a, a^{†}, N$ displayed above.

[F1]

$⟨ e_{i}, e_{j} ⟩ = δ_{i j}$ , $x = \sum_{n} ⟨ x, e_{n} ⟩ e_{n}$ with $\sum_{n} ∣ ⟨ x, e_{n} ⟩ ∣^{2} = ∥ x ∥^{2}$ for every $x \in H$ , and a series $\sum_{n} a_{n} e_{n}$ converges in $H$ exactly when $\sum_{n} ∣ a_{n} ∣^{2} < \infty$ , its sum then satisfying $∥ \sum_{n} a_{n} e_{n} ∥^{2} = \sum_{n} ∣ a_{n} ∣^{2}$ (Orthonormal families, complete orthonormal systems and Hilbert bases, Parseval equivalences for an orthonormal family).

[F2]

For a densely defined $T$ : $y \in D (T^{*})$ exactly when $x \mapsto ⟨ T x, y ⟩$ is bounded on $(D (T), ∥ \cdot ∥)$ , and then $T^{*} y$ is the unique $w$ with $⟨ T x, y ⟩ = ⟨ x, w ⟩$ for all $x \in D (T)$ ; $T^{*}$ is closed; $T$ is closed exactly when $T = (T^{*})^{*}$ is an adjoint (Adjoint of a densely defined operator, The adjoint is well defined, closed, and reverses inclusions).

[F3]

[F4]

The self-adjoint diagonal operator of Diagonal adjoint and graph-core lemma with $λ_{n} = n$ is precisely $N$ on $D (N)$ , and its spectrum is the closure of ${n : n \in N} = N$ (Diagonal spectrum).

[F5]

In the concrete case $H = L^{2} (R, d y)$ with $e_{n} = g_{n}$ , the operators $a_{P}, a_{P}^{†}$ of Hermite recurrence act on the finite expansions by $a_{P} g_{n} = \sqrt{n} g_{n - 1}$ and $a_{P}^{†} g_{n} = \sqrt{n + 1} g_{n + 1}$ , and $P = span {g_{n}}$ (Hermite recurrence(e),(f)).

Proof

1.1givenF1F3algebra

[Well-definedness and densities] For $x \in D (a)$ the coefficients $\sqrt{m + 1} c_{m + 1}$ are square summable with $\sum_{m} (m + 1) ∣ c_{m + 1} ∣^{2} = \sum_{n \geq 1} n ∣ c_{n} ∣^{2} < \infty$ , so the series $a x = \sum_{m} \sqrt{m + 1} c_{m + 1} e_{m}$ converges in $H$ and $∥ a x ∥^{2} = \sum_{n \geq 1} n ∣ c_{n} ∣^{2}$ by [F1]; the same computation gives the formulas for $a^{†} x$ and $N x$ . Linearity is coefficientwise. Every finite combination of the $e_{n}$ lies in all three domains, and $D_{0}$ is dense in $H$ by [F1], so the three domains are dense.

1.2givenF1F2F3algebra

[ $a^{*} = a^{†}$ ] First let $y \in D (a^{*})$ and $w : = a^{*} y$ . Testing the defining identity $⟨ a x, y ⟩ = ⟨ x, w ⟩$ on $x = e_{m}$ ( $m \geq 1$ ) gives $\sqrt{m} \overline{⟨ y, e_{m - 1} ⟩} = \overline{⟨ w, e_{m} ⟩}$ , that is $\sqrt{m} ⟨ y, e_{m - 1} ⟩ = ⟨ w, e_{m} ⟩$ ; hence $\sum_{m \geq 1} m ∣ ⟨ y, e_{m - 1} ⟩ ∣^{2} = \sum_{m \geq 1} ∣ ⟨ w, e_{m} ⟩ ∣^{2} \leq ∥ w ∥^{2} < \infty$ ; testing $x = e_{0}$ gives $⟨ e_{0}, w ⟩ = 0$ , hence $⟨ w, e_{0} ⟩ = 0$ . Therefore $y \in D (a^{†})$ and $a^{†} y = \sum_{m \geq 1} \sqrt{m} ⟨ y, e_{m - 1} ⟩ e_{m} = \sum_{m \geq 1} ⟨ w, e_{m} ⟩ e_{m} = w$ by [F1]. Thus $a^{*} \subseteq a^{†}$ . Conversely let $y \in D (a^{†})$ and $x \in D (a)$ ; then $⟨ a x, y ⟩ = \sum_{m \geq 0} \sqrt{m + 1} c_{m + 1} \overline{d_{m}}$ with $d_{m} = ⟨ y, e_{m} ⟩$ , absolutely convergent by [F3] since $\sum_{m} (m + 1) ∣ c_{m + 1} ∣^{2} = ∥ a x ∥^{2} < \infty$ and $\sum_{m} ∣ d_{m} ∣^{2} = ∥ y ∥^{2}$ , and the same series equals $\sum_{n \geq 1} c_{n} \overline{\sqrt{n} d_{n - 1}} = ⟨ x, a^{†} y ⟩$ ; hence $∣ ⟨ a x, y ⟩ ∣ = ∣ ⟨ x, a^{†} y ⟩ ∣ \leq ∥ x ∥ ∥ a^{†} y ∥$ , so the functional is bounded on $(D (a), ∥ \cdot ∥)$ , so $y \in D (a^{*})$ and $a^{*} y = a^{†} y$ . Therefore $a^{*} = a^{†}$ .

1.3givenF1F2F3algebra

[ $(a^{†})^{*} = a$ and closedness] Let $y \in D ((a^{†})^{*})$ and $w : = (a^{†})^{*} y$ ; testing against $x = e_{m}$ with $m \geq 0$ gives $\sqrt{m + 1} ⟨ y, e_{m + 1} ⟩ = ⟨ w, e_{m} ⟩$ , so $\sum_{m \geq 0} (m + 1) ∣ ⟨ y, e_{m + 1} ⟩ ∣^{2} = \sum_{m} ∣ ⟨ w, e_{m} ⟩ ∣^{2} < \infty$ , that is $\sum_{n \geq 1} n ∣ ⟨ y, e_{n} ⟩ ∣^{2} < \infty$ ; thus $y \in D (a)$ and $a y = \sum_{n \geq 1} \sqrt{n} ⟨ y, e_{n} ⟩ e_{n - 1} = \sum_{m \geq 0} ⟨ w, e_{m} ⟩ e_{m} = w$ . Conversely for $y \in D (a)$ and $x \in D (a^{†})$ the same two absolutely convergent series show $⟨ a^{†} x, y ⟩ = ⟨ x, a y ⟩$ , so $y \in D ((a^{†})^{*})$ with $(a^{†})^{*} y = a y$ . Hence $(a^{†})^{*} = a$ . Since $a = (a^{†})^{*}$ and $a^{†} = a^{*}$ are adjoints of densely defined operators, [F2] makes both closed.

1.4givenF1algebra

[Kernels] If $a x = 0$ then $\sqrt{m + 1} c_{m + 1} = 0$ for all $m$ , so $c_{n} = 0$ for $n \geq 1$ and $x = c_{0} e_{0}$ ; conversely $a e_{0} = 0$ . If $a^{†} x = 0$ then $\sqrt{m} c_{m - 1} = 0$ for all $m \geq 1$ , so $c_{n} = 0$ for all $n$ and $x = 0$ .

1.5givenF1algebra

[Products, commutator and exact domain] For $x = \sum_{n} c_{n} e_{n}$ , membership in $D (a a^{†})$ means $\sum_{n} (n + 1) ∣ c_{n} ∣^{2} < \infty$ and $\sum_{m \geq 1} m ∣ \sqrt{m} c_{m - 1} ∣^{2} = \sum_{n} (n + 1)^{2} ∣ c_{n} ∣^{2} < \infty$ . The second condition implies the first, and, for $x \in H$ , is equivalent to $\sum_{n} n^{2} ∣ c_{n} ∣^{2} < \infty$ since $n^{2} \leq (n + 1)^{2} \leq 2 n^{2} + 2$ . On this domain the coefficient of $e_{n}$ in $a a^{†} x$ is $(n + 1) c_{n}$ . Likewise $x \in D (a^{†} a)$ means $\sum_{n} n ∣ c_{n} ∣^{2} < \infty$ and $\sum_{m \geq 0} (m + 1) ∣ \sqrt{m + 1} c_{m + 1} ∣^{2} = \sum_{n \geq 1} n^{2} ∣ c_{n} ∣^{2} < \infty$ ; the latter implies the former, and the coefficient of $e_{n}$ in $a^{†} a x$ is $n c_{n}$ , including zero at $n = 0$ . Hence both product domains equal $D (N)$ , $a^{†} a = N$ , $a a^{†} = N + I$ there, and $[a, a^{†}] x = x$ . Their common domain is exactly $D (N)$ .

2.1step 1.1F4

[(e) self-adjointness and spectrum] By [F4] the operator with domain $D (N)$ and action $N e_{n} = n e_{n}$ is self-adjoint with spectrum $N$ , and step 1.1 identifies it with the diagonal operator of [F4]. The partial-sum statement is the graph-core clause of Diagonal adjoint and graph-core lemma with $λ_{n} = n$ .

3.1step 1.1step 1.3F5∎

[(f) closure of the concrete ladder] In the concrete case $H = L^{2} (R, d y)$ , $e_{n} = g_{n}$ and $P = span {g_{n}}$ by [F5]; on the finite expansion $φ = \sum_{n \leq M} c_{n} g_{n}$ the operator $a$ of this lemma acts by $a φ = \sum_{m \leq M - 1} \sqrt{m + 1} c_{m + 1} g_{m} = a_{P} φ$ , and similarly $a^{†} φ = a_{P}^{†} φ$ ; hence $a_{P} \subseteq a$ and $a_{P}^{†} \subseteq a^{†}$ . Since $a$ is closed by step 1.3 and $a_{P} \subseteq a$ , the closure satisfies $\overline{a_{P}} \subseteq a$ . Conversely, for $x \in D (a)$ the partial sums $x_{M} = \sum_{n \leq M} c_{n} g_{n} \in P$ satisfy $x_{M} \to x$ and $a_{P} x_{M} = a x_{M} \to a x$ by step 1.1, so $(x, a x) \in \overline{Γ (a_{P})}$ and $a \subseteq \overline{a_{P}}$ ; hence $a = \overline{a_{P}}$ , and the argument for $a^{†}$ is identical.

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Coherent-state convergence

Statement

Assume the Axiom of Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $H = L^{2} (R, d y)$ with inner product linear in the first argument, let $(g_{n})_{n \in N}$ be the Hermite family of Hermite recurrence, and let $a, a^{†}, N$ be the closed ladder operators and number operator of Closed ladder domains with domains $D (a), D (a^{†}), D (N)$ and $N = a^{†} a$ on $D (N)$ . For $z \in C$ put

$ψ_{z} : = e^{- ∣ z ∣^{2} / 2} \sum_{n = 0}^{\infty} \frac{z^{n}}{\sqrt{n!}} g_{n},$

the series being taken in $L^{2}$ . Then:

(a) the series converges in $L^{2}$ , $∥ ψ_{z} ∥ = 1$ , and $ψ_{z} = e^{- ∣ z ∣^{2} / 2} \sum_{n = 0}^{\infty} \frac{z^{n}}{\sqrt{n!}} g_{n}$ is the unique element of $L^{2}$ with coefficients $c_{n} = e^{- ∣ z ∣^{2} / 2} z^{n} / \sqrt{n!}$ in the basis $(g_{n})$ ;

(b) $ψ_{z} \in D (N^{k})$ for every $k \in N$ , where $N^{k}$ is the $k$ -fold iterate of $N$ with domain $D (N^{k}) = {x : \sum_{n} n^{2 k} ∣ ⟨ x, g_{n} ⟩ ∣^{2} < \infty}$ ; in particular

$\sum_{n} n^{2 k} ∣ c_{n} ∣^{2} = e^{- ∣ z ∣^{2}} \sum_{n = 0}^{\infty} \frac{n^{2 k} ∣ z ∣^{2 n}}{n!} < \infty;$

(c) $a ψ_{z} = z ψ_{z}$ (so $ψ_{z}$ is a unit eigenvector of the annihilation operator with eigenvalue $z$ ), and $∥ a ψ_{z} ∥^{2} = ∣ z ∣^{2}$ ;

(d) $⟨ ψ_{z}, ψ_{w} ⟩ = e^{- (∣ z ∣^{2} + ∣ w ∣^{2}) / 2 + z \overline{w}} = e^{- ∣ z - w ∣^{2} / 2} e^{i Im (z \overline{w})}$ for all $z, w \in C$ , so $∥ ψ_{z} - ψ_{w} ∥^{2} = 2 (1 - e^{- ∣ z - w ∣^{2} / 2} \cos (Im (z \overline{w})))$ and the map $z \mapsto ψ_{z}$ is injective;

(e) $⟨ N ψ_{z}, ψ_{z} ⟩ = ∣ z ∣^{2}$ and $∥ a^{†} ψ_{z} ∥^{2} = ∣ z ∣^{2} + 1$ , and $ψ_{0} = g_{0}$ .

The completeness of $(g_{n})$ (Hermite completeness) is used for uniqueness from the prescribed coefficients in (a) and to invoke the closed ladder operators of Closed ladder domains on a complete orthonormal family. Series convergence and normalization use orthonormality and Hilbert-space completeness; the operator and moment conclusions use the supplied closed ladder domains.

Facts & Assumptions

Given: the Hermite family $(g_{n})$ with its ladder operators $a, a^{†}, N$ ; a complex number $z$ ; the numbers $c_{n} = e^{- ∣ z ∣^{2} / 2} z^{n} / \sqrt{n!}$ .

[F1]

$⟨ g_{n}, g_{m} ⟩ = δ_{n m}$ (Hermite orthogonality(a)), and $L^{2}$ is a complex Hilbert space under Countable Choice ( $L^{2}$ with the integral pairing is a Hilbert space). Finite orthogonal sums satisfy $∥ \sum_{n \in E} a_{n} g_{n} ∥^{2} = \sum_{n \in E} ∣ a_{n} ∣^{2}$ . Thus a series $\sum_{n} a_{n} g_{n}$ converges exactly when $\sum_{n} ∣ a_{n} ∣^{2} < \infty$ : finite tail norms give the Cauchy criterion and Hilbert-space completeness gives the limit; conversely convergence bounds the partial norms. Continuity of the norm and inner product gives the sum's squared norm and its coefficients.

[F2]

On $D (N)$ one has $a^{†} a = N$ , $a a^{†} = N + I$ ; $D (a a^{†}) = D (a^{†} a) = D (N)$ ; and $a$ is closed with $a^{*} = a^{†}$ (Closed ladder domains(b),(d)).

[F3]

$e^{t} = \sum_{n \geq 0} t^{n} / n!$ for complex $t$ , the series converging absolutely (The complex exponential by its power series, The complex exponential series converges absolutely for every complex argument); powers and integer powers of complex numbers are those of the library's power conventions. The addition law $e^{v + w} = e^{v} e^{w}$ and agreement with the real exponential are $\exp (z + w) = \exp z \exp w$ , and the complex exponential extends the real exponential; for real $x, y$ , $e^{x + i y} = e^{x} (\cos y + i \sin y)$ and $∣ e^{x + i y} ∣ = e^{x}$ are $\exp (x + i y) = e^{x} (\cos y + i \sin y)$ , $∣ \exp (x + i y) ∣ = e^{x}$ , and $e^{i π} + 1 = 0$ . Positivity and $e^{- x} = 1 / e^{x}$ are The exponential is positive and satisfies $\exp (- x) = 1 / \exp (x)$ , and strict increase is The exponential function is strictly increasing.

[F4]

If $0 \leq a_{n + 1} \leq \frac{1}{2} a_{n}$ for all $n \geq N$ , then $\sum_{n \geq N} a_{n} \leq 2 a_{N}$ , by comparison with the geometric series $\sum_{k} 2^{- k} = 2$ (For $∣ r ∣ < 1$ , $\sum_{k \geq 0} r^{k} = 1 / (1 - r)$ , and for $∣ r ∣ \geq 1$ the series diverges).

[F5]

$a g_{n} = \sqrt{n} g_{n - 1}$ for $n \geq 1$ with $a g_{0} = 0$ ; $a^{†} g_{n} = \sqrt{n + 1} g_{n + 1}$ (Hermite recurrence(e)).

[F6]

The Hermite family is complete: any $h \in L^{2}$ with $⟨ h, g_{n} ⟩ = 0$ for every $n$ is zero (Hermite completeness(b)).

Proof

1.1givenF1F3F6

[Convergence and normalization] For $N \geq M$ , the partial sums $S_{N} : = e^{- ∣ z ∣^{2} / 2} \sum_{n \leq N} \frac{z^{n}}{\sqrt{n!}} g_{n}$ satisfy $∥ S_{N} - S_{M} ∥^{2} = e^{- ∣ z ∣^{2}} \sum_{M < n \leq N} \frac{∣ z ∣^{2 n}}{n!}$ by [F1]. As $N, M \to \infty$ the right side tends to $0$ because $\sum_{n} \frac{∣ z ∣^{2 n}}{n!} = e^{∣ z ∣^{2}} < \infty$ by [F3]; hence $(S_{N})$ is Cauchy and $ψ_{z} : = \lim_{N} S_{N} \in L^{2}$ exists, and taking limits in the same identity gives $∥ ψ_{z} ∥^{2} = e^{- ∣ z ∣^{2}} \sum_{n} \frac{∣ z ∣^{2 n}}{n!} = 1$ . For fixed $n$ , inner-product continuity gives $⟨ ψ_{z}, g_{n} ⟩ = \lim_{M \to \infty} ⟨ S_{M}, g_{n} ⟩ = c_{n}$ . If $x \in L^{2}$ has these same coefficients, then $⟨ x - ψ_{z}, g_{n} ⟩ = 0$ for every $n$ , so $x = ψ_{z}$ by completeness [F6].

2.1step 1.1F1F2F4algebra

[Number domains] For $k = 0$ use $N^{0} = I$ , $D (N^{0}) = H$ and $n^{0} = 1$ , including $n = 0$ ; step 1.1 proves the assertion. Now fix $k \geq 1$ and let $a_{n} : = n^{2 k} ∣ z ∣^{2 n} / n! \geq 0$ . If $z = 0$ , $ψ_{0} = g_{0}$ , which lies in every $D (N^{k})$ and all terms $a_{n}$ with $n \geq 1$ vanish. If $z \neq 0$ , for $n \geq 1$ , $\frac{a_{n + 1}}{a_{n}} = (\frac{n + 1}{n})^{2 k} \frac{∣ z ∣^{2}}{n + 1} \to 0$ , so there is $N$ with $a_{n + 1} \leq \frac{1}{2} a_{n}$ for $n \geq N$ ; by [F4], $\sum_{n \geq N} a_{n} \leq 2 a_{N} < \infty$ , and the finitely many initial terms are finite. Hence $\sum_{n} n^{2 k} ∣ c_{n} ∣^{2} = e^{- ∣ z ∣^{2}} \sum_{n} a_{n} < \infty$ . Since $D (N^{k}) = {x : \sum_{n} n^{2 k} ∣ ⟨ x, g_{n} ⟩ ∣^{2} < \infty}$ (inducting the domain formula of Closed ladder domains(a): $N^{j} x$ has coefficients $n^{j} ⟨ x, g_{n} ⟩$ , and the weight $n^{2 k}$ dominates every lower weight for $n \geq 1$ ), this says $ψ_{z} \in D (N^{k})$ .

2.2step 1.1F2F5algebra

[ $a ψ_{z} = z ψ_{z}$ ] Each partial sum $S_{N}$ is a finite combination of the $g_{n}$ , so $S_{N} \in D (N) \subseteq D (a)$ ; by [F5], $a S_{N} = e^{- ∣ z ∣^{2} / 2} \sum_{n = 1}^{N} z^{n} \sqrt{n} g_{n - 1} / \sqrt{n!} = z e^{- ∣ z ∣^{2} / 2} \sum_{m = 0}^{N - 1} \frac{z^{m}}{\sqrt{m!}} g_{m} = z S_{N - 1}$ . As $N \to \infty$ , $S_{N} \to ψ_{z}$ and $a S_{N} \to z ψ_{z}$ by step 1.1; since $a$ is closed [F2], $ψ_{z} \in D (a)$ and $a ψ_{z} = z ψ_{z}$ . Taking norms, $∥ a ψ_{z} ∥ = ∣ z ∣$ by (a).

2.3step 1.1F1F3algebra

[Inner products] For $z, w \in C$ , $⟨ ψ_{z}, ψ_{w} ⟩ = e^{- (∣ z ∣^{2} + ∣ w ∣^{2}) / 2} \sum_{n} \frac{z^{n} {\overline{w}}^{n}}{n!}$ by [F1] applied to the two convergent series, and $\sum_{n} \frac{(z \overline{w})^{n}}{n!} = e^{z \overline{w}}$ by [F3] applied to the series of the complex number $z \overline{w}$ ; hence $⟨ ψ_{z}, ψ_{w} ⟩ = e^{- (∣ z ∣^{2} + ∣ w ∣^{2}) / 2 + z \overline{w}} = e^{- ∣ z - w ∣^{2} / 2} e^{i Im (z \overline{w})}$ . Its modulus is $e^{- ∣ z - w ∣^{2} / 2}$ . Then $∥ ψ_{z} - ψ_{w} ∥^{2} = ∥ ψ_{z} ∥^{2} + ∥ ψ_{w} ∥^{2} - 2 Re ⟨ ψ_{z}, ψ_{w} ⟩ = 2 (1 - e^{- ∣ z - w ∣^{2} / 2} \cos (Im (z \overline{w})))$ . When $z \neq w$ the overlap modulus is strictly less than $1$ , so this distance is positive; when $z = w$ it is zero; the map is injective.

3.1step 1.1step 2.1F1F2F3algebra∎

[Number and creation moments] By step 2.1 with $k = 1$ , $ψ_{z} \in D (N)$ and $N ψ_{z}$ has coefficients $n c_{n}$ ; hence $⟨ N ψ_{z}, ψ_{z} ⟩ = \sum_{n} n ∣ c_{n} ∣^{2} = e^{- ∣ z ∣^{2}} \sum_{n} \frac{n ∣ z ∣^{2 n}}{n!}$ . The series reindexes as $∣ z ∣^{2} \sum_{m \geq 0} \frac{∣ z ∣^{2 m}}{m!} = ∣ z ∣^{2} e^{∣ z ∣^{2}}$ by [F3], so the value is $∣ z ∣^{2}$ . Next, $∥ a^{†} ψ_{z} ∥^{2} = ⟨ a^{†} ψ_{z}, a^{†} ψ_{z} ⟩ = ⟨ ψ_{z}, a a^{†} ψ_{z} ⟩ = ⟨ ψ_{z}, (N + I) ψ_{z} ⟩ = ∣ z ∣^{2} + 1$ using [F2] and $ψ_{z} \in D (N)$ . Finally $ψ_{0} = g_{0}$ since all coefficients with $n \geq 1$ vanish and $∥ ψ_{0} ∥ = 1$ .

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Coherent evolution and variance

Statement

Assume the Axiom of Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Fix $m, ω, ℏ > 0$ and let $ℓ = \sqrt{ℏ / (m ω)}$ , $D$ , $ψ_{n} = D g_{n}$ , $H_{o s c} = \frac{ℏ ω}{2} D A D^{- 1}$ be as in Oscillator scaling and exact realization, with $(g_{n})$ the Hermite family and $a, a^{†}, N$ the closed ladder and number operators of Closed ladder domains; put $a_{ψ} : = D a D^{- 1}$ , $a_{ψ}^{†} : = D a^{†} D^{- 1}$ , $N_{ψ} : = D N D^{- 1}$ , acting on $L^{2} (R, d x)$ with the transported domains, and for $z \in C$ let $ψ_{z}^{(y)} \in L^{2} (R, d y)$ be the coherent vector of Coherent-state convergence and define $ψ_{z} : = D ψ_{z}^{(y)} = e^{- ∣ z ∣^{2} / 2} \sum_{n} z^{n} ψ_{n} / \sqrt{n!} \in L^{2} (R, d x)$ . Finally set

$Q : = M_{x}, D (Q) = {f \in L^{2} (R, d x) : x f \in L^{2}}, P : = F_{2}^{*} M_{2 π ℏ ξ} F_{2}, D (P) = {f : ξ F_{2} f \in L^{2}},$

the self-adjoint position and momentum operators (Real L2 multipliers and unitary transport, Plancherel theorem; the conjugation is discharged in the proof).

Then:

(a) $H_{o s c} = ℏ ω (N_{ψ} + \frac{1}{2} I)$ as operators, with common domain $D (H_{o s c}) = D (N_{ψ})$ ;

(b) the formula

$U (t) f : = \sum_{n} e^{- i ω (n + 1 / 2) t} ⟨ f, ψ_{n} ⟩ ψ_{n} (t \in R, f \in L^{2} (R, d x))$

defines a strongly continuous one-parameter unitary group $U$ on $L^{2} (R, d x)$ : each $U (t)$ is a surjective linear isometry, $U (0) = I$ and $U (t + s) = U (t) U (s)$ , and the generator of $U$ is $- \frac{i}{ℏ} H_{o s c}$ , i.e. the limit $\lim_{t \to 0} \frac{1}{t} (U (t) f - f)$ exists exactly for $f \in D (H_{o s c})$ and equals $- \frac{i}{ℏ} H_{o s c} f$ ;

(d) on the orthonormal basis one has

$Q ψ_{n} = \frac{ℓ}{\sqrt{2}} (\sqrt{n + 1} ψ_{n + 1} + \sqrt{n} ψ_{n - 1}), P ψ_{n} = \frac{ℏ}{i ℓ \sqrt{2}} (\sqrt{n} ψ_{n - 1} - \sqrt{n + 1} ψ_{n + 1}),$

with the convention $ψ_{- 1} : = 0$ ; consequently, for every $z \in C$ ,

$Q ψ_{z} = \frac{ℓ}{\sqrt{2}} (a_{ψ} + a_{ψ}^{†}) ψ_{z}, P ψ_{z} = \frac{ℏ}{i ℓ \sqrt{2}} (a_{ψ} - a_{ψ}^{†}) ψ_{z};$

(e) with means $⟨ Q ⟩_{z} : = ⟨ Q ψ_{z}, ψ_{z} ⟩$ , $⟨ P ⟩_{z} : = ⟨ P ψ_{z}, ψ_{z} ⟩$ and variances $Var (Q)_{z} : = ∥ (Q - ⟨ Q ⟩_{z} I) ψ_{z} ∥^{2}$ , $Var (P)_{z} : = ∥ (P - ⟨ P ⟩_{z} I) ψ_{z} ∥^{2}$ ,

$⟨ Q ⟩_{z} = \sqrt{2} ℓ Re z, ⟨ P ⟩_{z} = \frac{\sqrt{2} ℏ}{ℓ} Im z, Var (Q)_{z} = \frac{ℓ^{2}}{2}, Var (P)_{z} = \frac{ℏ^{2}}{2 ℓ^{2}},$

so $Var (Q)_{z} Var (P)_{z} = ℏ^{2} / 4$ for every $z$ (the value produced by these exact computations; no independent uncertainty-bound statement is asserted here);

(f) after the evolution, with $ψ_{z} (t) : = U (t) ψ_{z}$ and $z (t) : = z e^{- i ω t}$ ,

$⟨ Q ⟩_{z (t)} = \sqrt{2} ℓ Re z (t), ⟨ P ⟩_{z (t)} = \frac{\sqrt{2} ℏ}{ℓ} Im z (t), Var (Q)_{z (t)} = \frac{ℓ^{2}}{2}, Var (P)_{z (t)} = \frac{ℏ^{2}}{2 ℓ^{2}} .$

The parameters $m, ω, ℏ$ enter as positive real numbers and no physical identification of $Q, P, H_{o s c}$ with measured quantities is made.

Facts & Assumptions

Given: the unitary $D$ , the complete orthonormal family $(ψ_{n})$ of eigenfunctions of $H_{o s c}$ , the operators $a_{ψ}, a_{ψ}^{†}, N_{ψ}$ transported from Closed ladder domains, and the coherent vectors $ψ_{z}$ .

[F1]

$(g_{n})$ is a complete orthonormal family of $L^{2} (R, d y)$ ; $D$ is a bijective isometry with inverse $D^{- 1} g (y) = ℓ^{1 / 2} g (ℓ y)$ ; $(ψ_{n})$ is a complete orthonormal family of $L^{2} (R, d x)$ ; $D (H_{o s c}) = {f : \sum_{n} ℏ^{2} ω^{2} (n + \frac{1}{2})^{2} ∣ c_{n} ∣^{2} < \infty}$ with $c_{n} = ⟨ f, ψ_{n} ⟩$ , $H_{o s c} ψ_{n} = ℏ ω (n + \frac{1}{2}) ψ_{n}$ , and $σ (H_{o s c}) = {ℏ ω (n + \frac{1}{2})}$ (Oscillator scaling and exact realization).

[F2]

On $D (N_{ψ})$ , $a_{ψ}^{†} a_{ψ} = N_{ψ}$ and $a_{ψ} a_{ψ}^{†} = N_{ψ} + I$ ; $a_{ψ}$ is closed; $a_{ψ} ψ_{z} = z ψ_{z}$ ; $ψ_{z} \in D (N_{ψ}^{k})$ for every $k$ , $⟨ ψ_{z}, ψ_{w} ⟩ = e^{- (∣ z ∣^{2} + ∣ w ∣^{2}) / 2 + z \overline{w}}$ , $∥ a_{ψ} ψ_{z} ∥^{2} = ∣ z ∣^{2}$ , $∥ a_{ψ}^{†} ψ_{z} ∥^{2} = ∣ z ∣^{2} + 1$ , $⟨ N_{ψ} ψ_{z}, ψ_{z} ⟩ = ∣ z ∣^{2}$ . These are the facts of Closed ladder domains(b),(d) and Coherent-state convergence, transported by the unitary $D$ .

[F3]

$Y = \frac{a + a^{†}}{\sqrt{2}}$ and $- i \partial_{y} = \frac{a - a^{†}}{i \sqrt{2}}$ on the polynomial-Gaussian space $P : = {p g_{0} : p \in C [y]}$ (denoted $P$ in the supplier), whence $\sqrt{2} y g_{n} = \sqrt{n} g_{n - 1} + \sqrt{n + 1} g_{n + 1}$ and $\sqrt{2} g_{n}^{'} = \sqrt{n} g_{n - 1} - \sqrt{n + 1} g_{n + 1}$ (Hermite recurrence(g), Hermite orthogonality(b)).

[F4]

$M_{x}$ is self-adjoint on $D (M_{x}) = {f : x f \in L^{2}}$ for the real function $x$ (Real L2 multipliers and unitary transport); $F_{2}$ is unitary (Plancherel theorem); if $T$ is self-adjoint on $K$ and $U : K \to H$ is a bijective linear isometry, then $U T U^{*}$ is self-adjoint on $U D (T)$ (proved in step 1.3); and $D (P) = W^{1, 2} (R)$ with $P f = - i ℏ f^{'}$ there (Integer-order W^{k,2} and H^k agree with equivalent norms, Distributional derivatives are polynomial Fourier multipliers).

[F5]

$e^{i x} = 1 + i x + o (x)$ as $x \to 0$ and $∣ e^{i x} - 1 - i x ∣ \leq 2 ∣ x ∣$ for $x \in R$ : the complex exponential is entire with derivative itself at every point (The complex exponential is entire and its complex derivative is itself, The complex exponential by its power series); the estimate follows from $e^{i x} - 1 = \int_{0}^{1} i x e^{i s x} d s$ , obtained by the chain rule and the fundamental theorem of calculus, applied to real and imaginary parts (The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g (c)$ , then $f \circ g$ is differentiable at $c$ with $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ , The second fundamental theorem: if $G$ is differentiable on $[a, b]$ with $G^{'} = f$ and $f$ is integrable, then $\int_{a}^{b} f = G (b) - G (a)$ ), so $∣ e^{i x} - 1 ∣ \leq ∣ x ∣$ ; and $∥ f ∥^{2} = \sum_{n} ∣ ⟨ f, ψ_{n} ⟩ ∣^{2}$ with the finite-subset net convergence of the expansion (Parseval equivalences for an orthonormal family).

[F6]

$∥ u v ∥_{1} \leq ∥ u ∥_{2} ∥ v ∥_{2}$ and $∣ ⟨ u, v ⟩ ∣ \leq ∥ u ∥ ∥ v ∥$ (Cauchy-Schwarz inequality for $L^{2}$ , The space $L^{p} (μ)$ as the quotient by null functions, Complex Lp classes and Euclidean test-function conventions).

Proof

1.1givenF1algebra

[(a) Hamiltonian as a number operator] Let $f \in D (H_{o s c})$ with $c_{n} = ⟨ f, ψ_{n} ⟩$ ; by [F1] this means $\sum_{n} (n + \frac{1}{2})^{2} ∣ c_{n} ∣^{2} < \infty$ , which is equivalent to $\sum_{n} n^{2} ∣ c_{n} ∣^{2} < \infty$ because $n^{2} \leq 4 (n + \frac{1}{2})^{2}$ and $(n + \frac{1}{2})^{2} \leq 2 n^{2} + \frac{1}{2}$ for all $n \geq 0$ ; hence $D (H_{o s c}) = D (N_{ψ})$ by [F1] and the transported domain formula, and $H_{o s c} f = \sum_{n} ℏ ω (n + \frac{1}{2}) c_{n} ψ_{n} = ℏ ω (\sum_{n} n c_{n} ψ_{n} + \frac{1}{2} f) = ℏ ω (N_{ψ} f + \frac{1}{2} f)$ .

1.2givenF5

[(b) unitarity of $U (t)$ ] For $f \in L^{2}$ the coefficients $c_{n} = ⟨ f, ψ_{n} ⟩$ are square summable, so the series in (b) converges and $∥ U (t) f ∥^{2} = \sum_{n} ∣ c_{n} ∣^{2} = ∥ f ∥^{2}$ by [F5]; $U (t)$ is linear and isometric. Its range contains every $ψ_{n} = U (t) (e^{i ω (n + 1 / 2) t} ψ_{n})$ , hence is dense; an isometry has closed range, so $U (t)$ is surjective: it is a bijective isometry. Also $U (t) ψ_{n} = e^{- i ω (n + 1 / 2) t} ψ_{n}$ shows $U (t) U (t)^{*}$ fixes every $ψ_{n}$ , hence equals the identity on the dense span; by continuity $U (t)^{*} = U (t)^{- 1}$ , and $U (t)^{- 1} = U (- t)$ .

1.3givenF3F4

[(d) the position and momentum operators] The conjugation claim of [F4]: if $T = T^{*}$ on $K$ and $U : K \to H$ is a bijective isometry, let $T^{'} : = U T U^{*}$ on $D (T^{'}) : = U D (T)$ . For $x, y \in D (T^{'})$ , writing $x = U ξ$ , $y = U η$ with $ξ, η \in D (T)$ , $⟨ T^{'} x, y ⟩ = ⟨ T ξ, η ⟩ = ⟨ ξ, T η ⟩ = ⟨ x, T^{'} y ⟩$ , so $T^{'}$ is symmetric. If $y \in D ((T^{'})^{*})$ then for all $ξ \in D (T)$ , $⟨ T ξ, U^{*} y ⟩ = ⟨ T^{'} U ξ, y ⟩ = ⟨ U ξ, (T^{'})^{*} y ⟩ = ⟨ ξ, U^{*} (T^{'})^{*} y ⟩$ , so $U^{*} y \in D (T^{*}) = D (T)$ and $T^{*} U^{*} y = U^{*} (T^{'})^{*} y$ ; hence $y \in U D (T) = D (T^{'})$ and $(T^{'})^{*} y = U T U^{*} y = T^{'} y$ : $D ((T^{'})^{*}) \subseteq D (T^{'})$ and $T^{'}$ is self-adjoint. Applying this with $U = F_{2}^{*}$ and the real multiplier $T = M_{2 π ℏ ξ}$ (self-adjoint by [F4]) makes $P$ self-adjoint with the stated domain, and [F4] identifies $D (P) = W^{1, 2}$ and $P f = - i ℏ f^{'}$ there. The position operator $Q = M_{x}$ is self-adjoint by [F4]. For the basis vectors, $Q ψ_{n} (x) = x ℓ^{- 1 / 2} g_{n} (x / ℓ) = ℓ D (y g_{n}) (x) = \frac{ℓ}{\sqrt{2}} D (\sqrt{n} g_{n - 1} + \sqrt{n + 1} g_{n + 1}) (x)$ by [F3], which is the first displayed formula with $ψ_{- 1} = 0$ ; and $P ψ_{n} = - i ℏ ψ_{n}^{'}$ with $ψ_{n}^{'} = ℓ^{- 1} D (g_{n}^{'}) = \frac{ℓ^{- 1}}{\sqrt{2}} D (\sqrt{n} g_{n - 1} - \sqrt{n + 1} g_{n + 1})$ , giving the second formula.

2.1step 1.2F5

[(b) group and strong continuity] $U (t + s) f = \sum_{n} e^{- i ω (n + 1 / 2) (t + s)} c_{n} ψ_{n} = U (t) U (s) f$ first on every finite linear combination of the basis vectors by scalar multiplication of their phases, and then on all $f$ by density and continuity of the bounded operators $U (t + s)$ and $U (t) U (s)$ from step 1.2; $U (0) = I$ . For continuity at $s$ : $∥ U (t) f - U (s) f ∥^{2} = \sum_{n} ∣ e^{- i ω (n + 1 / 2) t} - e^{- i ω (n + 1 / 2) s} ∣^{2} ∣ c_{n} ∣^{2}$ , and for $t \to s$ the summands converge to $0$ pointwise in $n$ while being bounded by $4 ∣ c_{n} ∣^{2}$ , so the sum tends to $0$ (the tail $\sum_{n > N} 4 ∣ c_{n} ∣^{2}$ is arbitrarily small and finitely many summands converge).

2.2step 1.2F1

[(c) evolution of coherent vectors] By the definition of $ψ_{z}$ in the $(ψ_{n})$ -basis, $U (t) ψ_{z} = e^{- ∣ z ∣^{2} / 2} \sum_{n} \frac{z^{n}}{\sqrt{n!}} e^{- i ω (n + 1 / 2) t} ψ_{n} = e^{- i ω t / 2} \cdot e^{- ∣ z ∣^{2} / 2} \sum_{n} \frac{(z e^{- i ω t})^{n}}{\sqrt{n!}} ψ_{n} = e^{- i ω t / 2} ψ_{z e^{- i ω t}}$ , the series converging by step 1.2 and the definition of $ψ_{w}$ .

2.3step 1.3F2F6algebra

[(d),(e) action on coherent vectors and the moments] Since $ψ_{z} \in D (N_{ψ}) \subseteq D (a_{ψ}^{†}) = D (a_{ψ}) \cap D (a_{ψ}^{†})$ and $a_{ψ}, a_{ψ}^{†}$ are (transports of) the closed operators of [F2], the operators $\frac{ℓ}{\sqrt{2}} (a_{ψ} + a_{ψ}^{†})$ and $\frac{ℏ}{i ℓ \sqrt{2}} (a_{ψ} - a_{ψ}^{†})$ agree with $Q$ and $P$ on every $ψ_{n}$ by step 1.3, and $Q, P$ are closed; the partial sums $ψ_{z}^{(N)} = e^{- ∣ z ∣^{2} / 2} \sum_{n \leq N} \frac{z^{n}}{\sqrt{n!}} ψ_{n}$ converge to $ψ_{z}$ and their $Q$ -images converge: $∥ (a_{ψ} + a_{ψ}^{†}) (ψ_{z}^{(N)} - ψ_{z}) ∥^{2} \leq 2 (∥ a_{ψ} (\cdot) ∥^{2} + ∥ a_{ψ}^{†} (\cdot) ∥^{2}) \leq 4 \sum_{n > N} (n + 1) ∣ c_{n} ∣^{2} \to 0$ , and likewise for $a_{ψ} - a_{ψ}^{†}$ ; hence the displayed identities for $Q ψ_{z}$ and $P ψ_{z}$ hold, and $Q ψ_{z}, P ψ_{z} \in L^{2}$ . Therefore $⟨ Q ⟩_{z} = \frac{ℓ}{\sqrt{2}} (z + \overline{z}) = \sqrt{2} ℓ Re z$ and $⟨ P ⟩_{z} = \frac{ℏ}{i ℓ \sqrt{2}} (z - \overline{z}) = \frac{\sqrt{2} ℏ}{ℓ} Im z$ . For the variances, $Var (Q)_{z} = ∥ Q ψ_{z} ∥^{2} - ⟨ Q ⟩_{z}^{2}$ because the mean is real, and $∥ Q ψ_{z} ∥^{2} = \frac{ℓ^{2}}{2} ∥ (a_{ψ} + a_{ψ}^{†}) ψ_{z} ∥^{2} = \frac{ℓ^{2}}{2} (∥ a_{ψ} ψ_{z} ∥^{2} + ∥ a_{ψ}^{†} ψ_{z} ∥^{2} + 2 Re ⟨ a_{ψ} ψ_{z}, a_{ψ}^{†} ψ_{z} ⟩)$ . Now $⟨ a_{ψ} ψ_{z}, a_{ψ}^{†} ψ_{z} ⟩ = ⟨ z ψ_{z}, a_{ψ}^{†} ψ_{z} ⟩ = z \overline{⟨ a_{ψ}^{†} ψ_{z}, ψ_{z} ⟩} = z^{2}$ because $⟨ a_{ψ}^{†} ψ_{z}, ψ_{z} ⟩ = ⟨ ψ_{z}, a_{ψ} ψ_{z} ⟩ = \overline{z}$ ; and $∥ a_{ψ} ψ_{z} ∥^{2} = ∣ z ∣^{2}$ , $∥ a_{ψ}^{†} ψ_{z} ∥^{2} = ∣ z ∣^{2} + 1$ by [F2]. So $∥ Q ψ_{z} ∥^{2} = \frac{ℓ^{2}}{2} (2 ∣ z ∣^{2} + 1 + 2 Re z^{2}) = \frac{ℓ^{2}}{2} ((z + \overline{z})^{2} + 1)$ and $Var (Q)_{z} = \frac{ℓ^{2}}{2}$ . The same computation with a minus sign gives $∥ P ψ_{z} ∥^{2} = \frac{ℏ^{2}}{2 ℓ^{2}} (2 ∣ z ∣^{2} + 1 - 2 Re z^{2})$ and $Var (P)_{z} = \frac{ℏ^{2}}{2 ℓ^{2}}$ because $⟨ P ⟩_{z}^{2} = \frac{2 ℏ^{2}}{ℓ^{2}} (Im z)^{2} = \frac{ℏ^{2}}{2 ℓ^{2}} (2 ∣ z ∣^{2} - 2 Re z^{2})$ ; the product is $\frac{ℓ^{2}}{2} \cdot \frac{ℏ^{2}}{2 ℓ^{2}} = \frac{ℏ^{2}}{4}$ . No minimality among all states is asserted; the displayed values are those computed for the coherent vectors.

3.1step 1.1step 1.2step 2.1F5

[(b) generator and its exact domain] Let $f \in D (H_{o s c})$ and $g_{t} : = \frac{1}{t} (U (t) f - f) + \frac{i}{ℏ} H_{o s c} f = \sum_{n} ε_{n} (t) c_{n} ψ_{n}$ with $ε_{n} (t) = \frac{e^{- i ω (n + 1 / 2) t} - 1}{t} + i ω (n + \frac{1}{2})$ . By [F5], $∣ ε_{n} (t) ∣ \leq 2 ω (n + \frac{1}{2})$ , so $\sum_{n} ∣ ε_{n} (t) ∣^{2} ∣ c_{n} ∣^{2} \leq 4 ω^{2} \sum_{n} (n + \frac{1}{2})^{2} ∣ c_{n} ∣^{2} < \infty$ by step 1.1, and $ε_{n} (t) \to 0$ as $t \to 0$ for each $n$ by $e^{i x} = 1 + i x + o (x)$ ; the same dominated-convergence argument as in step 2.1 gives $∥ g_{t} ∥ \to 0$ . Hence $t \mapsto U (t) f$ is differentiable at $0$ with derivative $- \frac{i}{ℏ} H_{o s c} f$ ; the general $t$ follows from the group law and $U (t) f \in D (H_{o s c})$ (the coefficients $e^{- i ω (n + 1 / 2) t} c_{n}$ have the same moduli as $c_{n}$ ). Conversely, if $\frac{1}{t} (U (t) f - f) \to g$ in $L^{2}$ , then for every $n$ , $⟨ \frac{1}{t} (U (t) f - f), ψ_{n} ⟩ = \frac{e^{- i ω (n + 1 / 2) t} - 1}{t} c_{n} \to - i ω (n + \frac{1}{2}) c_{n}$ while these pairings converge to $⟨ g, ψ_{n} ⟩$ ; hence $⟨ g, ψ_{n} ⟩ = - i ω (n + \frac{1}{2}) c_{n}$ for every $n$ and $\sum_{n} (n + \frac{1}{2})^{2} ∣ c_{n} ∣^{2} = ω^{- 2} \sum_{n} ∣ ⟨ g, ψ_{n} ⟩ ∣^{2} = ω^{- 2} ∥ g ∥^{2} < \infty$ by [F5], so $f \in D (H_{o s c})$ and $g = - \frac{i}{ℏ} H_{o s c} f$ .

4.1step 2.2step 2.3∎

[(f) evolved moments] By step 2.2 and the definitions, $⟨ Q ⟩_{z (t)} = ⟨ Q (e^{- i ω t / 2} ψ_{z (t)}), e^{- i ω t / 2} ψ_{z (t)} ⟩ = ⟨ Q ψ_{z (t)}, ψ_{z (t)} ⟩ = \sqrt{2} ℓ Re z (t)$ , and similarly $⟨ P ⟩_{z (t)} = \frac{\sqrt{2} ℏ}{ℓ} Im z (t)$ ; the variances in step 2.3 depend on $z$ only through the value of the displayed constant formulas, which are independent of $z$ , so they are unchanged: $Var (Q)_{z (t)} = ℓ^{2} / 2$ and $Var (P)_{z (t)} = ℏ^{2} / (2 ℓ^{2})$ .

TheoremStatement: AI-adaptedProof: AI-adaptedOpen item page →

Multidimensional oscillator completeness

Statement

Assume the Axiom of Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )) and let $d \geq 1$ . For each $j \in {1, \dots, d}$ put $K_{j}^{y} : = L^{2} (R, d y_{j})$ and take a copy $(g_{n}^{(j)})_{n \in N}$ of the Hermite family of Hermite recurrence in $K_{j}^{y}$ . Define the left-associated completed tensor products by $K_{1}^{y} : = K_{1}^{y}$ and $K_{r}^{y} : = K_{r - 1}^{y} \hat{\otimes} K_{r}^{y}$ for $2 \leq r \leq d$ . Let $Θ_{y}^{(1)} : K_{1}^{y} \to L^{2} (R, d y_{1})$ be the identity; recursively, for $2 \leq r \leq d$ , let

$Θ_{y}^{(r)} : = Θ_{r - 1, 1}^{y} \circ (Θ_{y}^{(r - 1)} \otimes I_{K_{r}^{y}}) : K_{r}^{y} ⟶ L^{2} (R^{r}, d y),$

where $Θ_{r - 1, 1}^{y} : L^{2} (R^{r - 1}, d y) \hat{\otimes} K_{r}^{y} \to L^{2} (R^{r}, d y)$ is the two-factor Euclidean product unitary of Euclidean L² product unitary: L²(R^a)⊗̂L²(R^b) ≅ L²(R^{a+b}), and $Θ_{y}^{(r - 1)} \otimes I_{K_{r}^{y}}$ is the bounded tensor map of Bounded tensor maps: extension, norm estimate, adjoint and commutation. Thus $Θ_{y}^{(r)}$ is a specified unitary, with $Θ_{y}^{(r)} (u \otimes v) (y^{'}, y_{r}) = Θ_{y}^{(r - 1)} u (y^{'}) v (y_{r})$ almost everywhere. For $N = (n_{1}, \dots, n_{d}) \in N^{d}$ , set $G_{N}^{y, 1} : = g_{n_{1}}^{(1)}$ and recursively $G_{N}^{y, r} : = G_{N}^{y, r - 1} \otimes g_{n_{r}}^{(r)}$ ; define $g_{N} : = Θ_{y}^{(d)} G_{N}^{y, d} \in L^{2} (R^{d}, d y)$ .

Let $φ : N \to N^{d}$ list the multi-indices by increasing total degree $M = \sum_{j} n_{j}$ and, within each finite level, in lexicographic order. By For $m \geq 1$ the number of weak compositions of $n$ into $m$ parts is $(\binom{n + m - 1}{m - 1})$ , and the number of compositions is $(\binom{n - 1}{m - 1})$ for $n \geq 1$ each level is finite and the levels exhaust $N^{d}$ , so this specifies a bijection. All sums indexed by $N^{d}$ below are taken in this order; nonnegative sums equivalently mean the supremum of finite subsums.

The functions $g_{N}$ have smooth product representatives from the recursive construction. These representatives are unique: a continuous difference that is nonzero at a point remains nonzero on a ball of positive Lebesgue measure. The coordinate derivatives in (b) and (c) are taken on these representatives and their scaled products.

Then:

(a) $(g_{N})_{N \in N^{d}}$ is a complete orthonormal family of $L^{2} (R^{d}, d y)$ : $⟨ g_{N}, g_{M} ⟩ = δ_{N M}$ and every $f \in L^{2} (R^{d}, d y)$ has $f = \sum_{N \in N^{d}} ⟨ f, g_{N} ⟩ g_{N}$ with $∥ f ∥^{2} = \sum_{N \in N^{d}} ∣ ⟨ f, g_{N} ⟩ ∣^{2}$ , the sum being the supremum over finite subfamilies (Orthonormal families, complete orthonormal systems and Hilbert bases, Parseval equivalences for an orthonormal family);

(b) with the eigenvalues $λ_{N} : = \sum_{j = 1}^{d} (2 n_{j} + 1) = 2 M + d$ , $M : = \sum_{j} n_{j}$ , let $A_{d}$ be the diagonal operator of Diagonal adjoint and graph-core lemma for the enumerated family $e_{m} : = g_{φ (m)}$ and eigenvalue sequence $λ_{φ (m)}$ . Then $A_{d}$ is self-adjoint with

$A_{d} g_{N} = λ_{N} g_{N}, D (A_{d}) = {f : \sum_{N \in N^{d}} λ_{N}^{2} ∣ ⟨ f, g_{N} ⟩ ∣^{2} < \infty}, σ (A_{d}) = {2 M + d : M \in N}$

(the spectrum by Diagonal spectrum); the algebraic span of the $g_{N}$ is a core, and on that span

$A_{d} = \sum_{j = 1}^{d} (- \partial_{y_{j}}^{2} + y_{j}^{2})$

as operators, the $j$ -th summand acting in the $j$ -th coordinate;

(c) for real $m_{j}, ω_{j}, ℏ > 0$ put $ℓ_{j} : = \sqrt{ℏ / (m_{j} ω_{j})}$ and let $D_{j} : K_{j}^{y} \to K_{j}^{x} : = L^{2} (R, d x_{j})$ be the one-dimensional unitary of Oscillator scaling and exact realization. Define $K_{1}^{x} : = K_{1}^{x}$ and $K_{r}^{x} : = K_{r - 1}^{x} \hat{\otimes} K_{r}^{x}$ for $2 \leq r \leq d$ , and define $Θ_{x}^{(1)}$ and $Θ_{x}^{(r)}$ by the same recursion as $Θ_{y}^{(r)}$ , using the two-factor Euclidean product unitaries on the $x$ -coordinate spaces. Let $D_{1} : = D_{1}$ and recursively $D_{r} : = D_{r - 1} \otimes D_{r} : K_{r}^{y} \to K_{r}^{x}$ ; each $D_{r}$ is unitary by Bounded tensor maps: extension, norm estimate, adjoint and commutation. Put $D_{d} : = Θ_{x}^{(d)} D_{d} (Θ_{y}^{(d)})^{- 1} : L^{2} (R^{d}, d y) \to L^{2} (R^{d}, d x)$ . For $N \in N^{d}$ , define $G_{N}^{x, 1} : = D_{1} g_{n_{1}}^{(1)} = : ψ_{n_{1}}^{(1)}$ and recursively $G_{N}^{x, r} : = G_{N}^{x, r - 1} \otimes ψ_{n_{r}}^{(r)}$ ; then $ψ_{N} : = D_{d} g_{N} = Θ_{x}^{(d)} G_{N}^{x, d}$ . Let $H$ be the diagonal operator of Diagonal adjoint and graph-core lemma for the enumerated family $ψ_{φ (m)}$ and eigenvalue sequence $E_{φ (m)}$ , where $E_{N} : = ℏ \sum_{j = 1}^{d} ω_{j} (n_{j} + \frac{1}{2})$ . Then $H$ is self-adjoint with

$H ψ_{N} = E_{N} ψ_{N}, D (H) = {f : \sum_{N} E_{N}^{2} ∣ ⟨ f, ψ_{N} ⟩ ∣^{2} < \infty}, σ (H) = \overline{{E_{N} : N \in N^{d}}},$

the span of the $ψ_{N}$ is a core, and on that core

$H = \sum_{j = 1}^{d} (- \frac{ℏ^{2}}{2 m_{j}} \partial_{x_{j}}^{2} + \frac{m_{j} ω_{j}^{2}}{2} x_{j}^{2});$

(d) if $ω_{1} = \dots = ω_{d} = ω$ , then $σ (H) = {ℏ ω (M + \frac{d}{2}) : M \in N}$ , and the eigenvalue $ℏ ω (M + \frac{d}{2})$ has multiplicity

$∣ {(n_{1}, \dots, n_{d}) \in N^{d} : n_{1} + \dots + n_{d} = M} ∣ = (\binom{M + d - 1}{d - 1});$

in particular for $d = 3$ the multiplicity is $(M + 1) (M + 2) / 2$ (For $m \geq 1$ the number of weak compositions of $n$ into $m$ parts is $(\binom{n + m - 1}{m - 1})$ , and the number of compositions is $(\binom{n - 1}{m - 1})$ for $n \geq 1$ ). All parameters are positive real numbers; no physical interpretation is asserted.

Facts & Assumptions

Given: the dimension $d \geq 1$ ; per-coordinate Hermite families $(g_{n}^{(j)})$ ; the recursively defined left-associated tensor products and product unitaries $Θ_{y}^{(r)}, Θ_{x}^{(r)}$ ; the eigenvalues $λ_{N} = 2 M + d$ and $E_{N} = ℏ \sum_{j} ω_{j} (n_{j} + \frac{1}{2})$ .

[F1]

Each $(g_{n}^{(j)})$ is a complete orthonormal family of $L^{2} (R, d y_{j})$ (Hermite completeness), and $(- \partial_{y_{j}}^{2} + y_{j}^{2}) g_{n}^{(j)} = (2 n + 1) g_{n}^{(j)}$ (Hermite orthogonality(b)).

[F2]

If $(e_{a})_{a \in A}$ and $(f_{b})_{b \in B}$ are orthonormal families whose closed linear spans are $H$ and $K$ , then $(κ (e_{a} \otimes f_{b}))$ is a complete orthonormal family of the completed tensor product; the two-factor Euclidean maps $Θ_{a, b}$ are surjective linear isometries with $Θ_{a, b} (u \otimes v) (x, y) = u (x) v (y)$ a.e. (Hilbert tensor construction: inner product, completion and product bases(c),(d), Euclidean L² product unitary: L²(R^a)⊗̂L²(R^b) ≅ L²(R^{a+b})).

[F3]

The diagonal lemma and diagonal spectrum for a complete orthonormal family and real eigenvalues $λ_{N}$ (Diagonal adjoint and graph-core lemma, Diagonal spectrum).

[F4]

The one-dimensional scaling operator $D_{j}$ is a bijective isometry with $D_{j} g_{n}^{(j)} = ψ_{n}^{(j)}$ , and $H_{o s c}^{(j)} : = (ℏ ω_{j} / 2) D_{j} A_{j} D_{j}^{- 1}$ satisfies $H_{o s c}^{(j)} ψ_{n}^{(j)} = ℏ ω_{j} (n + \frac{1}{2}) ψ_{n}^{(j)}$ and $H_{o s c}^{(j)} f = - \frac{ℏ^{2}}{2 m_{j}} f^{''} + \frac{m_{j} ω_{j}^{2}}{2} x^{2} f$ for $f$ in the span of the $ψ_{n}^{(j)}$ (Oscillator scaling and exact realization).

[F5]

For each $M \in N$ and $d \geq 1$ the number of weak compositions of $M$ into $d$ parts is $(\binom{M + d - 1}{d - 1})$ ; in particular each level ${N : \sum_{j} n_{j} = M}$ is finite and $⋃_{M} {N : \sum_{j} n_{j} = M} = N^{d}$ (For $m \geq 1$ the number of weak compositions of $n$ into $m$ parts is $(\binom{n + m - 1}{m - 1})$ , and the number of compositions is $(\binom{n - 1}{m - 1})$ for $n \geq 1$ ).

[F6]

If $S$ and $T$ are unitary maps between Hilbert spaces, then their bounded tensor map $S \otimes T$ is unitary on the completed tensor products (Bounded tensor maps: extension, norm estimate, adjoint and commutation).

Proof

1.1givenF1F2F3

[(a) product basis] For $d = 1$ , $Θ_{y}^{(1)}$ is the identity and the claim is the one-dimensional Hermite basis theorem. Inductively, the product-basis conclusion of [F2] applied to $K_{r - 1}^{y}$ and $K_{r}^{y}$ makes $(G_{N}^{y, r})_{(n_{1}, \dots, n_{r}) \in N^{r}}$ , indexed only by the first $r$ coordinates, a complete orthonormal family in $K_{r}^{y}$ ; the unitary $Θ_{y}^{(r)}$ transports it to the product family in $L^{2} (R^{r}, d y)$ . Thus $(g_{N})$ is complete orthonormal. Applying [F3] gives the stated expansion and Parseval identity.

1.2givenF3F5

[(b) dimensionless operator] By [F5] each level ${N : \sum_{j} n_{j} = M}$ is finite and the levels exhaust $N^{d}$ . Order each level lexicographically and concatenate these finite lists in increasing $M$ ; this gives a bijection $φ : N \to N^{d}$ . Apply [F3] to $e_{m} : = g_{φ (m)}$ and $λ_{m} : = λ_{φ (m)}$ : the resulting operator $A_{d}$ is self-adjoint with the stated domain and $A_{d} g_{N} = λ_{N} g_{N}$ , and its spectrum is the closure of ${λ_{N} : N \in N^{d}} = {2 M + d : M \in N}$ , which is closed because it is a discrete set of reals and its only accumulation point would be $+ \infty$ ; by the diagonal lemma the algebraic span of the $g_{N}$ is a core. This is (b), except the differential expression, proved in the next step.

2.1step 1.2F1

[(b) differential expression] Each one-dimensional Hermite class has its unique polynomial-Gaussian smooth representative by Hermite recurrence, and the recursive product maps represent $g_{N}$ almost everywhere by the smooth product $\prod_{j} g_{n_{j}}^{(j)} (y_{j})$ . Thus its coordinate derivatives are defined pointwise. For each $j$ , $- \partial_{y_{j}}^{2}$ and multiplication by $y_{j}^{2}$ act only on the $j$ -th factor: for $g_{N} = \prod_{j} g_{n_{j}}^{(j)}$ one has $(- \partial_{y_{j}}^{2} + y_{j}^{2}) g_{N} = ((- \partial_{y_{j}}^{2} + y_{j}^{2}) g_{n_{j}}^{(j)}) \prod_{k \neq j} g_{n_{k}}^{(k)} = (2 n_{j} + 1) g_{N}$ by [F1]. Summing over $j$ gives $\sum_{j} (- \partial_{y_{j}}^{2} + y_{j}^{2}) g_{N} = λ_{N} g_{N} = A_{d} g_{N}$ ; both sides are linear, so the identity extends to the span, which is the core of step 1.2.

2.2step 1.1F3F4F6

[(c) anisotropic operator] By induction and [F6], every $D_{r} = D_{r - 1} \otimes D_{r}$ is unitary; hence $D_{d} = Θ_{x}^{(d)} D_{d} (Θ_{y}^{(d)})^{- 1}$ is a composition of unitaries. The identity of the two-factor maps on elementary tensors gives $ψ_{N} = D_{d} g_{N} = Θ_{x}^{(d)} G_{N}^{x, d}$ , so $(ψ_{N})$ is a complete orthonormal family by step 1.1 and unitarity. Apply [F3] to $(ψ_{N})$ and $E_{N}$ : $H$ is self-adjoint with the displayed domain and spectrum, $H ψ_{N} = E_{N} ψ_{N}$ , and the span of the $ψ_{N}$ is a core. Each $ψ_{N}$ has the smooth product representative $\prod_{j} ψ_{n_{j}}^{(j)} (x_{j})$ because the Hermite functions are polynomial-Gaussian and the one-dimensional scaling is explicit. Differentiating this product in coordinate $x_{j}$ and applying the one-dimensional identity [F4] to its $j$ -th factor gives $(- \frac{ℏ^{2}}{2 m_{j}} \partial_{x_{j}}^{2} + \frac{m_{j} ω_{j}^{2}}{2} x_{j}^{2}) ψ_{N} = ℏ ω_{j} (n_{j} + \frac{1}{2}) ψ_{N}$ . Summing over $j$ gives $\sum_{j} (- \frac{ℏ^{2}}{2 m_{j}} \partial_{x_{j}}^{2} + \frac{m_{j} ω_{j}^{2}}{2} x_{j}^{2}) ψ_{N} = E_{N} ψ_{N} = H ψ_{N}$ , and linearity extends the identity to the core.

3.1step 2.2F3F5∎

[(d) isotropic multiplicities] Set $ω_{j} = ω$ . Then $E_{N} = ℏ ω (M + \frac{d}{2})$ with $M = \sum_{j} n_{j}$ , so ${E_{N}} = {ℏ ω (M + \frac{d}{2}) : M \in N}$ , a closed discrete set; by step 2.2 the spectrum is its closure, which is itself. Since $(ψ_{N})$ is an orthonormal family and $H ψ_{N} = E_{N} ψ_{N}$ , the eigenspace for $ℏ ω (M + \frac{d}{2})$ contains the span of all $ψ_{N}$ with $\sum_{j} n_{j} = M$ ; conversely, any element of $D (H)$ orthogonal to that span has all its nonzero coefficients on $ψ_{N}$ with $\sum_{j} n_{j} \neq M$ , so by the diagonal form $H f = \sum_{N} E_{N} ⟨ f, ψ_{N} ⟩ ψ_{N}$ of [F3] it lies in $\ker (H - ℏ ω (M + \frac{d}{2}) I)$ only if those coefficients vanish. Hence the multiplicity is the number of $N \in N^{d}$ with $\sum_{j} n_{j} = M$ , which is $(\binom{M + d - 1}{d - 1})$ by [F5]; for $d = 3$ this reads $(\binom{M + 2}{2}) = (M + 1) (M + 2) / 2$ .

5 · Examples, experiments and counterexamples

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Qm Math Oscillator

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

Diagonal adjoint and graph-core lemma

Statement

Facts & Assumptions

Proof

Diagonal spectrum

Statement

Facts & Assumptions

Proof

Hermite recurrence

Statement

Facts & Assumptions

Proof

Hermite orthogonality

Statement

Facts & Assumptions

Proof

Hermite completeness

Statement

Facts & Assumptions

Proof

Oscillator scaling and exact realization

Statement

Facts & Assumptions

Proof

Closed ladder domains

Statement

Facts & Assumptions

Proof

Coherent-state convergence

Statement

Facts & Assumptions

Proof

Coherent evolution and variance

Statement

Facts & Assumptions

Proof

Multidimensional oscillator completeness

Statement

Facts & Assumptions

Proof

5 · Examples, experiments and counterexamples

Sources