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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 $(\binom{M + 2}{2}) = (M + 1) (M + 2) / 2$ .

Multidimensional oscillator completeness

Statement

Facts & Assumptions

Proof

Depends on

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