Physics theoremStatement: AI-adaptedProof: AI-adaptedprecheck pass

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Interval models: discrete spectra, complete eigenbases, min-max bounds and nodal counts

Physical scope

The declared confined interval models of the pair with Dirichlet and separated Robin boundary conditions and a real continuous potential on the interval: self-adjointness of the assigned energy operators, discrete spectra with complete orthonormal eigenbases, the min-max comparison of the Dirichlet energies with the free box energies, simplicity of the eigenvalues, the nodal counts of the Dirichlet eigenfunctions, and the Robin eigenvalue condition. The statement is conditional on P1, P2, P4, P5, the declared boundary conditions and the cited mathematics; it asserts no laboratory realisation of a box or wall, no measured spectrum and no empirical result.

Statement

Adopt the state and composition postulate P1, the observable and probability postulate P2, the autonomous-dynamics postulate P4 and the kinematics and massive-particle model postulate P5 (State and composition postulate P1: Hilbert spaces, preparations and composite systems, Observable and probability postulate P2: self-adjoint observables, spectral probability and finite unsharp outcomes, Autonomous dynamics postulate P4: self-adjoint generators and unitary evolution, Kinematics and massive-particle model postulate P5: Newtonian frames and one-particle operators), read through the declared interval models The declared interval models with Dirichlet and Robin boundary conditions, and the declared zero-range (delta) model with $0 < L < \infty$ , $m > 0$ , $ℏ > 0$ , and assume the Axiom of Countable Choice and the Axiom of Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ ), The Axiom of Choice). For a real $V \in C ([0, L]; R)$ write $∥ V ∥_{\infty} : = \max_{x \in [0, L]} ∣ V (x) ∣$ and put $\tilde{V} : = (2 m / ℏ^{2}) V$ , so that the assigned energy operators of the definition satisfy $H_{D, V} = (ℏ^{2} / 2 m) (A_{D} + M_{\tilde{V}})$ and $H_{α β, V} = (ℏ^{2} / 2 m) A_{α β, \tilde{V}}$ . Then:

(a) Dirichlet interval model. $(H_{D, V}, J)$ is an assigned sharp observable; $H_{D, V}$ is self-adjoint with compact resolvent and has a complete orthonormal family $(ψ_{n})_{n \geq 1}$ of eigenvectors. Its eigenvalues, listed with multiplicity, form a sequence $E_{1} \leq E_{2} \leq \dots$ with $E_{n} \to \infty$ and $σ (H_{D, V}) = {E_{n} : n \geq 1}$ , and no other spectral points; each eigenspace is one-dimensional, so the list is strict, $E_{1} < E_{2} < \dots$ , and the eigenvector belonging to $E_{n}$ has exactly $n - 1$ zeros in $(0, L)$ . The min-max comparison $∣ E_{n} - \frac{ℏ^{2} π^{2} n^{2}}{2 m L^{2}} ∣ \leq ∥ V ∥_{\infty} (n \geq 1)$ holds. If $V = 0$ , a complete orthonormal family of eigenpairs of $H_{D, 0}$ is $(\frac{ℏ^{2} π^{2} n^{2}}{2 m L^{2}}, e_{n})$ with $e_{n} (x) = \sqrt{2 / L} \sin (n π x / L)$ , $n \geq 1$ , a complete orthonormal family of $L^{2} (0, L; C)$ . Every eigenvector at the displayed energy is a nonzero scalar multiple of the corresponding $e_{n}$ .

(b) Robin interval model. For real $α, β$ (the declared boundary parameters in $m^{- 1}$ ), $(H_{α β, V}, J)$ is an assigned sharp observable; $H_{α β, V}$ is self-adjoint with compact resolvent; it has a complete orthonormal eigenfamily, every eigenspace is one-dimensional, its spectrum consists exactly of its eigenvalues, which can be listed strictly increasingly as $E_{1} < E_{2} < \dots$ with $E_{n} \to \infty$ , and its essential spectrum is empty. With $λ_{n} : = 2 m E_{n} / ℏ^{2}$ and $u_{a} (\cdot, λ)$ the unique $C^{2}$ solution of $- u^{''} + \tilde{V} u = λ u$ with $u (0) = 1$ , $u^{'} (0) = α$ , one has $u_{a} (L, λ_{n}) \neq 0$ and $β = \frac{u_{a}^{'} (L, λ_{n})}{u_{a} (L, λ_{n})} (n \geq 1) .$

(c) Scope. The statement is conditional on P1, P2, P4, P5, on the declared boundary conditions and on the cited mathematics. The boundary conditions and parameters are declared model assumptions; the eigenvalues are read in joules when $V$ and $ℏ$ enter in the units of Dimensions, SI unit conventions and quantity types of the framework. No laboratory wall, box or surface is identified with the boundary conditions, no measured spectrum is asserted, and no completeness of any continuum or scattering object is claimed.

Facts & Assumptions

Given: The postulates P1, P2, P4, P5 and the choice hypotheses; $0 < L < \infty$ , $m > 0$ , $ℏ > 0$ ; real $V \in C ([0, L]; R)$ and $\tilde{V} = (2 m / ℏ^{2}) V$ ; real $α, β$ ; the operators $H_{D, V} = (ℏ^{2} / 2 m) (A_{D} + M_{\tilde{V}})$ and $H_{α β, V} = (ℏ^{2} / 2 m) A_{α β, \tilde{V}}$ of The declared interval models with Dirichlet and Robin boundary conditions, and the declared zero-range (delta) model clauses (b), (c).

[F1]

Dirichlet Laplacian and its bounded perturbations. Let $A_{D}$ be the Dirichlet Laplacian with $D (A_{D}) = {u \in H^{2} (0, L) : u^{*} (0) = u^{*} (L) = 0}$ and $A_{D} u = - D^{2} u$ , and let $\tilde{V} \in C ([0, L]; R)$ . Then $A : = A_{D} + M_{\tilde{V}}$ on $D (A_{D})$ is self-adjoint; $(A - z)^{- 1}$ is compact for every $z \in ρ (A)$ ; $A$ has a complete orthonormal family of eigenvectors; all eigenspaces are finite dimensional; the eigenvalues $E_{1} \leq E_{2} \leq \dots$ , listed with multiplicity, satisfy $E_{n} \to \infty$ and $σ_{e s s} (A) = \emptyset$ ; and with $λ_{n} = (n π / L)^{2}$ one has $∣ E_{n} - λ_{n} ∣ \leq ∥ \tilde{V} ∥_{\infty}$ for every $n$ (Bounded perturbations of the interval Dirichlet Laplacian clauses (a)-(c)).

[F2]

Dirichlet eigenvalues, simplicity and nodes. An $E \in R$ is a Dirichlet eigenvalue of $- D^{2} + \tilde{V}$ when there is a nonzero $w \in H^{2} (0, L)$ with $w^{*} (0) = w^{*} (L) = 0$ and $- w^{''} + \tilde{V} w = E w$ pointwise; the Dirichlet eigenvalues are exactly the numbers $E_{k}$ of the strictly increasing sequence of Dirichlet eigenvalues, simplicity and the nodal theorem on an interval, each with one-dimensional eigenspace $C u (\cdot, E_{k})$ , the solution $u (\cdot, E)$ of $- u^{''} + \tilde{V} u = E u$ , $u (0) = 0$ , $u^{'} (0) = 1$ ; and $u (\cdot, E_{k})$ has exactly $k - 1$ zeros in $(0, L)$ (Dirichlet eigenvalues, simplicity and the nodal theorem on an interval clauses (a), (b), (d)). The eigenvalue equation of the operator $A = A_{D} + M_{\tilde{V}}$ is, by its definition on $D (A_{D})$ , exactly the condition that $w$ be a nonzero Dirichlet eigenfunction: $w \in H^{2} (0, L)$ , $w^{*} (0) = w^{*} (L) = 0$ , $- D^{2} w + \tilde{V} w = E w$ (Identification of the Dirichlet Laplacian with the sine diagonal operator, One-dimensional Sobolev spaces on intervals).

[F3]

Free Dirichlet box. The diagonal operator $A_{D}$ of Diagonal adjoint and graph-core lemma with the sine family $e_{n}$ and eigenvalues $λ_{n} = (n π / L)^{2}$ equals the Dirichlet Laplacian $B = - D^{2}$ on $D (B) = {u \in H^{2} (0, L) : u^{*} (0) = u^{*} (L) = 0}$ ; $A_{D}$ is self-adjoint and $σ (A_{D}) = {λ_{n} : n \geq 1}$ with eigenvector $e_{n}$ for $λ_{n}$ ; $(e_{n})_{n \geq 1}$ is a complete orthonormal family of $L^{2} (0, L; C)$ (Identification of the Dirichlet Laplacian with the sine diagonal operator, Completeness of the Dirichlet sine system on an interval).

[F4]

Robin realisation. For real $α, β$ and $\tilde{V} \in C ([0, L]; R)$ , the operator $A_{α β, \tilde{V}}$ with $D (A_{α β, \tilde{V}}) = {u \in H^{2} (0, L) : u^{' *} (0) = α u^{*} (0), u^{' *} (L) = β u^{*} (L)}$ and $A_{α β, \tilde{V}} u = - D^{2} u + \tilde{V} u$ is self-adjoint with compact resolvent; it has a complete orthonormal family of eigenvectors; every eigenspace is one-dimensional; the spectrum consists exactly of the eigenvalues, which can be listed strictly increasingly $λ_{1} < λ_{2} < \dots$ with $λ_{n} \to \infty$ and $σ_{e s s} = \emptyset$ ; and for an eigenvalue $λ$ the eigenvector is a nonzero multiple of the solution $u_{a} (\cdot, λ)$ of $- u^{''} + \tilde{V} u = λ u$ , $u (0) = 1$ , $u^{'} (0) = α$ , and $β = u_{a}^{'} (L, λ) / u_{a} (L, λ)$ , with $u_{a} (L, λ) \neq 0$ (Green resolvent for separated Robin boundary conditions, Simple complete spectrum of the separated Robin realisation).

[F5]

Compact resolvent and eigenbasis. For a self-adjoint $A$ with a compact resolvent at some real point, a complete orthonormal eigenfamily exists, every eigenspace is finite dimensional, $∣ μ_{n} ∣ \to \infty$ and $σ (A) = {μ_{n}}$ with $σ_{e s s} (A) = \emptyset$ ; moreover every resolvent is compact (Self-adjoint operators with compact resolvent: eigenbasis and discrete spectrum).

[F6]

Positive scaling. For self-adjoint $A$ and $c > 0$ , $c A$ is self-adjoint; $z \in ρ (c A)$ iff $z / c \in ρ (A)$ with $(c A - z)^{- 1} = c^{- 1} (A - z / c)^{- 1}$ ; compactness of resolvents is preserved; the eigenfamily of $A$ is an eigenfamily of $c A$ with eigenvalues scaled by $c$ , and $σ (c A) = c σ (A)$ (Positive scalar multiples of self-adjoint operators: domains, resolvents, spectra and eigenbases clauses (a)-(d)).

[F7]

Assigned observables. An assigned sharp observable is a densely defined self-adjoint operator together with a declared unit; the joule is the framework's energy unit (Assigned sharp observables and the outcome law of the framework clause (a), Dimensions, SI unit conventions and quantity types of the framework clause (d)4).

Proof

technique · direct; transfer the mathematical interval operators to the declared physical energy operators along the positive scaling factor $\hbar^{2}/2m$, identify eigenvalues with the Dirichlet condition, and read the nodal counts and the Robin condition from the suppliers

Given: The data and hypotheses of the statement and [F1]-[F7].

1.1givenF1F6algebra

[Dirichlet operator after scaling] The mathematical operator $A = A_{D} + M_{\tilde{V}}$ is self-adjoint with compact resolvent, complete orthonormal eigenfamily and min-max eigenvalues $E_{n}$ , and $σ_{e s s} (A) = \emptyset$ with $∣ E_{n} - λ_{n} ∣ \leq ∥ \tilde{V} ∥_{\infty}$ by [F1]. Since $H_{D, V} = (ℏ^{2} / 2 m) A$ with $c : = ℏ^{2} / 2 m > 0$ , [F6] gives that $H_{D, V}$ is self-adjoint, that its resolvents are compact, that the same complete orthonormal family consists of eigenvectors with eigenvalues $E_{n} = c E_{n}$ , and that $σ (H_{D, V}) = c σ (A) = {E_{n}}$ with no other spectral points; moreover the eigenvectors of $H_{D, V}$ are exactly those of $A$ by clause (d) of [F6]. The min-max bound gives $∣ E_{n} - c λ_{n} ∣ = c ∣ E_{n} - λ_{n} ∣ \leq c ∥ \tilde{V} ∥_{\infty} = ∥ V ∥_{\infty}$ , since $c ∥ \tilde{V} ∥_{\infty} = (ℏ^{2} / 2 m) (2 m / ℏ^{2}) ∥ V ∥_{\infty} = ∥ V ∥_{\infty}$ .

2.1step 1.1F1F2algebra

[Dirichlet eigenvalues are simple with the nodal counts] By [F2] the eigenvalue condition for $A$ is exactly the Dirichlet eigenvalue condition, so the eigenvalue set of $A$ is the strictly increasing sequence $(E_{k})$ of the nodal theorem, each eigenspace being one-dimensional and the eigenvector for $E_{k}$ having $k - 1$ zeros in $(0, L)$ . Therefore the min-max list $(E_{n})$ of step 1.1 is this strict sequence, the eigenspaces of $H_{D, V}$ are one-dimensional, and the eigenvector belonging to $E_{n} = c E_{n}$ has exactly $n - 1$ zeros in $(0, L)$ .

2.2step 1.1F4F5F6

[Robin model] By [F4] the mathematical operator $A_{α β, \tilde{V}}$ is self-adjoint with compact resolvent, complete orthonormal eigenfamily, one-dimensional eigenspaces, strictly increasing eigenvalue list $λ_{n}$ with $λ_{n} \to \infty$ and no essential spectrum, and eigenvalue condition $β = u_{a}^{'} (L, λ) / u_{a} (L, λ)$ with $u_{a} (L, λ) \neq 0$ . Scaling by $c = ℏ^{2} / 2 m$ via [F6] preserves all these properties with eigenvalues $E_{n} = c λ_{n}$ , strictly increasing with $E_{n} \to \infty$ , and the eigenvalue condition at $λ_{n} = 2 m E_{n} / ℏ^{2}$ is unchanged, giving (b); the compact-resolvent transfer used here is the general statement of [F5].

3.1step 2.1F3F6algebra

[Free box] If $V = 0$ then $\tilde{V} = 0$ and $A = A_{D}$ , so by [F3] the eigenvalues of $A$ are $λ_{n} = (n π / L)^{2}$ with eigenvectors $e_{n}$ ; after scaling by $c = ℏ^{2} / 2 m$ the eigenvalues of $H_{D, 0}$ are $c λ_{n} = ℏ^{2} π^{2} n^{2} / (2 m L^{2})$ with the same complete orthonormal eigenvectors $e_{n}$ , and the nodal count $n - 1$ is read explicitly from the zeros $x = k L / n$ , $k = 1, \dots, n - 1$ , of $e_{n}$ in $(0, L)$ .

4.1step 1.1step 2.2F7∎

[Observable reading and scope] Both $H_{D, V}$ and $H_{α β, V}$ are densely defined, self-adjoint and read their spectral coordinate in joules by [F7], so each is an assigned sharp observable in the sense of clause (a) of the definition of assigned observables; no further physical reading of the eigenvalues is made, and the exclusions of (c) are those stated in clause (e) of The declared interval models with Dirichlet and Robin boundary conditions, and the declared zero-range (delta) model. This completes the proof of (a), (b) and (c).

Remarks

Where the boundary conditions enter. The Dirichlet and Robin conditions are declared data of the interval models; the mathematics of this theorem begins with the operators $A_{D} + M_{\tilde{V}}$ and $A_{α β, \tilde{V}}$ on their displayed domains, so the theorem transfers no property of the line model to the interval models and proves no limiting relation between Dirichlet and Robin conditions.
The min-max bound is a comparison, not a measurement. It bounds the Dirichlet energies of the perturbed box by the free box energies with an additive constant $∥ V ∥_{\infty}$ ; the bound is stated for the energies in joules and involves no data.

Interval models: discrete spectra, complete eigenbases, min-max bounds and nodal counts

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources