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Uniform magnetic fields, flux topology and Pauli coupling

1 · Prerequisites

2 · Summary

This page develops three related pieces of magnetic nonrelativistic quantum mechanics under explicit model assumptions: the closed-form treatment of a smooth vector potential that may grow at infinity, the topology of flux on an ideal periodic ring, and the formal spin-half Pauli coupling. The SI minimal-coupling convention is $Π = - i ℏ \nabla - q A$ . In the planar Landau formulas, for $A = (A_{x}, A_{y}) : R^{2} \to R^{2}$ , $B$ denotes the scalar perpendicular component $B_{⊥} : R^{2} \to R$ , $B_{⊥} = \partial_{x} A_{y} - \partial_{y} A_{x}$ . In the three-dimensional Pauli model, for $A : R^{3} \to R^{3}$ , $B : R^{3} \to R^{3}$ is the curl vector $B = \nabla \times A$ .

The mathematical carrier starts with Magnetic quadratic-form data and Closability of smooth magnetic forms with locally integrable potentials. They define the magnetic quadratic form on $C_{c}^{\infty}$ and establish its closability for smooth real $A$ and a real locally integrable scalar potential bounded below. The proof treats the covariant derivatives as symmetric core operators and uses the weak derivative criterion for closability. It does not assume that $A$ is bounded. For the linear Landau gauge the form route is essential: $A_{y} = B x$ is unbounded, so the bounded-coefficient $H^{2}$ realization does not apply.

For a constant perpendicular field, Closed-form diagonalization in Landau gauge constructs the partial Fourier and translation unitary and diagonalizes the closed form into oscillator fibers with an unrestricted $L^{2}$ degeneracy variable. Exact planar Landau spectrum in the linear gauge then identifies the exact infinite-plane spinless spectrum conditionally on the stated form representation input. The infinite multiplicity is exact for this ideal plane model; no finite-area or finite-sample state count is asserted.

Closed-form diagonalization with free longitudinal motion proves that adding a free longitudinal coordinate adds the multiplier $ℏ^{2} k_{z}^{2} / (2 m)$ to each transverse band, including the required weighted-form density and closure argument. Its physical consequence is worked out on the companion page.

The ring section fixes the periodic Sobolev domain in Periodic Sobolev domain for a circle and proves the shifted Fourier spectrum in Spectrum and gauge periodicity of a periodic flux operator. Under the adopted zero-width ring constraint, Aharonov–Bohm ring spectrum and flux periodicity gives the energy levels and periodicity under one flux quantum. The theorem's Remarks fix the loop orientation and circulation convention. The calculation does not model a finite-width ring or an apparatus.

Finally, Unbounded vector potential and effective Pauli model states the SI minimal-coupling model and an optional effective spin-half form. The core algebra in Pauli square identity and the formal g=2 coupling yields the formal coefficient $g = 2$ in that effective model. It is not a measured electron $g$ -factor, and the core identity does not assert a closed-domain identity or a relativistic correction.

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

DefinitionDefinition: AI-adaptedProof: Not applicableverified 2026-10-22 (gpt-6.1-sol)Open item page →

Magnetic quadratic-form data

Definition

Assume Countable Choice and let $d \geq 1$ be an integer. Work in $H = L^{2} (R^{d}; C)$ with $⟨ u, v ⟩ = \int_{R^{d}} u \overline{v}$ , linear in the first variable, using the quotient and complex-function conventions of the linked Euclidean $L^{2}$ definitions. Fix real scalars $m > 0$ , $ℏ > 0$ , and $q$ . Let $A = (A_{1}, \dots, A_{d}) \in C^{\infty} (R^{d}; R^{d})$ be real and let $V \in L_{l o c}^{1} (R^{d}; R)$ satisfy $V \geq - v_{0}$ almost everywhere for some finite $v_{0} \geq 0$ . Put $D = C_{c}^{\infty} (R^{d}; C)$ and, for $u \in D$ , define $Π_{j}^{A} u = - i ℏ \partial_{j} u - q A_{j} u (1 \leq j \leq d) .$ For $u, v \in D$ define the sesquilinear form $q_{A, V} (u, v) = \frac{1}{2 m} \sum_{j = 1}^{d} ⟨ Π_{j}^{A} u, Π_{j}^{A} v ⟩ + \int_{R^{d}} V (x) u (x) \overline{v (x)} d x,$ and write $q_{A, V} [u] = q_{A, V} (u, u)$ . Each integral is finite: the test functions and their magnetic derivatives are bounded with compact support, and $V$ is integrable on compact sets. The form is Hermitian, densely defined, and bounded below by $- v_{0} ∥ u ∥_{2}^{2}$ .

For $ε_{0} > 0$ define its shifted form norm by $∥ u ∥_{q, v_{0}, ε_{0}}^{2} = q_{A, V} [u] + (v_{0} + ε_{0}) ∥ u ∥_{2}^{2} .$ The form is closable when every sequence $(u_{n}) \subset D$ with $∥ u_{n} ∥_{2} \to 0$ and $∥ u_{n} - u_{m} ∥_{q, v_{0}, ε_{0}} \to 0$ as $n, m \to \infty$ also satisfies $∥ u_{n} ∥_{q, v_{0}, ε_{0}} \to 0$ . This condition is independent of the choice of $ε_{0} > 0$ . When it holds, the closed form closure is the completion of $D$ in this norm, identified with its $L^{2}$ limits; the corresponding closed form is denoted ${\overline{q}}_{A, V}$ .

Definition references

Position and momentum operators on Euclidean L2

Complex Lp classes and Euclidean test-function conventions

The space $L^{p} (μ)$ as the quotient by null functions

The Axiom of Countable Choice ( ${A C}_{ω}$ )

DefinitionDefinition: AI-adaptedProof: Not applicableverified 2026-10-22 (gpt-6.1-sol)Open item page →

Periodic Sobolev domain for a circle

Definition

Assume the Axiom of Choice and Countable Choice. Let $S^{1} = R / (2 π Z)$ with angular coordinate $ϕ$ , and identify $L^{2} (S^{1}; C)$ with $L^{2} ((0, 2 π), d ϕ; C)$ ; the omitted endpoint has measure zero. Let $H^{2} (0, 2 π)$ be the interval Sobolev space whose elements and first two weak derivatives are in $L^{2}$ . For $f \in H^{2} (0, 2 π)$ use the continuous endpoint representatives of $f$ and $f^{'}$ , and denote their traces by $f^{*} (0), f^{*} (2 π)$ and $(f^{'})^{*} (0), (f^{'})^{*} (2 π)$ . Define the periodic second-order domain $H_{p e r}^{2} (S^{1}) = {f \in H^{2} (0, 2 π) : f^{*} (0) = f^{*} (2 π), (f^{'})^{*} (0) = (f^{'})^{*} (2 π)} .$ It is the domain used for the periodic realization of the second derivative; it imposes no boundary at the angular seam beyond periodic matching.

Definition references

The Axiom of Choice

The Axiom of Countable Choice ( ${A C}_{ω}$ )

One-dimensional Sobolev spaces on intervals

Endpoint trace bound and decay of H1 functions

LemmaStatement: AI-adaptedProof: AI-generatedprecheck passverified 2026-10-22 (gpt-6.1-sol)Open item page →

Closability of smooth magnetic forms with locally integrable potentials

Statement

Assume Countable Choice and Dependent Choice. Let $d, m, ℏ, A, V, v_{0}$ and $q_{A, V}$ be the magnetic form on the test core, with $m, ℏ > 0$ , $A \in C^{\infty} (R^{d}; R^{d})$ , $V \in L_{l o c}^{1}$ real-valued and $V \geq - v_{0}$ almost everywhere. Then $q_{A, V}$ is closable on $C_{c}^{\infty} (R^{d}; C)$ in the shifted form norm $∥ u ∥_{q, v_{0}, ε_{0}}^{2} = q_{A, V} [u] + (v_{0} + ε_{0}) ∥ u ∥_{2}^{2}$ for every $ε_{0} > 0$ . Its closure is a densely defined closed Hermitian form bounded below by $- v_{0}$ .

Statement references

Magnetic quadratic-form data

The Axiom of Countable Choice ( ${A C}_{ω}$ )

The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -indexed chain

Facts & Assumptions

Given: Countable Choice, Dependent Choice, $d \geq 1$ , $m, ℏ > 0$ , real $q$ , smooth real $A$ , real $V \in L_{l o c}^{1}$ with $V \geq - v_{0}$ , and the core form $q_{A, V}$ .

[A1]

Countable Choice and Dependent Choice are the assumptions of the Hilbert-space, Fubini, and compact-interval integration-by-parts inputs (The Axiom of Countable Choice ( ${A C}_{ω}$ ), The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -indexed chain).

[F1]

The complex $L^{2}$ pairing is a Hilbert inner product, $C_{c}^{\infty}$ is dense in $L^{2}$ , and Cauchy–Schwarz holds. The weak-derivative convention uses this same quotient $L^{2}$ space (Magnetic quadratic-form data, Position and momentum operators on Euclidean L2, Cauchy-Schwarz inequality for $L^{2}$ ).

[F2]

Each $Π_{j}^{A}$ maps the core to itself and is symmetric there. On a compact rectangle this follows from one-variable integration by parts on each coordinate slice, Fubini, and the reality of $A_{j}$ (Integration by parts for absolutely continuous functions, Fubini's theorem for L^1 functions on a sigma-finite product).

[F3]

If $W = V + v_{0}$ , then $W \geq 0$ , $W \in L_{l o c}^{1}$ , and $W^{1 / 2} ϕ \in L^{2}$ for every $ϕ \in C_{c}^{\infty}$ : on a compact support, $∣ ϕ ∣$ is bounded and $\int W ∣ ϕ ∣^{2} < \infty$ (Magnetic quadratic-form data).

Proof

1.1givenF1F3algebra

Fix $ε_{0} > 0$ and put $W = V + v_{0}$ . For every $u \in C_{c}^{\infty}$ , $∥ u ∥_{q, v_{0}, ε_{0}}^{2} = \frac{1}{2 m} \sum_{j = 1}^{d} ∥ Π_{j}^{A} u ∥_{2}^{2} + ∥ W^{1 / 2} u ∥_{2}^{2} + ε_{0} ∥ u ∥_{2}^{2} .$ This identity shows that if $(u_{n})$ is Cauchy in the shifted form norm, then each sequence $(Π_{j}^{A} u_{n})$ and $(W^{1 / 2} u_{n})$ is Cauchy in $L^{2}$ .

2.1givenA1F1F2F3step 1.1

Suppose additionally that $∥ u_{n} ∥_{2} \to 0$ . Let $w_{j}$ be the $L^{2}$ limit of $Π_{j}^{A} u_{n}$ . For any $ϕ \in C_{c}^{\infty}$ , symmetry [F2] and Cauchy–Schwarz give $⟨ w_{j}, ϕ ⟩ = \lim_{n} ⟨ Π_{j}^{A} u_{n}, ϕ ⟩ = \lim_{n} ⟨ u_{n}, Π_{j}^{A} ϕ ⟩ = 0.$ Density in [F1] implies $w_{j} = 0$ . If $w_{0}$ is the $L^{2}$ limit of $W^{1 / 2} u_{n}$ , then [F3] and Cauchy–Schwarz similarly give $⟨ w_{0}, ϕ ⟩ = \lim_{n} ⟨ u_{n}, W^{1 / 2} ϕ ⟩ = 0$ for every test function $ϕ$ , so density gives $w_{0} = 0$ .

3.1A1F1step 1.1step 2.1algebra∎

Every term on the right side of the norm identity in step 1.1 therefore tends to zero, including $ε_{0} ∥ u_{n} ∥_{2}^{2}$ . Thus $∥ u_{n} ∥_{q, v_{0}, ε_{0}} \to 0$ , which is exactly the closability criterion in Magnetic quadratic-form data. The completion is densely embedded in $L^{2}$ because it contains the dense core, and the criterion makes that embedding injective; its form is closed by construction and retains the lower bound $- v_{0}$ .

LemmaStatement: AI-adaptedProof: AI-generatedprecheck passverified 2026-10-22 (gpt-6.1-sol)Open item page →

Closed-form diagonalization in Landau gauge

Statement

Assume Countable Choice and Dependent Choice, $m, ℏ > 0$ , and real $q, B$ with $q B \neq 0$ . On $L^{2} (R^{2}; C)$ close the nonnegative form $q_{B} [u] = \frac{1}{2 m} (∥ - i ℏ \partial_{x} u ∥_{2}^{2} + ∥ (- i ℏ \partial_{y} - q B x) u ∥_{2}^{2}), u \in C_{c}^{\infty} (R^{2}; C) .$ Put $ω_{c} = ∣ q B ∣ / m$ , $E_{n} = ℏ ω_{c} (n + \frac{1}{2})$ , and let $(ψ_{n})_{n \geq 0}$ be the complete normalized eigenfunctions of the one-dimensional oscillator with mass $m$ , frequency $ω_{c}$ , and action constant $ℏ$ in its exact self-adjoint oscillator realization. For Schwartz $u$ define $F_{y} u (x, k) = (2 π)^{- 1 / 2} \int_{R} e^{- i k y} u (x, y) d y$ , and use its unitary $L^{2}$ extension. Put $x_{0} (k) = ℏ k / (q B)$ and $(U u) (ξ, k) = F_{y} u (ξ + x_{0} (k), k)$ . Then $U$ is unitary from $L^{2} (R_{x, y}^{2})$ onto $L^{2} (R_{ξ, k}^{2})$ . Writing $U u (ξ, k) = \sum_{n = 0}^{\infty} f_{n} (k) ψ_{n} (ξ), f_{n} \in L^{2} (R_{k}),$ the form closure domain and value are exactly $D ({\overline{q}}_{B}) = {u : \sum_{n = 0}^{\infty} E_{n} ∥ f_{n} ∥_{L^{2} (k)}^{2} < \infty}, {\overline{q}}_{B} [u] = \sum_{n = 0}^{\infty} E_{n} ∥ f_{n} ∥_{L^{2} (k)}^{2} .$ Thus the form is the diagonal oscillator form with an unrestricted $L^{2} (k)$ degeneracy factor; it asserts no finite-area state count.

Statement references

Magnetic quadratic-form data

The Axiom of Countable Choice ( ${A C}_{ω}$ )

The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -indexed chain

Oscillator scaling and exact realization

Dominated convergence

Facts & Assumptions

Given: Countable Choice and Dependent Choice, $m, ℏ > 0$ , $q B \neq 0$ , the form $q_{B}$ , and the oscillator eigenfamily $(ψ_{n})$ with energies $E_{n}$ .

[F1]

The magnetic form on $C_{c}^{\infty}$ is the form of Magnetic quadratic-form data with $d = 2$ , $A = (0, B x)$ and $V = 0$ .

[F2]

The $2 π$ -frequency Fourier transform on $L^{1}$ has kernel $e^{- 2 π i y ξ}$ (Fourier transform on complex L1 classes), and its extension to $L^{2}$ is unitary (Plancherel theorem). The wave-number convention in the Statement is the unitary rescaling $F_{y} g (k) = (2 π)^{- 1 / 2} \hat{g} (k / (2 π))$ ; for Schwartz $g$ it satisfies $F_{y} (- i \partial_{y} g) = k F_{y} g$ by integration by parts, and Fourier transform preserves Schwartz space (Fourier transform is a topological automorphism of Schwartz space). The Euclidean $L^{2}$ product identification and Fubini apply (Euclidean L² product unitary: L²(R^a)⊗̂L²(R^b) ≅ L²(R^{a+b}), Fubini's theorem for L^1 functions on a sigma-finite product); the integration-by-parts identity used for the derivative multiplier is Integration by parts for absolutely continuous functions.

[F3]

For $ω_{c} > 0$ , let $H_{o s c}$ be the exact self-adjoint oscillator realization in Oscillator scaling and exact realization. It has complete orthonormal basis $(ψ_{n})$ , eigenvalues $E_{n}$ , and domain $D (H_{o s c}) = {g : \sum_{n = 0}^{\infty} E_{n}^{2} ∣ ⟨ g, ψ_{n} ⟩ ∣^{2} < \infty} .$ On its polynomial-Gaussian core it has differential expression $L = - ℏ^{2} \partial_{ξ}^{2} / (2 m) + m ω_{c}^{2} ξ^{2} / 2$ ; in particular, $L ψ_{n} = E_{n} ψ_{n}$ on that core.

[F4]

Translation of one Lebesgue variable preserves its integral, and integration by parts holds for smooth compactly supported functions (A C^1 diffeomorphism satisfies the change-of-variables formula for L^1 functions, Integration by parts for absolutely continuous functions).

[F5]

Smooth compactly supported cutoffs equal to one on expanding balls and with first derivatives bounded by $C / R$ are obtained by rescaling the standard smooth step function (The standard smooth step function).

[F6]

Tonelli's theorem permits integration of the nonnegative oscillator energy series term by term over $(ξ, k)$ with Lebesgue measure (Tonelli's theorem for nonnegative measurable functions on a sigma-finite product).

[F7]

For every finite $p$ and dimension, compactly supported smooth functions are dense in $L^{p} (R^{d})$ ; in particular this holds for $L^{2} (R_{k})$ (Compactly supported smooth functions are dense in W^{k,p}(R^n)).

[F8]

The weighted sequence space in the Statement is complete: it is $L^{2}$ on $N_{0} \times R_{k}$ for the sigma-finite measure $(n, k) \mapsto E_{n} d # (n) d k$ , and $L^{2}$ of any measure space is Hilbert under Countable Choice ( $L^{2}$ with the integral pairing is a Hilbert space).

[F9]

On any measure space, measurable complex-valued functions converging almost everywhere and bounded in modulus by a single nonnegative measurable $L^{1}$ function converge in integral (indeed in $L^{1}$ ). Dominated convergence

[A1]

Countable Choice is assumed by the unitary Fourier, product-space, and oscillator suppliers; Dependent Choice is an exact hypothesis of the absolutely-continuous integration-by-parts supplier (The Axiom of Countable Choice ( ${A C}_{ω}$ ), The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -indexed chain, Integration by parts for absolutely continuous functions).

Proof

1.1givenA1F2F4construct

On the dense tensor core $a (x) b (y)$ with $a, b \in S (R)$ , $F_{y}$ is the identity on the first factor and the unitary Fourier transform on the second, by [F2] and the Euclidean $L^{2}$ product identification in [F2]. It extends uniquely to a unitary on $L^{2} (R^{2})$ . For each $k$ , translation $x = ξ + x_{0} (k)$ preserves the $x$ -integral by [F4]; Fubini then shows that $W g (ξ, k) = g (ξ + x_{0} (k), k)$ is an isometry with inverse $W^{- 1} h (x, k) = h (x - x_{0} (k), k)$ . Hence $U = W F_{y}$ is unitary.

2.1givenA1F2F3F4F6step 1.1algebra

Let $u \in C_{c}^{\infty} (R^{2})$ and write $F (ξ, k) = U u (ξ, k)$ . For each fixed $k$ , $F (\cdot, k)$ is smooth and compactly supported in $ξ$ . Plancherel in $y$ , the derivative identity in [F2], and the substitution $x = ξ + x_{0} (k)$ give $q_{B} [u] = \int_{R^{2}} (\frac{ℏ^{2}}{2 m} ∣ \partial_{ξ} F ∣^{2} + \frac{(q B)^{2}}{2 m} ξ^{2} ∣ F ∣^{2}) d ξ d k .$ For a compactly supported smooth $g (ξ)$ , put $L g = - ℏ^{2} g^{''} / (2 m) + m ω_{c}^{2} ξ^{2} g / 2 \in L^{2}$ . Integrating by parts twice on a finite interval containing the support of $g$ gives $⟨ L g, ψ_{n} ⟩ = E_{n} ⟨ g, ψ_{n} ⟩$ , since $L ψ_{n} = E_{n} ψ_{n}$ and the boundary terms vanish. Completeness and Parseval therefore give $\sum_{n = 0}^{\infty} E_{n}^{2} ∣ ⟨ g, ψ_{n} ⟩ ∣^{2} = ∥ L g ∥_{2}^{2} < \infty .$ The exact domain characterization in [F3] now implies $g \in D (H_{o s c})$ , and its diagonal action gives $H_{o s c} g = L g$ . Hence $⟨ H_{o s c} g, g ⟩ = \sum_{n = 0}^{\infty} E_{n} ∣ ⟨ g, ψ_{n} ⟩ ∣^{2} .$ Integration by parts also identifies the left side with $\frac{1}{2 m} (ℏ^{2} ∥ g^{'} ∥_{2}^{2} + (q B)^{2} ∥ ξ g ∥_{2}^{2}) .$ Apply this identity to $F (\cdot, k)$ for almost every $k$ . Tonelli [F6] permits integrating the nonnegative series, so the displayed core form is the coefficient form on every transformed core vector.

3.1A1F1F2F3F4F5F6F7F9step 1.1step 2.1constructalgebra

For $N < \infty$ and $f_{0}, \dots, f_{N} \in C_{c}^{\infty} (R_{k})$ , $F_{N} (ξ, k) = \sum_{n = 0}^{N} f_{n} (k) ψ_{n} (ξ)$ has inverse $U^{- 1} F_{N} (x, y) = \frac{1}{\sqrt{2 π}} \sum_{n = 0}^{N} \int_{R} e^{i k y} f_{n} (k) ψ_{n} ⁣ (x - \frac{ℏ k}{q B}) d k .$ Write $v : = U^{- 1} F_{N}$ . The integrands have compact $k$ -support and are Schwartz in $x$ uniformly there by the Hermite form in [F3]; differentiating under the integral and integrating by parts in $k$ show rapid decay in $y$ , so the inverse is Schwartz. Define $χ_{R} (x, y) = \prod_{s \in {x, y}} σ (2 - s / R) σ (2 + s / R),$ using [F5]. These cutoffs are one on $[- R, R]^{2}$ , vanish outside $[- 2 R, 2 R]^{2}$ , and have $∣ \nabla χ_{R} ∣ \leq C / R$ . For each covariant derivative, $Π_{j}^{A} (χ_{R} v) = χ_{R} Π_{j}^{A} v - i ℏ (\partial_{j} χ_{R}) v .$ Take integer $R \geq 1$ . The step-function bounds in [F5] give $0 \leq χ_{R} \leq 1$ , and for every fixed $(x, y)$ , $χ_{R} (x, y) = 1$ once $R \geq \max {∣ x ∣, ∣ y ∣}$ , so $χ_{R} \to 1$ pointwise. Since $v$ is Schwartz and $A = (0, B x)$ is linear, $Π_{1}^{A} v = - i ℏ \partial_{x} v$ and $Π_{2}^{A} v = - i ℏ \partial_{y} v - q B x v$ are Schwartz and belong to $L^{2} (R^{2})$ . For each $w \in {v, Π_{1}^{A} v, Π_{2}^{A} v}$ , the functions $∣ (χ_{R} - 1) w ∣^{2}$ are measurable, tend pointwise to zero, and are bounded by the single measurable $L^{1}$ majorant $∣ w ∣^{2}$ . Applying [F9] on the Lebesgue measure space $R^{2}$ gives $∥ (χ_{R} - 1) w ∥_{2} \to 0$ . The derivative-cutoff term separately satisfies $∥ ℏ (\partial_{j} χ_{R}) v ∥_{2} \leq C ℏ ∥ v ∥_{2} / R \to 0$ . Thus the displayed product identity and the triangle inequality give $∥ Π_{j}^{A} (χ_{R} v - v) ∥_{2} \to 0$ for both $j$ , as well as $∥ χ_{R} v - v ∥_{2} \to 0$ . The form identity of step 2.1 also holds for Schwartz functions: the Fourier and shear maps preserve Schwartz regularity, and for each Schwartz $g (ξ)$ one has $L g \in L^{2}$ ; its integration-by-parts pairings with the polynomial-Gaussian $ψ_{n}$ have vanishing boundary terms at both infinities. The same Parseval calculation and exact diagonal operator-domain criterion in [F3] therefore give $g \in D (H_{o s c})$ and the same oscillator energy identity. All resulting nonnegative energies can again be integrated by [F6]. Apply this extended identity to the Schwartz difference $χ_{R} v - v$ . Its weighted coefficient norm squared is exactly $\frac{1}{2 m} \sum_{j = 1}^{2} ∥ Π_{j}^{A} (χ_{R} v - v) ∥_{2}^{2}$ , which tends to zero by the preceding bounds. Since $χ_{R} v \in C_{c}^{\infty}$ , each $F_{N}$ therefore lies in the weighted-norm closure of the transformed original core, without assuming closability in advance. Finite coefficient families are dense in the weighted target: truncate its convergent nonnegative series, then approximate the remaining finitely many $L^{2} (k)$ coefficients by $C_{c}^{\infty} (k)$ using [F7].

4.1F1F2F3F8step 2.1step 3.1algebra∎

Let $K_{E}$ be the coefficient space ${(f_{n}) : \sum_{n} E_{n} ∥ f_{n} ∥_{2}^{2} < \infty}$ with squared norm $\sum_{n} E_{n} ∥ f_{n} ∥_{2}^{2}$ , and let $J : K_{E} ↪ L^{2} (N_{0} \times R, d # d k)$ be the identity inclusion. Step 2.1 maps the transformed core isometrically into $K_{E}$ . Since $E_{n} \geq E_{0} = ℏ ω_{c} / 2 > 0$ , $J$ is continuous and injective, with $∥ J f ∥_{2}^{2} \leq E_{0}^{- 1} \sum_{n} E_{n} ∥ f_{n} ∥_{2}^{2}$ . The coefficient map is unitary on the underlying $L^{2}$ space by [F2] and [F3]. Thus $q_{B} [u] \geq E_{0} ∥ u ∥_{2}^{2}$ on the core. For every $ε > 0$ , $q_{B} [u] \leq q_{B} [u] + ε ∥ u ∥_{2}^{2} \leq (1 + ε / E_{0}) q_{B} [u],$ so each shifted form norm is equivalent to $q_{B} [u]^{1 / 2}$ . Suppose a core sequence is Cauchy in $q_{B} [u]^{1 / 2}$ and tends to zero in $L^{2}$ . Its weighted coefficient image converges in $K_{E}$ to some $f$ ; unweighted $L^{2}$ convergence and continuity of $J$ give $J f = 0$ . Injectivity of $J$ gives $f = 0$ , so $q_{B} [u] \to 0$ along that sequence. The equivalence above then gives convergence to zero in the defining shifted form norm, proving closability of the core form. By [F8] the target is complete, so the isometric image of the form completion is closed there. Step 3.1 places a dense subspace of that target in the image; hence the image is the whole weighted space and the form closure is exactly the stated one.

Remarks

The diagram records the unitary changes of representation used in the proof; the argument establishes each arrow and does not rely on the diagram.

Here $H$ is the coefficient map $F \mapsto (⟨ F (\cdot, k), ψ_{n} ⟩)_{n \geq 0}$ , which is unitary by the complete oscillator basis. The linear Landau potential is unbounded, so no bounded-coefficient $H^{2}$ realization is invoked.

TheoremStatement: AI-adaptedProof: AI-generatedprecheck passverified 2026-10-22 (gpt-6.1-sol)Open item page →

Spectrum and gauge periodicity of a periodic flux operator

Statement

Assume the Axiom of Choice, Countable Choice, and Dependent Choice. Let $κ > 0$ and $α \in R$ . On $H = L^{2} (S^{1}; C) = L^{2} ((0, 2 π), d ϕ; C)$ use the periodic domain $H_{p e r}^{2} (S^{1})$ defined by periodic endpoint traces and define $T_{α} f = κ (- i \partial_{ϕ} - α)^{2} f, D (T_{α}) = H_{p e r}^{2} (S^{1}) .$ Then $T_{α}$ is self-adjoint and the normalized characters $e_{n} (ϕ) = (2 π)^{- 1 / 2} e^{i n ϕ}$ , $n \in Z$ , form a complete orthonormal eigenbasis with eigenvalues $λ_{n} = κ (n - α)^{2}$ . Its spectrum is the closed set ${λ_{n} : n \in Z}$ and its spectral measures are purely atomic on that set. For every $j \in Z$ , multiplication by $e^{i j ϕ}$ unitarily intertwines $T_{α}$ with $T_{α + j}$ . A smooth periodic unit-modulus function $u$ satisfying $u^{- 1} (- i \partial_{ϕ} - α) u = - i \partial_{ϕ}$ exists exactly when $α \in Z$ .

Statement references

Periodic Sobolev domain for a circle

All self-adjoint interval Laplacian boundary conditions

Fourier coefficients and trigonometric polynomials on the torus

The trigonometric system is complete in $L^{2}$ of the torus

Kato-Rellich theorem

Spectral theorem for unbounded self-adjoint operators (PVM form)

Facts & Assumptions

Given: the Choice, Countable Choice, and Dependent Choice axioms; $κ > 0$ , $α \in R$ , the Hilbert space $H$ , and the periodic domain in the Statement.

[A1]

The Choice, Countable Choice, and Dependent Choice hypotheses are those of the interval realization, Fourier-basis, Kato–Rellich, and unbounded spectral suppliers (The Axiom of Choice, The Axiom of Countable Choice ( ${A C}_{ω}$ ), The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -indexed chain).

[F1]

For $f \in H^{2} (0, 2 π)$ , the periodic endpoint traces are those of Periodic Sobolev domain for a circle. The interval Laplacian with the swap matrix $U = (\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array})$ has exactly these periodic boundary conditions and is self-adjoint by All self-adjoint interval Laplacian boundary conditions.

[F2]

The characters $e_{n}^{T} ([t]) = e^{2 π i n t}$ form a complete orthonormal basis of $L^{2} (T, d m_{T})$ , where $T = R / Z$ has normalized Haar probability measure (Fourier coefficients and trigonometric polynomials on the torus, The trigonometric system is complete in $L^{2}$ of the torus). The map $W : L^{2} (T, d m_{T}) \to L^{2} ((0, 2 π), d ϕ), (W g) (ϕ) = (2 π)^{- 1 / 2} g ([ϕ / (2 π)])$ is unitary: in the fundamental-domain coordinate $t \in [0, 1)$ , $∥ W g ∥_{2}^{2} = \frac{1}{2 π} \int_{0}^{2 π} ∣ g ([ϕ / (2 π)]) ∣^{2} d ϕ = \int_{0}^{1} ∣ g ([t]) ∣^{2} d t = ∥ g ∥_{2}^{2},$ and its inverse is $(W^{- 1} h) ([t]) = \sqrt{2 π} h (2 π t)$ almost everywhere. Moreover $W e_{n}^{T} = (2 π)^{- 1 / 2} e^{i n ϕ} = e_{n}$ . Thus the exact characters in the Statement are a complete orthonormal basis for the stated Lebesgue measure convention.

[F3]

If $A$ is self-adjoint and $B$ is symmetric on $D (A)$ with $∥ B f ∥ \leq a ∥ A f ∥ + b ∥ f ∥$ for some $a < 1$ , then $A + B$ on $D (A)$ is self-adjoint (Kato-Rellich theorem, Relative boundedness with respect to an operator).

[F4]

A self-adjoint operator has a unique regular spectral projection-valued measure, and its domain and spectral integrals are given by the unbounded spectral theorem (Spectral theorem for unbounded self-adjoint operators (PVM form)). For any self-adjoint $T$ with a complete orthonormal eigenbasis $(e_{n})_{n \in Z}$ and eigenvalues $λ_{n}$ , this gives spectral projections equal to sums of the rank-one projections onto the $e_{n}$ and $D (T) = {f = \sum_{n \in Z} c_{n} e_{n} : \sum_{n \in Z} ∣ λ_{n} ∣^{2} ∣ c_{n} ∣^{2} < \infty} .$ On this domain, $T f = \sum_{n \in Z} λ_{n} c_{n} e_{n}$ in $L^{2}$ .

[F5]

Integration by parts on the interval holds for the periodic Sobolev functions and their endpoint traces; in particular $P = - i \partial_{ϕ}$ is symmetric on the stated domain (Integration by parts and the endpoint-vanishing space on intervals).

Proof

1.1A1F1F2F5construct

Set $A_{0} = - κ \partial_{ϕ}^{2}$ on $H_{p e r}^{2} (S^{1})$ . In the boundary coordinates $Γ_{0} f = (f (0), f (2 π))$ and $Γ_{1} f = (f^{'} (0), - f^{'} (2 π))$ , the unitary swap matrix in [F1] makes the boundary equation equivalent to $f (0) = f (2 π)$ and $f^{'} (0) = f^{'} (2 π)$ : adding and subtracting its two scalar equations gives these two equalities. Hence $- \partial_{ϕ}^{2}$ is self-adjoint on this domain, and multiplication by the positive real $κ$ shows that $A_{0}$ is self-adjoint. Let $P = - i \partial_{ϕ}$ . Integration by parts [F5] and equality of the endpoint traces show that $P$ is symmetric on $H_{p e r}^{2}$ . For $f = \sum_{n} c_{n} e_{n}$ in this domain, integration by parts gives $⟨ P f, e_{n} ⟩ = n c_{n}$ and $⟨ A_{0} f, e_{n} ⟩ = κ n^{2} c_{n}$ . Parseval for [F2] therefore implies $∥ P f ∥_{2}^{2} = \sum_{n} n^{2} ∣ c_{n} ∣^{2}$ and $∥ A_{0} f ∥_{2}^{2} = κ^{2} \sum_{n} n^{4} ∣ c_{n} ∣^{2}$ .

2.1A1F2F3step 1.1algebra

On $D (A_{0})$ write $T_{α} = A_{0} + B_{α}$ with $B_{α} = - 2 κ α P + κ α^{2} I$ . This operator is symmetric because $P$ is symmetric and $α, κ$ are real. For every $ε > 0$ and integer $n$ , $∣ n ∣ \leq ε n^{2} + (4 ε)^{- 1}$ ; the coefficient identities in step 1.1 and the triangle inequality give $∥ B_{α} f ∥_{2} \leq a ∥ A_{0} f ∥_{2} + b ∥ f ∥_{2}$ with any prescribed $a > 0$ after choosing $ε$ sufficiently small and then a finite $b$ . Applying [F3] with $a < 1$ proves that $T_{α}$ is self-adjoint on the stated periodic domain.

3.1A1F2F4step 2.1algebra

Each $e_{n}$ lies in the periodic domain and direct differentiation gives $T_{α} e_{n} = κ (n - α)^{2} e_{n}$ . Completeness [F2] makes this the full orthonormal eigenbasis. The associated spectral projections are therefore the sums of the character projections whose eigenvalues lie in the chosen Borel set, so every scalar spectral measure is supported on ${κ (n - α)^{2} : n \in Z}$ . This set has no finite accumulation point because $κ > 0$ and $∣ n - α ∣ \to \infty$ . If $z$ is outside it, $\inf_{n} ∣ λ_{n} - z ∣ > 0$ . For $f = \sum_{n} c_{n} e_{n}$ , where $c_{n} = ⟨ f, e_{n} ⟩$ , the diagonal map $R_{z} f = \sum_{n} c_{n} (λ_{n} - z)^{- 1} e_{n}$ is bounded on $H$ . Also $\sup_{n} ∣ λ_{n} / (λ_{n} - z) ∣ < \infty$ , so the domain characterization in [F4] shows that $R_{z}$ maps into $D (T_{α})$ . Coefficientwise, $(T_{α} - z) R_{z} = I$ and $R_{z} (T_{α} - z) = I$ on $D (T_{α})$ , so $z$ is in the resolvent. If $z = λ_{n}$ then $e_{n}$ is a nonzero kernel vector. Hence the spectrum is exactly the displayed set and there is no continuous spectral measure.

4.1givenF2step 3.1algebra∎

Let $j \in Z$ and $U_{j} f = e^{i j ϕ} f$ . It is unitary, preserves the periodic $H^{2}$ domain, and $(P - α - j) U_{j} f = U_{j} (P - α) f$ by differentiation; squaring gives $T_{α + j} U_{j} = U_{j} T_{α}$ . If a smooth periodic unit-modulus function $u$ satisfies the displayed conjugacy, then $(P - α) u = 0$ , or $u^{'} = i α u$ . The solutions are $u (ϕ) = c e^{i α ϕ}$ with $∣ c ∣ = 1$ , which are periodic exactly when $e^{2 π i α} = 1$ , equivalently $α \in Z$ . Conversely, for integral $α$ this multiplier is single-valued and periodic.

Remarks

The diagram expresses the integer-flux unitary intertwining. Its arrow labels are the multiplication operators proved in step 4.1; the proof above is independent of the diagram.

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Closed-form diagonalization with free longitudinal motion

Statement

Assume Countable Choice and Dependent Choice, $m, ℏ > 0$ , and real $q, B$ with $q B \neq 0$ . Let $c = ℏ^{2} / (2 m)$ , $ω_{c} = ∣ q B ∣ / m$ , and $E_{n} = ℏ ω_{c} (n + \frac{1}{2})$ for $n \in N_{0}$ . On $L^{2} (R_{x, y, z}^{3}; C)$ close the nonnegative form $q_{B}^{(3)} [u] = \frac{1}{2 m} (∥ - i ℏ \partial_{x} u ∥_{2}^{2} + ∥ (- i ℏ \partial_{y} - q B x) u ∥_{2}^{2} + ∥ - i ℏ \partial_{z} u ∥_{2}^{2}), u \in C_{c}^{\infty} (R^{3}; C) .$ Use the wave-number Fourier convention $F_{y} g (k_{y}) = (2 π)^{- 1 / 2} \int e^{- i k_{y} y} g (y) d y, F_{z} g (k_{z}) = (2 π)^{- 1 / 2} \int e^{- i k_{z} z} g (z) d z .$ and set $U_{3} u (ξ, k_{y}, k_{z}) = F_{z} ⁣ [F_{y} u ⁣ (ξ + \frac{ℏ k_{y}}{q B}, k_{y}, z)] (k_{z}) .$ Then $U_{3}$ extends to a unitary map onto $L^{2} (R_{k_{y}, k_{z}}^{2}; ℓ^{2} (N_{0}))$ by expansion in the complete transverse oscillator basis $ψ_{n} (ξ)$ . If $U_{3} u (ξ, k_{y}, k_{z}) = \sum_{n = 0}^{\infty} f_{n} (k_{y}, k_{z}) ψ_{n} (ξ),$ the form closure domain and value are exactly $D ({\overline{q}}_{B}^{(3)}) = {u : \sum_{n = 0}^{\infty} \int_{R^{2}} (E_{n} + c k_{z}^{2}) ∣ f_{n} (k_{y}, k_{z}) ∣^{2} d k_{y} d k_{z} < \infty},$ ${\overline{q}}_{B}^{(3)} [u] = \sum_{n = 0}^{\infty} \int_{R^{2}} (E_{n} + c k_{z}^{2}) ∣ f_{n} (k_{y}, k_{z}) ∣^{2} d k_{y} d k_{z} .$

Statement references

Closed-form diagonalization in Landau gauge

The Axiom of Countable Choice ( ${A C}_{ω}$ )

The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -indexed chain

Plancherel theorem

Facts & Assumptions

Given: Countable Choice and Dependent Choice, $m, ℏ > 0$ , real $q, B$ with $q B \neq 0$ , and the three-dimensional core form in the Statement.

[A1]

Countable Choice is the hypothesis of the transverse form, product Fourier, and compactly-supported $L^{2}$ density inputs. The transverse form lemma also assumes Dependent Choice because its integration-by-parts supplier does (The Axiom of Countable Choice ( ${A C}_{ω}$ ), The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -indexed chain, Closed-form diagonalization in Landau gauge).

[F1]

The displayed core form is the $d = 3$ , $A = (0, B x, 0)$ instance of the magnetic form definition; in $d = 2$ the planar core form closes to the oscillator coefficient form with energies $E_{n}$ and complete normalized functions $ψ_{n}$ (Magnetic quadratic-form data, Closed-form diagonalization in Landau gauge).

[F2]

The partial Fourier transforms in $y, z$ are unitary on $L^{2}$ and send $- i \partial_{y}, - i \partial_{z}$ to multiplication by $k_{y}, k_{z}$ on Schwartz functions; the product-space identifications and their iterated use are valid (Fourier transform on complex L1 classes, Plancherel theorem, Fourier transform is a topological automorphism of Schwartz space, Euclidean L² product unitary: L²(R^a)⊗̂L²(R^b) ≅ L²(R^{a+b}), Integration by parts for absolutely continuous functions).

[F3]

Translation of the $x$ variable preserves Lebesgue measure (A C^1 diffeomorphism satisfies the change-of-variables formula for L^1 functions).

[F4]

$C_{c}^{\infty} (R^{2})$ is dense in $L^{2} (R^{2})$ , and smooth cutoffs equal to one on expanding boxes with first derivatives bounded by $C / R$ can be built from the standard smooth step function (Compactly supported smooth functions are dense in W^{k,p}(R^n), The standard smooth step function).

[F5]

Fubini and Tonelli apply to Lebesgue product spaces, including the nonnegative oscillator-energy series (Fubini's theorem for L^1 functions on a sigma-finite product, Tonelli's theorem for nonnegative measurable functions on a sigma-finite product).

[F6]

The weighted coefficient space is complete: identify it with $L^{2}$ on $N_{0} \times R^{2}$ for the measure $(n, k_{y}, k_{z}) \mapsto (E_{n} + c k_{z}^{2}) d # (n) d k_{y} d k_{z}$ . This measure is sigma-finite because the weight is finite on each bounded box and finite index set, and $L^{2}$ on a measure space is Hilbert under Countable Choice ( $L^{2}$ with the integral pairing is a Hilbert space).

[F7]

The oscillator eigenfunctions $(ψ_{n})$ are normalized Hermite functions, hence polynomial times Gaussian and Schwartz (Oscillator scaling and exact realization).

[F8]

Dominated convergence applies to the integrable weighted coefficient tails and to the $L^{2}$ tails of Schwartz functions (Dominated convergence).

Proof

1.1A1F1F2F3construct

First apply the unitary planar map from [F1] at each fixed $z$ , then the partial Fourier transform in $z$ from [F2]. On the dense product core both maps are unitary; the Euclidean $L^{2}$ product identifications in [F2] extend their composition uniquely to a unitary on $L^{2} (R^{3})$ . The inverse uses the inverse Fourier transforms and the opposite translation, which is measure preserving by [F3]. The coefficient map in the transverse variable is unitary because $(ψ_{n})$ is a complete orthonormal basis. Thus $U_{3}$ is unitary onto the stated coefficient space.

1.2A1F4F8constructalgebra

Let $f = (f_{n})$ have finite weighted norm and fix $ε > 0$ . Since the defining nonnegative series converges, choose $N$ so the norm of its tail over $n > N$ is below $ε / 3$ . For the finitely many indices $n \leq N$ , the weighted integrals outside $K_{R} = [- R, R]_{k_{y}} \times [- R, R]_{k_{z}}$ tend to zero as $R \to \infty$ by dominated convergence, so choose one $R$ making their combined tail norm below $ε / 3$ . Define $η_{R} (k_{y}, k_{z}) = \prod_{s \in {k_{y}, k_{z}}} σ (R + 1 - s) σ (R + 1 + s) .$ It equals one on $K_{R}$ , is between zero and one, and has support in $K_{R + 1}$ by [F4]. For each $n \leq N$ , truncate $f_{n}$ to $K_{R}$ and approximate that truncation in unweighted $L^{2}$ by $φ_{n} \in C_{c}^{\infty} (R^{2})$ using [F4]. The function $g_{n} = η_{R} φ_{n}$ is smooth and supported in $K_{R + 1}$ , and on that box $E_{n} + c k_{z}^{2} \leq E_{n} + c (R + 1)^{2}$ . Since $η_{R} = 1$ on $K_{R}$ and $0 \leq η_{R} \leq 1$ , choose each of the finitely many $L^{2}$ errors small enough that the weighted norm difference between $g_{n}$ and the truncation has combined norm below $ε / 3$ . Set $g_{n} = 0$ for $n > N$ . The triangle inequality then gives a finite family $g_{0}, \dots, g_{N} \in C_{c}^{\infty} (R^{2})$ within $ε$ of $f$ in the weighted coefficient norm.

2.1A1F1F2F5step 1.1algebra

For $u \in C_{c}^{\infty} (R^{3})$ , every $z$ -slice $u (\cdot, \cdot, z)$ is a planar test function. Apply the form identity in [F1] on those slices and integrate in $z$ by Fubini. The $z$ derivative commutes with the planar transform; after Fourier transformation in $z$ , Plancherel and the derivative identity in [F2] turn $∥ - i ℏ \partial_{z} u ∥_{2}^{2} / (2 m)$ into the multiplier $c k_{z}^{2}$ . Tonelli [F5] permits summing the nonnegative transverse energies and integrating in $(k_{y}, k_{z})$ . Therefore $q_{B}^{(3)} [u] = \sum_{n = 0}^{\infty} \int_{R^{2}} (E_{n} + c k_{z}^{2}) ∣ f_{n} (k_{y}, k_{z}) ∣^{2} d k_{y} d k_{z} .$ This makes the transformed core an isometric subspace of the stated weighted coefficient space.

2.2A1F2F4F7F8step 1.2constructalgebra

For the finite family from step 1.2 put $G (ξ, k_{y}, k_{z}) = \sum_{n = 0}^{N} g_{n} (k_{y}, k_{z}) ψ_{n} (ξ)$ . Its inverse is $v (x, y, z) = \frac{1}{2 π} \sum_{n = 0}^{N} \int_{R^{2}} e^{i (k_{y} y + k_{z} z)} g_{n} (k_{y}, k_{z}) ψ_{n} ⁣ (x - \frac{ℏ k_{y}}{q B}) d k_{y} d k_{z} .$ Each coefficient is smooth and compactly supported in $(k_{y}, k_{z})$ , and each $ψ_{n}$ is a polynomial times a Gaussian by [F7]. Every derivative of the integrand is smooth with compact wave-number support and has uniformly rapid decay in $x$ because $k_{y}$ remains in a bounded set. Repeated integration by parts in $k_{y}, k_{z}$ gives arbitrary polynomial decay in $y, z$ ; the same estimates hold after any fixed $x, y, z$ derivatives. Thus $v$ is Schwartz. Define $χ_{R} (x, y, z) = \prod_{s \in {x, y, z}} σ (2 - s / R) σ (2 + s / R),$ using the smooth step from [F4]. These cutoffs equal one on $[- R, R]^{3}$ , vanish outside $[- 2 R, 2 R]^{3}$ , and satisfy $∣ \nabla χ_{R} ∣ \leq C / R$ . For each covariant derivative, $Π_{j} (χ_{R} v) = χ_{R} Π_{j} v - i ℏ (\partial_{j} χ_{R}) v,$ where $Π_{x} = - i ℏ \partial_{x}$ , $Π_{y} = - i ℏ \partial_{y} - q B x$ , and $Π_{z} = - i ℏ \partial_{z}$ . Dominated convergence [F8] sends $(χ_{R} - 1) Π_{j} v$ and $(χ_{R} - 1) v$ to zero in $L^{2}$ , while the added term has norm at most $C ℏ ∥ v ∥_{2} / R$ . Hence $χ_{R} v$ approximates $v$ in the form norm and belongs to the original compactly supported core. Thus each finite smooth coefficient family from step 1.2 lies in the transformed form closure.

3.1F1F6step 1.2step 2.1step 2.2algebra∎

Step 2.1 embeds the transformed core isometrically into the weighted coefficient space. Because the form completion is complete and the weighted target is complete by [F6], the isometric image of the completion is closed in that target. Steps 1.2 and 2.2 show that a dense subspace of the entire weighted space lies in the image, so the image is the whole target. This identifies the completion and its quadratic form as stated.

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Unbounded vector potential and effective Pauli model

Physical scope

A nonrelativistic point-particle model with a static smooth external vector potential and locally integrable lower-bounded scalar potential on Euclidean R^d, d >= 1, in SI units. The planar field is the scalar perpendicular component and the three-dimensional field is the curl vector. The optional spin-half Pauli coupling is a three-dimensional effective model term only when its form is closable and semibounded; no measured g-factor or relativistic correction is asserted.

Postulate

Use SI units with mass $m > 0$ in kilograms, charge $q \in R$ in coulombs, action constant $ℏ > 0$ in joule-seconds, vector-potential components $A_{j}$ in tesla-metres, and scalar potential-energy function $V$ in joules. For each integer $d \geq 1$ , let $A \in C^{\infty} (R^{d}; R^{d})$ be real and let $V \in L_{l o c}^{1} (R^{d}; R)$ be bounded below. Adopt the general-dimensional minimal-coupling assignment whose core form is $q_{A, V} [u] = \frac{1}{2 m} \sum_{j = 1}^{d} ∥ (- i ℏ \partial_{j} - q A_{j}) u ∥_{2}^{2} + \int_{R^{d}} V ∣ u ∣^{2}, u \in C_{c}^{\infty} (R^{d}; C) .$ Use its closed form closure and the unique self-adjoint representing Hamiltonian. For dimensional form norms use the fixed reference energy $E_{0} = 1 J$ : apply the closed-form representation to $q_{A, V} / E_{0}$ and multiply the representing operator by $E_{0}$ .

The magnetic-field notation is dimension-specific. In $d = 2$ , with coordinates $(x, y)$ and $A = (A_{x}, A_{y})$ , define the scalar perpendicular field $B_{⊥} (x, y) = \partial_{x} A_{y} (x, y) - \partial_{y} A_{x} (x, y),$ in teslas. Thus $A = (0, B x)$ gives $B_{⊥} = B$ for a constant signed scalar $B$ . In $d = 3$ , for coordinates $(x, y, z)$ and $A = (A_{x}, A_{y}, A_{z})$ , define the field vector $B : R^{3} \to R^{3}$ in teslas by $B = (\partial_{y} A_{z} - \partial_{z} A_{y}, \partial_{z} A_{x} - \partial_{x} A_{z}, \partial_{x} A_{y} - \partial_{y} A_{x}) = \nabla \times A .$ No magnetic-field object is assigned by this notation in $d = 1$ .

For an optional spin-half degree of freedom in three dimensions, define $σ_{1} = (\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array}), σ_{2} = (\begin{array}{cc} 0 & - i \\ i & 0 \end{array}), σ_{3} = (\begin{array}{cc} 1 & 0 \\ 0 & - 1 \end{array}), S_{j} = \frac{ℏ}{2} σ_{j} .$ For a core spinor $u = (u_{1}, u_{2}) \in C_{c}^{\infty} (R^{3}; C^{2})$ , define the spin-independent form $q_{A, V}^{(2)} [u] = q_{A, V} [u_{1}] + q_{A, V} [u_{2}] .$ For a declared dimensionless real coupling $g$ , adopt the effective Pauli core form $q_{A, V, g}^{P} [u] = q_{A, V}^{(2)} [u] - \frac{g q}{2 m} \int_{R^{3}} u (x)^{*} (S \cdot B (x)) u (x) d x,$ where $u^{*}$ is conjugate transpose and $(S \cdot B) (x) = \sum_{j = 1}^{3} S_{j} B_{j} (x)$ for the three-dimensional field vector just defined. Include this term only in model instances where the resulting form is closable and bounded below, and use its closed form closure and representing Hamiltonian. No universal semiboundedness for arbitrary $A, V, g$ is asserted.

Postulate references

David Tong, Applications of Quantum Mechanics, §§1.1.1 and 1.5

V. P. Nair, Notes on Quantum Mechanics, §§11.2–11.3

Aram Harrow, MIT 8.06 Lecture Notes, Chapter 4, §3.1

Scope and non-derivation

The core form data and the closability argument are given in Magnetic quadratic-form data and Closability of smooth magnetic forms with locally integrable potentials. The form-representation interface Semibounded closed-form representation and perturbation specifies the unique self-adjoint operator associated with the adopted closed form. The generic closability supplier assumes Countable Choice and Dependent Choice; the representation supplier assumes the Axiom of Choice and Countable Choice. These are the mathematical hypotheses of the construction, not additional physical postulates.

The mathematical identity in Pauli square identity and the formal g=2 coupling is a formal-core identity. This postulate does not identify $g$ with an electron measurement, derive the anomalous magnetic moment, or use the Dirac or quantum-electrodynamic theories. The linear Landau gauge is covered by the local closed-form construction; no bounded-coefficient $H^{2}$ hypothesis is imposed.

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Exact planar Landau spectrum in the linear gauge

Physical scope

Exact spectral prediction of the adopted spinless nonrelativistic point-particle model on the infinite plane, for a constant nonzero perpendicular field in the unbounded linear Landau gauge. It makes no finite-sample degeneracy count, boundary claim, or empirical assertion.

Statement

Assume the Axiom of Choice, Countable Choice, and Dependent Choice, and the adopted spinless minimal-coupling form model. Let $m > 0$ be mass in kilograms, $q \neq 0$ charge in coulombs, $B \neq 0$ the constant signed perpendicular field component in teslas, and $ℏ > 0$ in joule-seconds. Let $A : R^{2} \to R^{2}$ be $A (x, y) = (0, B x)$ in tesla-metres; then $B_{⊥} = \partial_{x} A_{y} - \partial_{y} A_{x} = B$ is its scalar perpendicular field. Take wavefunctions in $L^{2} (R^{2}; C)$ . Set $V = 0$ and define the Hamiltonian by the closed form of the core expression $q_{B} [u] = \frac{1}{2 m} (∥ - i ℏ \partial_{x} u ∥_{2}^{2} + ∥ (- i ℏ \partial_{y} - q B x) u ∥_{2}^{2}), u \in C_{c}^{\infty} (R^{2}; C) .$ The resulting Friedrichs Hamiltonian is unitarily equivalent, under partial Fourier transformation in $y$ followed by $x = ξ + ℏ k / (q B)$ , to the constant family of oscillators $h (k) = - \frac{ℏ^{2}}{2 m} \partial_{ξ}^{2} + \frac{(q B ξ)^{2}}{2 m}, k \in R,$ with its form-closed realization. Its spectrum is ${\frac{ℏ ∣ q B ∣}{m} (n + \frac{1}{2}) : n \in N_{0}} .$ Every listed eigenvalue has infinite multiplicity, and every spectral measure is purely atomic on this set; there is no continuous spectral measure. This is a conditional model deduction, not a claim that a material realizes the ideal infinite-plane Hamiltonian.

Statement references

Unbounded vector potential and effective Pauli model

Closed-form diagonalization in Landau gauge

Semibounded closed-form representation and perturbation

Oscillator scaling and exact realization

The Axiom of Choice

The Axiom of Countable Choice ( ${A C}_{ω}$ )

The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -indexed chain

Facts & Assumptions

Given: the Axiom of Choice, Countable Choice, Dependent Choice, the spinless model postulate, $m > 0$ , $q B \neq 0$ , $ℏ > 0$ , $A = (0, B x)$ , $V = 0$ , and the core form in the Statement. Use SI units.

[A1]

Adopt the spinless minimal-coupling assignment and the stated Friedrichs-form prescription (Unbounded vector potential and effective Pauli model).

[A2]

The Axiom of Choice and Countable Choice are hypotheses of the closed-form representation and unbounded spectral suppliers. Dependent Choice is required by the Landau form supplier (The Axiom of Choice, The Axiom of Countable Choice ( ${A C}_{ω}$ ), The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -indexed chain, Closed-form diagonalization in Landau gauge).

[F1]

The closed magnetic form is unitarily equivalent to the diagonal form $\sum_{n} E_{n} ∥ f_{n} ∥_{2}^{2}$ on coefficient families, with $E_{n} = ℏ ∣ q B ∣ (n + \frac{1}{2}) / m$ , under the explicitly normalized partial Fourier and translation map (Closed-form diagonalization in Landau gauge).

[F2]

The closed-form representation theorem applies to a dimensionless closed form. Write $E_{r e f} = 1 J$ for the postulate's reference energy and apply the theorem to $q / E_{r e f}$ . On the closed form domain, $(q / E_{r e f}) [F] + ∥ F ∥_{2}^{2} = E_{r e f}^{- 1} (q [F] + E_{r e f} ∥ F ∥_{2}^{2})$ , so normalization preserves its form domain, density, and closedness. The theorem's operator domain consists of form-domain vectors $F$ for which there is a coefficient-space vector $G$ satisfying $(q / E_{r e f}) (F, V) = ⟨ G, V ⟩$ for every form-domain vector $V$ . If $\hat{H}$ represents this normalized form, then the model Hamiltonian is $H = E_{r e f} \hat{H}$ on the same operator domain, so its spectral values are multiplied by $E_{r e f}$ . Here $E_{r e f}$ is distinct from the indexed Landau level $E_{0}$ used below. The theorem's exact hypotheses include the Axiom of Choice and Countable Choice (Unbounded vector potential and effective Pauli model, Semibounded closed-form representation and perturbation).

[F3]

The one-dimensional oscillator with $ω_{c} = ∣ q B ∣ / m$ has the complete Hermite eigenbasis and eigenvalues $E_{n} = ℏ ω_{c} (n + \frac{1}{2})$ (Oscillator scaling and exact realization).

[F4]

A self-adjoint operator has a unique regular spectral projection-valued measure (Spectral theorem for unbounded self-adjoint operators (PVM form)).

Proof

1.1givenA1A2F1F3

The hypotheses $m, ℏ > 0$ and $q B \neq 0$ make $ω_{c} = ∣ q B ∣ / m > 0$ and the translation $x_{0} (k) = ℏ k / (q B)$ well-defined. Apply [F1]: the unitary $U$ sends the closed form to the coefficient form $q [F] = \sum_{n = 0}^{\infty} E_{n} ∥ f_{n} ∥_{L^{2} (k)}^{2},$ where $F (ξ, k) = \sum_{n} f_{n} (k) ψ_{n} (ξ)$ and $(ψ_{n})$ is the complete oscillator basis from [F3]. In particular, $q [F] \geq E_{0} ∥ F ∥_{2}^{2} > 0$ for nonzero $F$ .

2.1A2F1F2step 1.1constructalgebra

Apply the closed-form representation theorem [F2] to the normalized form $\hat{q} = q / E_{r e f}$ . If $F = (f_{n})$ is in its representing operator domain, there is $G = (g_{n})$ in the coefficient Hilbert space such that $\hat{q} (F, V) = ⟨ G, V ⟩$ for every form-domain vector $V$ . Fix $n$ and take $V$ to have sole nonzero component $V_{n} = (E_{n} / E_{r e f}) f_{n} - g_{n}$ . It lies in the form domain because $f_{n}, g_{n} \in L^{2} (k)$ and $E_{n} / E_{r e f}$ is finite. The representation identity gives $0 = \hat{q} (F, V) - ⟨ G, V ⟩ = ∥ (E_{n} / E_{r e f}) f_{n} - g_{n} ∥_{2}^{2},$ so $g_{n} = (E_{n} / E_{r e f}) f_{n}$ for every $n$ . Since $G$ belongs to the Hilbert space, $\sum_{n} (E_{n} / E_{r e f})^{2} ∥ f_{n} ∥_{2}^{2} < \infty$ , equivalently $\sum_{n} E_{n}^{2} ∥ f_{n} ∥_{2}^{2} < \infty$ because $E_{r e f} > 0$ .

3.1A2F1F2step 2.1algebra

Conversely, suppose $F$ is in the form domain and $\sum_{n} E_{n}^{2} ∥ f_{n} ∥_{2}^{2} < \infty$ . Define $G_{n} = (E_{n} / E_{r e f}) f_{n}$ , which is a vector in the coefficient Hilbert space. For every form-domain vector $V = (v_{n})$ , Cauchy–Schwarz gives absolute convergence and $\hat{q} (F, V) = \sum_{n} (E_{n} / E_{r e f}) ⟨ f_{n}, v_{n} ⟩ = \sum_{n} ⟨ G_{n}, v_{n} ⟩ = ⟨ G, V ⟩ .$ Thus [F2] represents the normalized operator by $(f_{n}) \mapsto ((E_{n} / E_{r e f}) f_{n})$ on the stated domain. The model Hamiltonian is $E_{r e f}$ times this operator and therefore acts by $(f_{n}) \mapsto (E_{n} f_{n})$ on that same domain; uniqueness identifies it with the postulated Hamiltonian.

4.1A2F1F3F4step 3.1algebra∎

For each $n$ , the eigenspace at $E_{n}$ is $L^{2} (R_{k}) \otimes C ψ_{n}$ , which is infinite-dimensional; the energies are strictly increasing and tend to $+ \infty$ . If $z \notin {E_{n} : n \geq 0}$ , then $\inf_{n} ∣ E_{n} - z ∣ > 0$ and $\sup_{n} E_{n} / ∣ E_{n} - z ∣ < \infty$ . The diagonal inverse is therefore bounded and maps into the operator domain, so $z$ is in the resolvent. Each $E_{n}$ is an eigenvalue, and the complete coefficient decomposition together with [F4] shows that all spectral projections are sums of eigenspace projections. Hence every spectral measure is atomic on the displayed set and the spectrum is exactly that set.

Remarks

The linear potential $A_{y} = B x$ is unbounded. The proof works with its closable quadratic form and never applies a bounded-coefficient $H^{2}$ realization. The finite-rectangle degeneracy count in the cited treatments depends on boundary conditions; no exact hard-wall or finite-area count is claimed here.

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Aharonov–Bohm ring spectrum and flux periodicity

Physical scope

Exact spectrum of an ideal spinless particle constrained to a zero-width circle with static enclosed magnetic flux and periodic wavefunctions. Finite-width rings, interactions, apparatus, and experimental identification are outside scope.

Statement

Adopt the static ring model of a spinless particle of mass $m > 0$ kilograms and charge $q \neq 0$ coulombs constrained to a circle of radius $R > 0$ metres. Use $ℏ > 0$ in joule-seconds, flux $Φ$ in webers, and energies in joules. The angular coordinate is $ϕ \in R / (2 π Z)$ ; the single-valued wavefunction belongs to the periodic domain $H_{p e r}^{2} (S^{1})$ ; the scalar potential is zero; and a static tangential vector potential $A_{ϕ} = Φ / (2 π R)$ has circulation $\oint A \cdot d ℓ = Φ$ , where $Φ$ is flux in webers. Under the SI minimal-coupling assignment, the Hamiltonian is $H_{Φ} = \frac{ℏ^{2}}{2 m R^{2}} {(- i \partial_{ϕ} - α)}^{2}, α = \frac{q Φ}{2 π ℏ},$ on that periodic domain. Its energies are $E_{n} (Φ) = \frac{ℏ^{2}}{2 m R^{2}} (n - α)^{2}, n \in Z,$ with a complete periodic Fourier eigenbasis. For every $j \in Z$ , $Φ \mapsto Φ + 2 π ℏ j / q$ gives a unitarily equivalent Hamiltonian; a noninteger $α$ cannot be removed by a single-valued periodic gauge phase. These are conditional deductions of the ideal ring model.

Statement references

Unbounded vector potential and effective Pauli model

Periodic Sobolev domain for a circle

Spectrum and gauge periodicity of a periodic flux operator

Facts & Assumptions

Given: the ideal ring constraint, $m > 0$ , $q \neq 0$ , $R > 0$ , $ℏ > 0$ , static flux $Φ$ , zero scalar potential, periodic wavefunctions, and the SI minimal-coupling postulate.

[A1]

The ring is an ideal zero-width constraint with no radial or transverse degrees of freedom. The model adopts a single-valued periodic wavefunction and the periodic $H^{2}$ realization of the tangential minimal-coupling expression; the static flux is represented by $A_{ϕ} = Φ / (2 π R)$ .

[A2]

The Choice, Countable Choice, and Dependent Choice axioms are the exact mathematical hypotheses of the periodic spectral calculation (The Axiom of Choice, The Axiom of Countable Choice ( ${A C}_{ω}$ ), The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -indexed chain).

[F1]

The minimal-coupling covariant momentum is $- i ℏ \nabla - q A$ in SI units; $m$ is the mass parameter and $q$ the charge parameter (Unbounded vector potential and effective Pauli model).

[F2]

The angular Hilbert space and periodic $H^{2}$ domain are those of Periodic Sobolev domain for a circle.

[F3]

For $κ > 0$ and $α \in R$ , the operator $κ (- i \partial_{ϕ} - α)^{2}$ has complete eigenbasis $(2 π)^{- 1 / 2} e^{i n ϕ}$ , energies $κ (n - α)^{2}$ , integer-shift unitary equivalence, and no periodic gauge multiplier removing a noninteger $α$ (Spectrum and gauge periodicity of a periodic flux operator).

Proof

1.1givenA1A2F1algebra

On the circle, arclength is $s = R ϕ$ , so the tangential derivative is $\partial_{s} = R^{- 1} \partial_{ϕ}$ . The circulation of the constant tangential potential is $\int_{0}^{2 π} A_{ϕ} R d ϕ = Φ$ by [A1]. Therefore the covariant tangential momentum supplied by [F1] is $- i ℏ \partial_{s} - q A_{ϕ} = \frac{1}{R} (- i ℏ \partial_{ϕ} - \frac{q Φ}{2 π}) .$ Dividing its square by $2 m$ gives the Hamiltonian in the Statement and the dimensionless parameter $α = q Φ / (2 π ℏ)$ .

2.1A1A2F2F3step 1.1algebra∎

Put $κ = ℏ^{2} / (2 m R^{2}) > 0$ . The operator in the Statement is $κ (- i \partial_{ϕ} - α)^{2}$ , so [F2] and [F3] give its periodic domain, complete Fourier eigenbasis, energy list, and spectrum. If the flux is shifted by $2 π ℏ j / q$ , then $α$ shifts by $j$ and the unitary multiplication $U_{j} f = e^{i j ϕ} f$ intertwines the two Hamiltonians by [F3]. The same fact [F3] shows that a smooth single-valued periodic phase removing $α$ exists only for integral $α$ . Thus the spectrum and its gauge periodicity follow conditionally from the declared ideal ring model; no finite-width correction or measurement is asserted.

Remarks

The loop orientation is increasing $ϕ$ , and positive flux means the circulation $\oint A \cdot d ℓ = Φ$ in that orientation. The figure is a schematic of the excluded flux core and the constrained path; it is not a geometrical proof of the spectral statement.

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Pauli square identity and the formal g=2 coupling

Physical scope

Formal differential-operator identity on compactly supported two-component spinors for a smooth static SI vector potential in three-dimensional nonrelativistic Pauli-model kinematics. It asserts neither a closed-domain identity nor a measured particle g-factor.

Statement

Let $m > 0$ be mass in kilograms, $q \in R$ charge in coulombs, $ℏ > 0$ in joule-seconds, and let $A \in C^{\infty} (R^{3}; R^{3})$ be a real SI vector potential in tesla-metres, with $B = \nabla \times A$ in teslas. On the two-component core $C_{c}^{\infty} (R^{3}; C^{2})$ define $Π_{j} = - i ℏ \partial_{j} - q A_{j}$ . With the spin-half matrices and $S_{j} = ℏ σ_{j} / 2$ fixed in the effective Pauli model, the following identity holds on that core: ${(\sum_{j = 1}^{3} σ_{j} Π_{j})}^{2} = \sum_{j = 1}^{3} Π_{j}^{2} - q ℏ \sum_{j = 1}^{3} σ_{j} B_{j} .$ Consequently the quadratic Pauli expression $\frac{1}{2 m} (\sum_{j} σ_{j} Π_{j})^{2}$ equals the minimally coupled kinetic expression plus the spin term $- \frac{q}{m} S \cdot B$ , which is the formal effective Pauli coupling at $g = 2$ . This is a core identity only; it makes no claim about closed operator domains or the measured magnetic moment of an electron.

Statement references

Unbounded vector potential and effective Pauli model

Magnetic quadratic-form data

Divergence and curl of a $C^{1}$ vector field

Facts & Assumptions

Given: the effective spin-half model, $m > 0$ , real $q$ , $ℏ > 0$ , smooth real $A$ , $B = \nabla \times A$ , and compactly supported smooth two-component spinors.

[A1]

The spin matrices are $σ_{1} = (\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array}), σ_{2} = (\begin{array}{cc} 0 & - i \\ i & 0 \end{array}), σ_{3} = (\begin{array}{cc} 1 & 0 \\ 0 & - 1 \end{array}), S_{j} = \frac{ℏ}{2} σ_{j},$ and $g$ is a declared real dimensionless effective coupling (Unbounded vector potential and effective Pauli model).

[F1]

On the test core, $Π_{j} = - i ℏ \partial_{j} - q A_{j}$ ; the curl convention is $B_{k} = \sum_{i, j} ϵ_{k i j} \partial_{i} A_{j}$ (Magnetic quadratic-form data, Divergence and curl of a $C^{1}$ vector field).

[F2]

The ordinary product rule holds for the smooth coefficient products (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 $g (c) \neq 0$ ).

Proof

1.1A1F1F2algebra

Direct multiplication of the matrices in [A1] gives $σ_{j}^{2} = I_{2}$ for each $j$ , $σ_{1} σ_{2} = i σ_{3}$ , $σ_{2} σ_{3} = i σ_{1}$ , $σ_{3} σ_{1} = i σ_{2}$ , and the reversed products have the opposite sign. Hence $σ_{i} σ_{j} = δ_{i j} I_{2} + i \sum_{k = 1}^{3} ϵ_{i j k} σ_{k} .$ For a core spinor $u$ , the product rule [F2] and the definition of $Π_{j}$ give $[Π_{i}, Π_{j}] u = i q ℏ (\partial_{i} A_{j} - \partial_{j} A_{i}) u = i q ℏ \sum_{k = 1}^{3} ϵ_{i j k} B_{k} u .$ All terms remain compactly supported smooth spinors.

2.1step 1.1algebra

Since the Pauli matrices act only on the spin factor, they commute with each $Π_{j}$ . Expanding the square using step 1.1 gives ${(\sum_{i} σ_{i} Π_{i})}^{2} = \sum_{i} Π_{i}^{2} I_{2} + i \sum_{i, j, k} ϵ_{i j k} σ_{k} Π_{i} Π_{j} = \sum_{i} Π_{i}^{2} I_{2} + \frac{i}{2} \sum_{i, j, k} ϵ_{i j k} σ_{k} [Π_{i}, Π_{j}] .$ Substitute the commutator from step 1.1. The finite identity $\sum_{i, j} ϵ_{i j k} ϵ_{i j ℓ} = 2 δ_{k ℓ}$ follows by listing the two nonzero ordered pairs when $k = ℓ$ and none when $k \neq ℓ$ ; the resulting term is $- q ℏ \sum_{k} σ_{k} B_{k}$ . This proves the displayed Pauli square.

3.1A1step 2.1algebra∎

Dividing by $2 m$ and using $S_{j} = ℏ σ_{j} / 2$ from [A1] rewrites the spin term as $- q ℏ σ \cdot B / (2 m) = - q S \cdot B / m = - 2 q S \cdot B / (2 m)$ . Comparison with the adopted coupling $- g q S \cdot B / (2 m)$ identifies its formal coefficient as $g = 2$ . The conclusion is only the stated identity of core differential expressions; the postulate separately requires a closable semibounded form before assigning a self-adjoint Pauli Hamiltonian.

5 · Examples, experiments and counterexamples

None yet.

Uniform magnetic fields, flux topology and Pauli coupling

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

Magnetic quadratic-form data

Definition

Definition references

Periodic Sobolev domain for a circle

Definition

Definition references

Closability of smooth magnetic forms with locally integrable potentials

Statement

Statement references

Facts & Assumptions

Proof

Closed-form diagonalization in Landau gauge

Statement

Statement references

Facts & Assumptions

Proof

Remarks

Spectrum and gauge periodicity of a periodic flux operator

Statement

Statement references

Facts & Assumptions

Proof

Remarks

Closed-form diagonalization with free longitudinal motion

Statement

Statement references

Facts & Assumptions

Proof

Unbounded vector potential and effective Pauli model

Postulate

Postulate references

Scope and non-derivation

Exact planar Landau spectrum in the linear gauge

Statement

Statement references

Facts & Assumptions

Proof

Remarks

Aharonov–Bohm ring spectrum and flux periodicity

Statement

Statement references

Facts & Assumptions

Proof

Remarks

Pauli square identity and the formal g=2 coupling

Statement

Statement references

Facts & Assumptions

Proof

5 · Examples, experiments and counterexamples

Sources