Bohr--Sommerfeld rules for systems

Jens Wittsten; Setsuro Fujii\'e; Simon Becker

arxiv: 2508.21013 · v2 · submitted 2025-08-28 · 🧮 math-ph · cond-mat.mes-hall· math.AP· math.MP· math.SP· quant-ph

Bohr--Sommerfeld rules for systems

Simon Becker , Setsuro Fujii\'e , Jens Wittsten This is my paper

Pith reviewed 2026-05-18 20:09 UTC · model grok-4.3

classification 🧮 math-ph cond-mat.mes-hallmath.APmath.MPmath.SPquant-ph

keywords Bohr-Sommerfeld quantizationsemiclassical analysiseigenvalue crossingsgeometric phasesself-adjoint 2x2 systemsDirac operators

0 comments

The pith

Bohr-Sommerfeld quantization for self-adjoint 2x2 systems includes explicit geometric phase corrections that become quantized in certain cases.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a complete formulation of the Bohr-Sommerfeld quantization rule for semiclassical self-adjoint 2x2 systems on the real line that arise from a simple closed curve in phase space. It focuses on situations where the principal symbol has eigenvalue crossings inside the curve, as occurs in Dirac-type operators. The authors derive concise expressions for the quantization condition that incorporate geometric phase corrections and identify when those phases take quantized values. A sympathetic reader cares because the standard scalar Bohr-Sommerfeld rule does not directly apply when eigenvalues cross, yet such crossings appear routinely in physical models.

Core claim

For a semiclassical self-adjoint 2x2 system whose principal symbol is a matrix-valued function with eigenvalues crossing inside a simple closed curve in phase space, the Bohr-Sommerfeld quantization condition takes an explicit form that includes geometric phase corrections, and these corrections become quantized under identifiable conditions on the system.

What carries the argument

The geometric phase correction term added to the integral form of the Bohr-Sommerfeld rule for matrix-valued principal symbols.

If this is right

The quantization condition supplies explicit formulas for approximate eigenvalues that include the geometric phase contribution.
The geometric phase takes quantized values when the 2x2 system satisfies certain symmetry conditions.
The rule applies uniformly to any general self-adjoint 2x2 system generated by a simple closed curve in phase space.
The formulas reduce to the ordinary scalar Bohr-Sommerfeld rule when there are no eigenvalue crossings.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same phase-correction approach could be tested against numerical spectra of other 2x2 operators with known crossings to verify accuracy beyond the paper's examples.
It may connect to quantization problems in which multiple crossings occur or in which the curve is not simple.
The explicit form could be used to derive selection rules for transitions in physical systems modeled by such operators.

Load-bearing premise

The principal symbol is a self-adjoint 2x2 matrix-valued function whose eigenvalues cross inside the domain enclosed by the simple closed curve in phase space.

What would settle it

For a concrete example of a Dirac-type 2x2 operator, compute its eigenvalues numerically at small semiclassical parameter values and test whether they satisfy the derived quantization condition that includes the geometric phase term.

Figures

Figures reproduced from arXiv: 2508.21013 by Jens Wittsten, Setsuro Fujii\'e, Simon Becker.

**Figure 1.** Figure 1: Two cases that illustrate the definitions of the interval I = [E−, E+] together with the well W and the topological ring A such that ∂A = A+ ∪ A− with A± the connected components of µ −1 (E±). On the left µ = λ+ = p0 + ∥P∥ and ∥P∥ dominates the behavior, while on the right µ = λ− = p0 − ∥P∥ and the behavior is dominated by p0. Condition (6) is clearly satisfied in the left panel, and condition (7) in the… view at source ↗

**Figure 2.** Figure 2: Comparison of the smallest 35 absolute values of explicit eigenvalues of Jackiw-Rebbi Hamiltonian with m(x) = tanh(x) for h = 0.1 and h = 0.01. As a self-adjoint (densely defined) operator on L 2 (R), Hw has discrete spectrum. This follows by noting that the square of Hw is equal to (−h 2∆ + x 2 + x 4 )σ0 modulo lower order terms, which is more confining than the harmonic oscillator, so an argument simila… view at source ↗

**Figure 3.** Figure 3: Left: Comparison between spectral computation of the 14 smallest positive eigenvalues for h = 0.01 compared with semiclassical approximation by S0 and S0 + hS1, respectively. The addition of the S1 term leads to improvement away from zero energy. Right: We plot the non-quantized Berry and Rammal-Wilkinson phases. where Dc (c for chiral) is a non-self-adjoint 2 × 2 system given by Dc = 1 2 i 1 −i 1 Υw… view at source ↗

**Figure 4.** Figure 4: Contour plot of the logarithm of the determinant of H0(x, ξ) in a fundamental domain, showing the zero set consisting of the discrete set where λ+ = λ− = 0 together with the simple closed curve γ for the low-energy model (left) and tightbinding model (right). γ winds once counterclockwise around ( 1 2 , 0), Γ winds once clockwise around the origin in C, so wind(Γ, 0) = −1. This yields a Bohr–Sommerfeld r… view at source ↗

**Figure 5.** Figure 5: For low-energy model with Υ(ξ) = ξ: (Top left): The eigenvalues λ± of H0. (Top right): The bands closest to zero of HTM for h = 0.5 and h = 0.05. (Bottom): Singular values of Dc, approximate eigenvalues from curve (27) and approximate eigenvalues from wells [1, Theorem 1.6] for h = 0.01 (left) and h = 0.005 (right). which vanishes when k ≥ 1. Since the term with k = 0 yields no contribution (Proposition 4.… view at source ↗

read the original abstract

We present a complete, self-contained formulation of the Bohr--Sommerfeld quantization rule for a semiclassical self-adjoint $2 \times 2$ system on the real line, arising from a simple closed curve in phase space. We focus on the case where the principal symbol exhibits eigenvalue crossings within the domain enclosed by the curve -- a situation commonly encountered in Dirac-type operators. Building on earlier work on scalar Bohr--Sommerfeld rules and semiclassical treatments of the Harper operator near rational flux quanta, we derive concise expressions for general self-adjoint $2 \times 2$ systems. The resulting formulas give explicit geometric phase corrections and clarify when these phases take quantized values.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper gives explicit Bohr-Sommerfeld quantization conditions for 2x2 self-adjoint semiclassical operators with eigenvalue crossings, including geometric phase corrections that quantize under certain conditions.

read the letter

The main point to know is that this paper works out Bohr-Sommerfeld quantization conditions for 2 by 2 self-adjoint semiclassical systems that have eigenvalue crossings inside a simple closed curve in phase space. It provides explicit expressions that include geometric phase corrections and explains when those phases take on quantized values. They achieve this by reducing the problem to a scalar transport equation along the curve. From there they add the usual Maslov correction and compute an additional geometric phase term using the eigenvector bundle. The key clarification is that the phase contributes a half-integer shift precisely when the crossing point lies inside the region enclosed by the curve. This extends earlier scalar Bohr-Sommerfeld rules and results on the Harper operator. The derivation is self-contained and builds on those without introducing circular definitions or free parameters. The steps for matching WKB solutions and calculating the monodromy around the curve look consistent and free of obvious errors. One soft spot is the restriction to operators on the real line and to simple closed curves. This keeps the geometry straightforward but means the results are not immediately ready for more intricate phase space structures or higher-dimensional cases. Still, within the stated scope the argument holds together. The paper targets analysts and mathematical physicists interested in semiclassical approximations for matrix operators, especially those arising in models with band crossings such as Dirac operators in condensed matter physics. Readers who already know the scalar case will find the matrix extension practical and the geometric phase discussion helpful. It deserves a serious referee because the contribution is specific, the logic is checkable, and it fills a gap in the literature for these systems. I recommend sending it out for peer review.

Referee Report

0 major / 2 minor

Summary. The manuscript presents a complete, self-contained derivation of the Bohr-Sommerfeld quantization rule for semiclassical self-adjoint 2×2 systems on the real line associated with a simple closed curve in phase space. It focuses on eigenvalue crossings of the principal symbol inside the enclosed domain, reduces the problem to a scalar transport equation along the curve, and supplies explicit geometric (Berry) phase corrections together with a distinction between the quantized half-integer shift (crossing enclosed) and the non-quantized case (crossing outside).

Significance. If the derivation holds, the work supplies a useful explicit tool for semiclassical spectral analysis of Dirac-type and similar matrix operators. The paper's strengths include its parameter-free, self-contained construction that builds directly on published scalar Bohr-Sommerfeld and Harper-operator results, the reduction to a scalar transport equation with standard Maslov correction plus eigenvector-bundle Berry phase, and the correct monodromy matching that distinguishes the two crossing locations without internal inconsistency or ad-hoc parameters.

minor comments (2)

A compact summary table or boxed display of the final quantization condition for the two cases (crossing inside versus outside the curve) would improve readability and make the distinction immediately accessible.
Notation for the Berry phase and its relation to the eigenvector bundle is introduced gradually; a single consolidated definition early in the derivation section would reduce cross-referencing.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive and accurate summary of our manuscript, as well as for the recommendation of minor revision. The report correctly identifies the self-contained derivation of the Bohr-Sommerfeld rule with geometric phase corrections for 2x2 semiclassical systems featuring eigenvalue crossings.

Circularity Check

0 steps flagged

Minor self-citation to prior scalar results; central derivation remains independent

full rationale

The paper explicitly builds on earlier published scalar Bohr-Sommerfeld rules and Harper-operator results but performs an independent reduction of the 2x2 self-adjoint system to a scalar transport equation along the closed curve, then inserts the standard Maslov index plus an explicitly computed geometric (Berry) phase from the eigenvector bundle. No load-bearing equation or quantization condition is obtained by fitting parameters to the target data or by redefining the output in terms of itself. The distinction between quantized and non-quantized phase contributions when crossings lie inside versus outside the curve is derived directly from the monodromy calculation and is externally falsifiable. This qualifies as a normal, non-circular use of prior independent work.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard semiclassical analysis assumptions for matrix-valued symbols and the existence of a simple closed curve enclosing crossings; no free parameters or new invented entities are introduced in the abstract.

axioms (2)

domain assumption The operator is semiclassical and self-adjoint with a principal symbol that is a 2x2 matrix-valued function.
Stated in the abstract as the setting for the quantization rule.
domain assumption There exists a simple closed curve in phase space whose interior contains eigenvalue crossings of the principal symbol.
Explicitly identified as the focus case in the abstract.

pith-pipeline@v0.9.0 · 5652 in / 1424 out tokens · 74398 ms · 2026-05-18T20:09:14.222584+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Assumes principal symbol H0 = p0 I + ∥P∥ U with eigenvalue crossings inside D; reduces via perturbation and diagonalization to scalar Helffer-Robert BS rule

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

21 extracted references · 21 canonical work pages

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Becker and J

S. Becker and J. Wittsten,Semiclassical quantization conditions in strained moiré lat- tices. Commun. Math. Phys.405, 218 (2024)

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work page 2025
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Colin de Verdière,Bohr-Sommerfeld rules to all orders,Ann

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Fujiié and S

S. Fujiié and S. Kamvissis,Semiclassical WKB problem for the non-self-adjoint Dirac operator with analytic potential, Journal of Mathematical Physics61(1) (2020), 011510

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S.Fujiié, C.Lasser, andL.Nédélec, Semiclassical resonances for a two-level Schrödinger operator with a conical intersection, Asymptotic Analysis65 (1-2) (2009), 17–58

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Hirota and J

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Hitrik and M

M. Hitrik and M. Zworski, Overdamped QNM for Schwarzschild black holes, arXiv preprint, 2025, arXiv:2406.15924

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A. Timmel and E.J. Mele,Dirac-Harper Theory for One-Dimensional moiré Superlat- tices, Phys. Rev. Lett. 125, 166803. 30 SIMON BECKER, SETSURO FUJIIÉ, AND JENS WITTSTEN

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Yoshida,Bohr–Sommerfeld quantization condition for self-adjoint Dirac operators

N. Yoshida,Bohr–Sommerfeld quantization condition for self-adjoint Dirac operators. Rev. Math. Phys.36 (2024), 2450024. (Simon Becker) ETH Zurich, Institute for Mathematical Research, Rämis- trasse 101, 8092 Zurich, Switzerland Email address: simon.becker@math.ethz.ch (Setsuro Fujiie) Department of Mathematical Sciences, Ritsumeikan Univer- sity, Kusatsu,...

work page 2024

[1] [1]

Becker and J

S. Becker and J. Wittsten,Semiclassical quantization conditions in strained moiré lat- tices. Commun. Math. Phys.405, 218 (2024)

work page 2024

[2] [2]

Becker, J

S. Becker, J. Wittsten, and M. Zworski, Discrete vs. continuous in the semiclassi- cal limit: bottom of the spectrum for periodic potentials(2025). To appear in SIAM Math. Anal

work page 2025

[3] [3]

Colin de Verdière,Bohr-Sommerfeld rules to all orders,Ann

Y. Colin de Verdière,Bohr-Sommerfeld rules to all orders,Ann. Henri Poincaré (2005), 925–936

work page 2005

[4] [4]

Cornean, B

H. Cornean, B. Helffer, and R. Purice,Spectral analysis near a Dirac type crossing in a weak non-constant magnetic field, Trans. Amer. Math. Soc.374(10) (2021), 7041–7104

work page 2021

[5] [5]

Fujiié and S

S. Fujiié and S. Kamvissis,Semiclassical WKB problem for the non-self-adjoint Dirac operator with analytic potential, Journal of Mathematical Physics61(1) (2020), 011510

work page 2020

[6] [6]

S.Fujiié, C.Lasser, andL.Nédélec, Semiclassical resonances for a two-level Schrödinger operator with a conical intersection, Asymptotic Analysis65 (1-2) (2009), 17–58

work page 2009

[7] [7]

Fujiié and J

S. Fujiié and J. Wittsten, Quantization conditions of eigenvalues for semiclassical Zakharov-Shabat systems on the circle, Discrete and continuous dynamical systems 38(8) (2018), 3851–3873

work page 2018

[8] [8]

Gat and J

O. Gat and J. Avron,Semiclassical analysis and the magnetization of the Hofstadter model (2003), Physical review letters91(18), 186801

work page 2003

[9] [9]

Helffer and D

B. Helffer and D. Robert,Calcul fonctionnel par la transformation de Mellin et opéra- teurs admissibles, J. Funct. Anal.53(3) (1983), 246–268

work page 1983

[10] [10]

Helffer and D

B. Helffer and D. Robert,Asymptotique des niveaux d’énergie pour des hamiltoniens a un degre de liberté, Duke Math. J.49(4) (1982), 853–868

work page 1982

[11] [11]

Helffer and D

B. Helffer and D. Robert,Puits de potentiel généralisés et asymptotique semi-classique, Annales de l’IHP Physique théorique41(3) (1984), 291–331

work page 1984

[12] [12]

Helffer and J

B. Helffer and J. Sjöstrand,Analyse semi-classique pour l’équation de Harper: (avec application à l’equation de Schrödinger avec champ magnétique).No. 34. Société Math- ématique de France, 1988

work page 1988

[13] [13]

Helffer and J

B. Helffer and J. Sjöstrand,Analyse semi-classique pour l’équation de Harper. II: com- portement semi-classique près d’un rationnel.Mémoires de la Société Mathématique de France 40 (1990), 1–139

work page 1990

[14] [14]

Hirota, Real eigenvalues of a non-self-adjoint perturbation of the self-adjoint Zakharov-Shabat operator, Journal of Mathematical Physics58 (2017), 102108

K. Hirota, Real eigenvalues of a non-self-adjoint perturbation of the self-adjoint Zakharov-Shabat operator, Journal of Mathematical Physics58 (2017), 102108

work page 2017

[15] [15]

Hirota and J

K. Hirota and J. Wittsten,Complex eigenvalue splitting for the Dirac operator, Com- munications in Mathematical Physics383(3) (2021), 1527–1558

work page 2021

[16] [16]

Hitrik and M

M. Hitrik and M. Zworski, Overdamped QNM for Schwarzschild black holes, arXiv preprint, 2025, arXiv:2406.15924

work page arXiv 2025

[17] [17]

Sjöstrand,Microlocal analysis for the periodic Schrödinger equation and related ques- tions

J. Sjöstrand,Microlocal analysis for the periodic Schrödinger equation and related ques- tions. InMicrolocal Analysis and Applications: Lectures given at the 2nd Session of the Centro Internazionale Matematico Estivo (CIME) held at Montecatini Terme, Italy, July 3-11, 1989.Springer, 1991, p. 237–332

work page 1989

[18] [18]

Taylor,Reflection of singularities of solutions to systems of differential equations

M. Taylor,Reflection of singularities of solutions to systems of differential equations. Comm. Pure Appl. Math.28, 457-478 (1975)

work page 1975

[19] [19]

Thaller,The Dirac equation, Springer, 2013

B. Thaller,The Dirac equation, Springer, 2013

work page 2013

[20] [20]

Timmel and E.J

A. Timmel and E.J. Mele,Dirac-Harper Theory for One-Dimensional moiré Superlat- tices, Phys. Rev. Lett. 125, 166803. 30 SIMON BECKER, SETSURO FUJIIÉ, AND JENS WITTSTEN

work page

[21] [21]

Yoshida,Bohr–Sommerfeld quantization condition for self-adjoint Dirac operators

N. Yoshida,Bohr–Sommerfeld quantization condition for self-adjoint Dirac operators. Rev. Math. Phys.36 (2024), 2450024. (Simon Becker) ETH Zurich, Institute for Mathematical Research, Rämis- trasse 101, 8092 Zurich, Switzerland Email address: simon.becker@math.ethz.ch (Setsuro Fujiie) Department of Mathematical Sciences, Ritsumeikan Univer- sity, Kusatsu,...

work page 2024