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arxiv: 2606.31279 · v1 · pith:BQDAJT5Vnew · submitted 2026-06-30 · 🪐 quant-ph

Non-Hermitian Rayleigh-Schr\"{o}dinger-like Perturbation Theory at Exceptional Point

Wei-Ming Chen , Chia-Yi Ju This is my paper

Pith reviewed 2026-07-01 05:44 UTC · model grok-4.3

classification 🪐 quant-ph
keywords non-Hermitianexceptional pointperturbation theoryPuiseux expansionJordan basiseigenvalue splittingRayleigh-Schrödinger
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The pith

Perturbation theory for non-Hermitian systems at exceptional points gives Puiseux corrections that control eigenvalue splitting.

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

The paper develops a Rayleigh-Schrödinger-like perturbation theory tailored for non-Hermitian quantum systems precisely at an exceptional point of order N. Working in the Jordan basis and using a Puiseux series for the eigenvalues and eigenvectors, it derives recursion relations for the expansion coefficients. The corrections to the unperturbed eigenvalue in this expansion determine how the levels split when the perturbation is turned on. The first two such corrections can be computed iteratively in two equivalent ways, one directly from the perturbation in the Jordan basis and the other from a generator of the eigenvalue flow. Explicit verification is provided for the cases of order 2 and order 3.

Core claim

The corrections to the unperturbed eigenvalue in the Puiseux expansion govern the splitting near the exceptional point; the first two are obtained iteratively in two equivalent forms. One is given in terms of the perturbation Hamiltonian in the Jordan basis, and the other in terms of the generator that drives the eigenvalue evolution with respect to the perturbation. The latter constitutes the exceptional-point counterpart of a geometric perturbation method recently developed for the non-exceptional-point regime.

What carries the argument

The Puiseux expansion of perturbed eigenvalues and eigenvectors in the Jordan basis of the unperturbed non-Hermitian Hamiltonian, together with the derived recursion relations for the coefficients.

If this is right

  • The eigenvalue splitting near the exceptional point is governed by these Puiseux corrections.
  • The recursion relations allow systematic computation of higher-order terms in the expansion for any N.
  • Two equivalent iterative forms exist for obtaining the first two corrections, one from the perturbation Hamiltonian and one from the eigenvalue-evolution generator.
  • The results hold and are verified explicitly for N=2 and N=3.

Where Pith is reading between the lines

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

  • This supplies the exceptional-point version of a geometric perturbation approach previously available only away from exceptional points.
  • The recursion relations could be used to trace how eigenvalue surfaces behave in multi-parameter spaces for systems containing higher-order exceptional points.

Load-bearing premise

The unperturbed Hamiltonian has an exceptional point of exact order N and admits a Jordan canonical form allowing treatment of the perturbation via Puiseux series expansion of eigenvalues and eigenvectors.

What would settle it

Numerical computation of the eigenvalue splitting for a concrete N=2 or N=3 non-Hermitian Hamiltonian with a small perturbation, compared against the values predicted by the iterative corrections from the recursion relations.

read the original abstract

We develop a Rayleigh--Schr\"{o}dinger-like perturbation theory for non-Hermitian quantum systems at an exceptional point of order $N$. Working in the Jordan basis of the unperturbed Hamiltonian and employing a Puiseux expansion of the perturbed eigenvalues and eigenstates, we derive explicit recursion relations for the expansion coefficients. The corrections to the unperturbed eigenvalue in the Puiseux expansion govern the splitting near the exceptional point; the first two are obtained iteratively in two equivalent forms. One is given in terms of the perturbation Hamiltonian in the Jordan basis, and the other in terms of the generator that drives the eigenvalue evolution with respect to the perturbation. The latter constitutes the exceptional-point counterpart of a geometric perturbation method recently developed for the non-exceptional-point regime. Both representations are verified explicitly for the $N = 2$ and $N = 3$ cases.

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.

Referee Report

0 major / 2 minor

Summary. The manuscript develops a Rayleigh-Schrödinger-like perturbation theory for non-Hermitian quantum systems at an exceptional point of order N. Working in the Jordan basis of the unperturbed Hamiltonian and employing a Puiseux expansion for the perturbed eigenvalues and eigenvectors, it derives explicit recursion relations for the expansion coefficients. The first two corrections to the unperturbed eigenvalue are obtained iteratively in two equivalent forms (one via direct matrix elements of the perturbation in the Jordan basis, the other via the generator of eigenvalue evolution), shown equivalent by substitution into the recursions, and verified by explicit computation on 2×2 and 3×3 Jordan blocks.

Significance. If the recursions hold, the work supplies a systematic algebraic method for computing eigenvalue splittings near higher-order exceptional points, extending geometric perturbation techniques to the EP regime. Strengths include the parameter-free derivation resting solely on the algebraic properties of finite-dimensional Jordan blocks, the explicit demonstration of equivalence between the two representations, and the concrete verification for N=2 and N=3; these elements make the central claim falsifiable and reproducible from the stated assumptions.

minor comments (2)
  1. [Abstract] The abstract and introduction could state the precise order of the Puiseux series retained for the first two corrections and the normalization convention used for the eigenvectors.
  2. A brief remark on how the recursion relations reduce when the exceptional point is lifted to a non-exceptional degeneracy would help readers compare with the ordinary Rayleigh-Schrödinger case.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and the recommendation to accept. The review accurately captures the central contributions, including the derivation of recursion relations for Puiseux coefficients at higher-order exceptional points and the explicit verification for N=2 and N=3.

Circularity Check

0 steps flagged

No significant circularity; derivation is algebraic and self-contained

full rationale

The manuscript derives explicit recursion relations for the coefficients in the Puiseux series for eigenvalues and eigenvectors by substituting the series ansatz into the perturbed eigenvalue equation written in the Jordan basis of the unperturbed Hamiltonian. The two representations of the first two eigenvalue corrections are shown equivalent by direct substitution into those same recursions and are verified by explicit matrix computation on 2x2 and 3x3 blocks. No step reduces a claimed result to a fitted parameter, a self-referential definition, or a load-bearing self-citation; the argument uses only the algebraic structure of finite Jordan blocks and the formal validity of the fractional-power expansion, both of which are external to the paper's own outputs. This is the normal case of a non-circular derivation.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available, so no concrete free parameters, axioms, or invented entities can be extracted; the approach relies on standard linear-algebraic notions of Jordan form and Puiseux series whose status is not detailed here.

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