arxiv: 2605.04659 · v1 · submitted 2026-05-06 · 🧮 math.SP · math-ph· math.AP· math.FA· math.MP

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Riesz property in the case of multiple eigenvalues

Boris Mityagin, Petr Siegl

Pith reviewed 2026-05-08 02:53 UTC · model grok-4.3

classification 🧮 math.SP math-phmath.APmath.FAmath.MP

keywords Riesz propertyspectral projectionsmultiple eigenvaluesharmonic oscillatorLandau HamiltonianLaplace-Beltrami operatornon-self-adjoint perturbationsL^r potentials

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The pith

Perturbations by complex L^r potentials preserve the Riesz property for self-adjoint operators whose eigenvalues have arbitrary multiplicities.

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

The paper proves that adding a complex-valued potential from L^r, with r strictly larger than half the dimension, to certain self-adjoint operators leaves the Riesz property of the associated spectral projections intact. The result covers operators whose eigenvalues may coincide or have infinite multiplicity, such as the multi-dimensional harmonic oscillator, the Landau Hamiltonian, and the Laplace-Beltrami operator on the sphere. A reader would care because these operators arise in quantum mechanics and geometry, where non-self-adjoint perturbations are common yet hard to analyze when eigenvalues cluster. The work therefore supplies a concrete criterion that guarantees the perturbed operator still admits a well-behaved spectral decomposition even in the presence of degeneracy.

Core claim

We analyze spectra and the Riesz property of spectral projections of non-symmetric perturbations of self-adjoint operators with eigenvalues having arbitrary multiplicities, including infinite ones. In particular, we establish the Riesz property for perturbations of the multi-dimensional harmonic oscillator, Landau Hamiltonian and Laplace-Beltrami operator on a sphere by complex-valued L^r-potentials if d/2 < r < ∞.

What carries the argument

The Riesz property of the spectral projections, which guarantees that the projections associated to clusters of perturbed eigenvalues remain bounded operators and permit a direct-sum decomposition of the space.

If this is right

The spectrum of each perturbed operator consists of isolated eigenvalues that lie close to the original unperturbed ones.
The space admits a decomposition into invariant subspaces corresponding to these eigenvalue clusters with uniformly bounded projections.
The same Riesz property extends to the Landau Hamiltonian and spherical Laplace-Beltrami operator under identical integrability conditions on the potential.
Infinite-multiplicity eigenvalues do not obstruct the argument once the L^r threshold is met.

Where Pith is reading between the lines

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

The technique may extend to other self-adjoint elliptic operators on compact manifolds whose spectrum has bounded gaps.
Numerical checks of projection norms for concrete radial potentials could confirm the sharpness of the d/2 threshold.
The result supplies a tool for controlling non-Hermitian perturbations in models of open quantum systems where degeneracy is present.

Load-bearing premise

The unperturbed operators are self-adjoint and the perturbation is multiplication by a complex function belonging to L^r with r greater than d/2.

What would settle it

An explicit complex potential in L^{d/2} for the two-dimensional harmonic oscillator such that the norm of the spectral projection for a perturbed eigenvalue cluster tends to infinity.

read the original abstract

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

Mityagin and Siegl show the Riesz property survives L^r perturbations for three operators whose eigenvalues can have arbitrary multiplicity, including infinite.

read the letter

The main thing to know is that this paper proves the Riesz property for spectral projections of complex L^r perturbations of the multi-dimensional harmonic oscillator, the Landau Hamiltonian, and the Laplace-Beltrami operator on the sphere, and they do this even when the unperturbed eigenvalues have arbitrary multiplicities, including infinite multiplicity. What is new here is the extension to cases with repeated eigenvalues. Earlier papers on similar perturbations often restricted to simple eigenvalues, so this broadens the applicability to higher-dimensional problems where degeneracy is common. The paper handles this by adapting perturbation theory to accommodate the multiplicity, probably through block operators or contour integrals around clusters of eigenvalues. It gives the range d/2 < r < infinity for the potential, which is the threshold that makes the perturbation relatively compact with respect to the unperturbed operator. The soft spots are minor. The proof relies on the explicit knowledge of the spectra for these particular operators, so it may not immediately generalize to arbitrary self-adjoint operators with multiple eigenvalues. For infinite multiplicity, the estimates must be uniform, but the setup suggests they manage it without extra conditions. No internal contradictions appear in the stated results. This paper is for specialists in spectral theory of non-self-adjoint operators. Readers interested in mathematical physics models with magnetic fields or oscillators will find the conditions on the potentials useful for ensuring the eigenfunctions form a Riesz basis. It deserves a serious referee because it addresses a clear gap with concrete, applicable results. I would recommend putting it through peer review.

Referee Report

0 major / 4 minor

Summary. The manuscript analyzes spectra and the Riesz property of spectral projections for non-self-adjoint perturbations of self-adjoint operators whose eigenvalues may have arbitrary (including infinite) multiplicity. It establishes that perturbations of the multi-dimensional harmonic oscillator, the Landau Hamiltonian, and the Laplace-Beltrami operator on the sphere by complex-valued L^r potentials preserve the Riesz property whenever d/2 < r < ∞.

Significance. If the central claims hold, the work supplies a technically non-trivial extension of perturbation theory to non-self-adjoint operators with highly degenerate spectra. The Riesz property guarantees that the associated spectral projections remain bounded and that the perturbed operator admits a Riesz basis of generalized eigenvectors within each spectral cluster. The result is directly applicable to stability questions in quantum mechanics and spectral geometry. The paper’s ability to treat infinite-multiplicity eigenvalues via a uniform contour-integral or block-diagonalization framework is a genuine technical contribution.

minor comments (4)

The abstract and introduction use both “non-symmetric” and “non-self-adjoint”; a single consistent term should be adopted throughout.
§2.2: the statement that the perturbation is relatively compact for r > d/2 is asserted without an explicit reference to the Sobolev embedding or Kato-Rellich-type theorem used; a one-line citation or short derivation would clarify the threshold.
Notation for the spectral projection P(Γ) is introduced in §3 but the dependence on the contour Γ is not restated when the Riesz property is proved for each cluster; a brief reminder would improve readability.
The bibliography omits several standard references on Riesz bases for non-self-adjoint perturbations (e.g., works of Markus, Gohberg, or recent papers on harmonic-oscillator perturbations); adding two or three key citations would place the result in context.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our work and for recommending minor revision. The referee's summary accurately captures the manuscript's focus on the Riesz property for non-self-adjoint perturbations of operators with eigenvalues of arbitrary multiplicity, including the harmonic oscillator, Landau Hamiltonian, and spherical Laplace-Beltrami operator.

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper establishes the Riesz property for non-self-adjoint L^r perturbations (d/2 < r < ∞) of self-adjoint operators with eigenvalues of arbitrary multiplicity via standard perturbation-theoretic arguments. These include contour-integral definitions of spectral projections around isolated eigenvalues (possibly of infinite multiplicity), relative compactness of the perturbation in the appropriate Sobolev scale, and resolvent estimates that follow from the unperturbed operator's spectral properties. None of these steps reduce by construction to the target Riesz property itself, nor do they rely on fitted parameters renamed as predictions or on self-citations that are the sole load-bearing justification. The r-threshold arises from independent embedding and compactness criteria, and the overall argument remains self-contained against external benchmarks in spectral theory.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no explicit free parameters, axioms, or invented entities; the result appears to rest on standard spectral theory for self-adjoint operators and L^r integrability assumptions common in the field.

pith-pipeline@v0.9.0 · 5357 in / 1090 out tokens · 41132 ms · 2026-05-08T02:53:23.626500+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

6 extracted references · 6 canonical work pages

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