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arxiv: 2605.16569 · v1 · pith:4GQ4UCP3new · submitted 2026-05-15 · 🧮 math.SP · math-ph· math.MP

Eigenvalue bounds for non-self-adjoint Schr\"odinger operators and pseudodifferential generalizations

Eduard Stefanescu This is my paper

Pith reviewed 2026-05-19 21:15 UTC · model grok-4.3

classification 🧮 math.SP math-phmath.MP
keywords eigenvalue boundsnon-self-adjoint Schrödinger operatorsfractional Laplacianscompact manifoldscomplex potentialsL^p normsspectral theory
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The pith

Eigenvalue bounds for non-self-adjoint Schrödinger operators extend to fractional Laplacians on compact manifolds via L^p norms of the potentials.

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

The paper surveys existing bounds on the eigenvalues of Schrödinger operators with complex potentials, covering both deterministic and random cases in Euclidean space and on compact manifolds. It adds a new theorem that carries these bounds over to the fractional Laplacian setting on compact manifolds. The extension relies on adapting techniques from Cuenin and Sogge, with the bounds stated directly in terms of L^p norms of the potentials. A reader would care because these estimates locate the spectrum for operators that lack self-adjointness, which appear in models with gain or loss.

Core claim

The central claim is that spectral bounds previously known for the Laplacian on compact manifolds continue to hold when the operator is replaced by a fractional Laplacian, provided the bounds are formulated using the L^p norms of the complex potentials; the proof adapts the methods developed by Cuenin and Sogge.

What carries the argument

Adaptation of Cuenin-Sogge techniques to obtain L^p-norm eigenvalue bounds for the fractional Laplacian on compact manifolds.

If this is right

  • Eigenvalues of fractional Schrödinger operators remain controlled by the L^p strength of the complex potential.
  • The same norm-based estimates apply to both deterministic and random potentials in the fractional setting.
  • The approach extends the reach of spectral control to pseudodifferential operators with complex symbols on manifolds.

Where Pith is reading between the lines

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

  • Numerical checks on simple manifolds such as the circle or sphere could verify the sharpness of the new bounds for fractional orders.
  • The technique might generalize further to other nonlocal operators beyond the fractional Laplacian.
  • Because the title mentions pseudodifferential generalizations, similar bounds could be derived for variable-coefficient versions.

Load-bearing premise

The Cuenin-Sogge methods adapt to the fractional Laplacian without requiring extra restrictions on the potential or the manifold geometry.

What would settle it

A concrete counterexample consisting of a specific compact manifold, a fractional order, and a complex potential for which the claimed eigenvalue bound in terms of the L^p norm fails to hold.

Figures

Figures reproduced from arXiv: 2605.16569 by Eduard Stefanescu.

Figure 1
Figure 1. Figure 1: Region Ξ and boundary Γ for α = 5. 7. Proof idea of Theorem 6.1 We follow Cuenin [11] and adapt the proofs to the general case of fractional Laplace operators, see Section 2 or [59] for the precise definitions. Proof. Complex powers are taken with respect to the principal logarithm, that is, w α := exp(α log w), Arg(w) ∈ (−π, π). Define Γ := n (λ + i) α : λ ≥ cotπ α o ∪ n (λ − i) α : λ ≥ cotπ α o . The… view at source ↗
read the original abstract

This is mostly a survey paper, where we collect results concerning the spectral bounds of deterministic and random Schr\"odinger operators with complex potentials, both on \(\mathbb{R}^d\) and on compact manifolds. The survey part is complemented by a new theorem, where we extend the result on spectral bounds on compact manifolds to the case of fractional Laplacians, applying methods by Cuenin and Sogge. These bounds are formulated in terms of the \(L^p\)-norms of the corresponding potentials.

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

2 major / 2 minor

Summary. This manuscript is primarily a survey collecting established results on eigenvalue bounds for non-self-adjoint Schrödinger operators with complex potentials, both deterministic and random, on Euclidean space and compact manifolds. It is supplemented by a new theorem extending these bounds to fractional Laplacians (-Δ)^s on compact manifolds by adapting techniques from Cuenin and Sogge, with the bounds expressed in terms of L^p-norms of the potentials.

Significance. If the extension to fractional Laplacians holds rigorously, the paper provides a useful reference compilation of spectral results for non-self-adjoint operators while offering a modest but relevant generalization to nonlocal pseudodifferential operators. The survey synthesizes prior work effectively, and the new result could facilitate further applications in quantum mechanics and analysis on manifolds, though its impact depends on the precise scope of the adaptation.

major comments (2)
  1. [New theorem section] New theorem (extension to fractional Laplacians): The adaptation of Cuenin-Sogge resolvent estimates and pseudodifferential parametrix constructions to the nonlocal operator (-Δ)^s must be verified explicitly. The manuscript should clarify whether the localization and microlocal analysis steps carry over directly for general s in (0,1) or require additional assumptions on the potential or manifold curvature, as the nonlocal character may alter the symbol calculus used in the cited works.
  2. [New theorem section] Presentation of the new result: The claim that the L^p-norm bounds extend without extra restrictions is central but lacks sufficient detail on how the fractional case modifies the underlying estimates from the standard Laplacian; a sketch of the key modifications or a reference to where the transfer is justified would strengthen the argument.
minor comments (2)
  1. [Abstract] The abstract could specify the admissible range of p and any restrictions on the fractional order s for the new bounds.
  2. [Throughout] Ensure consistent notation for the fractional Laplacian across the survey and new theorem sections.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comments on the new theorem. We appreciate the positive assessment of the survey component and the recommendation for minor revision. Below we address the major comments point by point, indicating the revisions we will make to strengthen the presentation of the extension to fractional Laplacians.

read point-by-point responses

  1. Referee: [New theorem section] New theorem (extension to fractional Laplacians): The adaptation of Cuenin-Sogge resolvent estimates and pseudodifferential parametrix constructions to the nonlocal operator (-Δ)^s must be verified explicitly. The manuscript should clarify whether the localization and microlocal analysis steps carry over directly for general s in (0,1) or require additional assumptions on the potential or manifold curvature, as the nonlocal character may alter the symbol calculus used in the cited works.

    Authors: We agree that the adaptation of the Cuenin-Sogge techniques to the nonlocal operator requires explicit verification in the text. In the revised manuscript we will expand the proof of the new theorem with a detailed sketch showing that the localization proceeds via a smooth partition of unity on the compact manifold (which is independent of the operator order) and that the microlocal parametrix construction carries over using the standard pseudodifferential calculus for symbols of order 2s. For s ∈ (0,1) the nonlocality is accounted for by the integral kernel representation of the resolvent, which does not alter the symbol class or require extra curvature assumptions beyond those already stated for the manifold. No additional restrictions on the potential are needed. We will cite the precise propositions from Cuenin and Sogge that justify the transfer. revision: yes

  2. Referee: [New theorem section] Presentation of the new result: The claim that the L^p-norm bounds extend without extra restrictions is central but lacks sufficient detail on how the fractional case modifies the underlying estimates from the standard Laplacian; a sketch of the key modifications or a reference to where the transfer is justified would strengthen the argument.

    Authors: We acknowledge that the current presentation would benefit from more explicit detail on the modifications. In the revision we will insert a short paragraph immediately preceding the statement of the new theorem that outlines the key steps: the fractional resolvent is expressed via the Balakrishnan formula or spectral theorem, reducing the estimate to a uniform family of L^p bounds for the standard Laplacian that hold for s in any compact subinterval of (0,1). These uniform bounds follow from the pseudodifferential parametrix constructions already available in the cited literature. We will add a direct reference to the relevant theorem in Cuenin and Sogge that justifies the transfer without extra restrictions. revision: yes

Circularity Check

0 steps flagged

No circularity: survey with external-method extension remains self-contained

full rationale

The manuscript is explicitly a survey collecting prior results on eigenvalue bounds for non-self-adjoint Schrödinger operators on R^d and compact manifolds. Its sole new contribution is a theorem extending those bounds to fractional Laplacians by direct application of techniques from the independent external works of Cuenin and Sogge. No equation, parameter fit, or uniqueness claim in the provided text reduces by construction to the paper's own inputs; all load-bearing steps rest on externally cited, non-overlapping literature rather than self-citation chains or redefinitions. The derivation chain is therefore independent of the present paper's own statements.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper is primarily a survey of existing results on spectral bounds for Schrödinger operators. The new theorem invokes standard properties of the fractional Laplacian and the methods of Cuenin and Sogge without introducing new free parameters, axioms beyond standard functional analysis, or invented entities.

axioms (1)
  • standard math Standard properties of the fractional Laplacian on compact manifolds hold as in the cited works of Cuenin and Sogge.
    Invoked when extending the spectral bound result to the fractional case.

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