Convexity inequalities for eigenvalues and log-concavity of eigenfunctions

Paul Bryan , Julie Clutterbuck , Cale Rankin

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Pith reviewed 2026-05-09 18:39 UTC · model grok-4.3

classification 🧮 math.AP

keywords Brunn-Minkowski inequalityDirichlet eigenvalueslog-concavitySchrödinger operatoreigenfunctionsconvex domainsspectral inequalities

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The pith

New proofs establish the Brunn-Minkowski inequality for Dirichlet eigenvalues and the log-concavity of eigenfunctions.

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

This paper presents simple new proofs for two classical results on the Schrödinger operator. It proves that the first Dirichlet eigenvalue satisfies the Brunn-Minkowski inequality for a class of domains including C^{1,1} connected domains with convex potentials. It further establishes that the first Dirichlet eigenfunction is log-concave. In the special case of convex domains, the log-concavity follows immediately as a corollary of the inequality. Sympathetic readers would appreciate these accessible proofs as they connect domain geometry directly to eigenvalue behavior.

Core claim

We give simple new proofs of two well-known results for the Schrödinger operator: first, the Brunn-Minkowski inequality for Dirichlet eigenvalues and, second, the log-concavity of the first Dirichlet eigenfunction. Our proof of the first applies to a class of domains including C^{1,1} connected domains and convex potentials. In the special case of convex domains, the second result is a simple corollary.

What carries the argument

The Brunn-Minkowski inequality applied to the first Dirichlet eigenvalue of the Schrödinger operator with convex potential.

Load-bearing premise

The domains must belong to the class of C^{1,1} connected domains with convex potentials, or be convex in the corollary case.

What would settle it

A pair of C^{1,1} domains with convex potentials whose Minkowski combination violates the Brunn-Minkowski inequality for the first Dirichlet eigenvalue would disprove the result.

read the original abstract

We give simple new proofs of two well-known results for the Schr\"odinger operator: first, the Brunn--Minkowski inequality for Dirichlet eigenvalues and, second, the log-concavity of the first Dirichlet eigenfunction. Our proof of the first applies to a class of domains including $C^{1,1}$ connected domains and convex potentials. In the special case of convex domains, the second result is a simple corollary.

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 supplies simpler proofs of two known results on Schrödinger eigenvalues and eigenfunctions, with a modest extension to C^{1,1} domains.

read the letter

This paper supplies simpler proofs of two known results on Schrödinger eigenvalues and eigenfunctions, with a modest extension to C^{1,1} domains. The authors state upfront that the Brunn-Minkowski inequality for Dirichlet eigenvalues and the log-concavity of the first eigenfunction are already established, so the contribution sits in the arguments rather than the statements. They handle the first result for C^{1,1} connected domains with convex potentials and recover the second as a corollary on convex domains. That regularity step is the clearest incremental piece. The approach appears to rest on variational techniques, which keeps the logic direct and avoids heavy machinery. If the full proofs deliver on the simplicity claim, they could make these facts easier to teach or adapt inside the subfield. The soft spots are limited. Because the theorems are not new, the paper's reach depends on how much cleaner or more general the derivations turn out to be once the details are checked. The abstract gives no indication of circularity or unstated gaps, but the value is still bounded by the fact that these are refinements rather than breakthroughs. No load-bearing assumptions look obviously fragile from the stated scope. This is for readers already working in spectral geometry or elliptic PDEs who need an alternative reference for these inequalities. It is not essential for outsiders. I would bring it to a reading group if the group focuses on eigenvalue problems, but only as a short note rather than a main paper. It deserves peer review because the claims are clearly scoped, the domain class is explicit, and clean proofs of standard facts can still save time for others even if they do not reshape the field.

Referee Report

0 major / 0 minor

Summary. The manuscript claims to provide simple new proofs of two well-known results for the Schrödinger operator: the Brunn-Minkowski inequality for Dirichlet eigenvalues, applicable to a class of domains including C^{1,1} connected domains with convex potentials, and the log-concavity of the first Dirichlet eigenfunction on convex domains as a corollary.

Significance. If the proofs are valid, they offer simpler alternatives to existing results in spectral theory and convex geometry using standard variational techniques. The paper presents independent proofs of known statements with no free parameters, ad-hoc axioms, or circularity, which is a strength for accessibility and potential extensions within the stated regularity classes.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive report and the recommendation to accept the manuscript. The referee's assessment aligns with our view that the proofs are direct and avoid circularity or ad-hoc assumptions.

Circularity Check

0 steps flagged

No significant circularity; derivations are independent proofs

full rationale

The paper supplies new variational proofs of two known results (Brunn-Minkowski inequality for Dirichlet eigenvalues of the Schrödinger operator and log-concavity of the ground-state eigenfunction) under explicitly stated regularity hypotheses (C^{1,1} connected domains with convex potentials, or convex domains for the corollary). No equations reduce by construction to fitted inputs, no self-definitional loops appear, and no load-bearing uniqueness theorems or ansätze are imported solely via self-citation. The central claims rest on standard comparison principles and convexity arguments that are developed internally from the stated assumptions rather than presupposing the target inequalities.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

With only the abstract available, the precise free parameters, axioms, and invented entities cannot be extracted. The work relies on standard background from PDE theory and spectral geometry for the Schrödinger operator and Dirichlet eigenvalues.

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Works this paper leans on

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