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Semiclassical inequalities for Dirichlet and Neumann Laplacians on convex domains

Rupert L. Frank, Simon Larson

TL;DR

The article develops sharp semiclassical inequalities for Dirichlet and Neumann Laplacians on bounded convex domains, extending classical bounds to a range of gamma<1 and linking these to Riesz means and Weyl asymptotics. It introduces extrapolation techniques to transfer bounds across gamma, proves that convexity yields gamma_d^sharp < 1, and connects these results to natural shape-optimization questions, showing when optimizers converge to balls or degenerate into lower-dimensional, spaghetti-like structures. The work also analyzes partially semiclassical limits in collapsing convex domains, deriving a unified framework that separates semiclassical directions from quantum directions via cross-sections, and provides non-asymptotic improvements (via Amrein–Berthier-type uncertainty) that sharpen prior inequalities. Collectively, the paper ties spectral bounds to geometric optimization and asymptotics, offering precise thresholds and limiting shapes that illuminate Pólya-type conjectures in convex geometry and the spectral theory of Laplacians. The results have implications for spectral shape optimization and the understanding of how domain geometry governs high-energy spectral statistics.

Abstract

We are interested in inequalities that bound the Riesz means of the eigenvalues of the Dirichlet and Neumann Laplacians in terms of their semiclassical counterpart. We show that the classical inequalities of Berezin-Li-Yau and Kröger, valid for Riesz exponents $γ\geq 1$, extend to certain values $γ<1$, provided the underlying domain is convex. We also study the corresponding optimization problems and describe the implications of a possible failure of Pólya's conjecture for convex sets in terms of Riesz means. These findings allow us to describe the asymptotic behavior of solutions of a spectral shape optimization problem for convex sets.

Semiclassical inequalities for Dirichlet and Neumann Laplacians on convex domains

TL;DR

The article develops sharp semiclassical inequalities for Dirichlet and Neumann Laplacians on bounded convex domains, extending classical bounds to a range of gamma<1 and linking these to Riesz means and Weyl asymptotics. It introduces extrapolation techniques to transfer bounds across gamma, proves that convexity yields gamma_d^sharp < 1, and connects these results to natural shape-optimization questions, showing when optimizers converge to balls or degenerate into lower-dimensional, spaghetti-like structures. The work also analyzes partially semiclassical limits in collapsing convex domains, deriving a unified framework that separates semiclassical directions from quantum directions via cross-sections, and provides non-asymptotic improvements (via Amrein–Berthier-type uncertainty) that sharpen prior inequalities. Collectively, the paper ties spectral bounds to geometric optimization and asymptotics, offering precise thresholds and limiting shapes that illuminate Pólya-type conjectures in convex geometry and the spectral theory of Laplacians. The results have implications for spectral shape optimization and the understanding of how domain geometry governs high-energy spectral statistics.

Abstract

We are interested in inequalities that bound the Riesz means of the eigenvalues of the Dirichlet and Neumann Laplacians in terms of their semiclassical counterpart. We show that the classical inequalities of Berezin-Li-Yau and Kröger, valid for Riesz exponents , extend to certain values , provided the underlying domain is convex. We also study the corresponding optimization problems and describe the implications of a possible failure of Pólya's conjecture for convex sets in terms of Riesz means. These findings allow us to describe the asymptotic behavior of solutions of a spectral shape optimization problem for convex sets.
Paper Structure (22 sections, 30 theorems, 239 equations)

This paper contains 22 sections, 30 theorems, 239 equations.

Table of Contents

  1. Introduction and main results
  2. Weyl asymptotics and semiclassical inequalities
  3. Semiclassical inequalities on convex domains
  4. The optimization problems
  5. Consequences in asymptotic shape optimization
  6. Weyl asymptotics for Riesz means on collapsing convex sets
  7. Some comments on the proofs
  8. Improved inequalities
  9. Improved Berezin--Li--Yau and Kröger inequality
  10. A priori bounds in the Neumann case
  11. Extrapolating semiclassical inequalities for Riesz means
  12. Semiclassical inequalities in convex domains
  13. Proof of Theorems \ref{['thm: Extended range of semiclassical ineq Dir']} and \ref{['thm: Extended range of semiclassical ineq Neu']}
  14. A characterization of the critical Riesz exponent
  15. The optimization problems
  16. ...and 7 more sections

Key Result

Theorem 1.1

Fix any $d\geq 1$. There exists $\gamma <1$ such that for all open, bounded, convex $\Omega \subset \mathbb{R}^d$ and $\lambda \geq 0,$

Theorems & Definitions (58)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 1.4
  • Theorem 1.5
  • Proposition 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Remark 1.9
  • Proposition 2.1
  • Proposition 2.2
  • ...and 48 more