Semiclassical inequalities for Dirichlet and Neumann Laplacians on convex domains

2024/10/07 by Frank, Rupert L., Larson, Simon · 2 citations
#FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.2410.04769

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≥ 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.

Cited by

• Improved Berezin-Li-Yau inequality and Kröger inequality and consequences
• Pólya's conjecture up to ε-loss and quantitative estimates for the remainder of Weyl's law

Related

• Two-term, asymptotically sharp estimates for eigenvalue means of the Laplacian
• Complementary asymptotically sharp estimates for eigenvalue means of Laplacians
• Two new universal inequalities for Neumann eigenvalues of the Laplacian on a planar convex domain
• Semiclassical estimates for eigenvalue means of Laplacians on spheres
• A universal inequality for Neumann eigenvalues of the Laplacian on a convex domain in Euclidean space