Lieb--Thirring inequalities on manifolds with constant negative curvature

TL;DR

This paper establishes Lieb--Thirring inequalities on hyperbolic manifolds with constant negative curvature, providing bounds on the discrete spectrum and a Pólya-type inequality with numerical evidence in specific domains.

Contribution

It extends Lieb--Thirring inequalities to negatively curved manifolds and introduces a Pólya-type inequality with numerical validation.

Findings

Discrete spectrum below the continuous spectrum on hyperbolic manifolds

A Pólya-type inequality with a non-sharp constant

Numerical evidence supporting the Pólya inequality in a 2D domain

Abstract

In this short note we prove Lieb--Thirring inequalities on manifolds with negative constant curvature. The discrete spectrum appears below the continuous spectrum $(d-1)^2/4, \infty)$, where $d$ is the dimension of the hyperbolic space. As an application we obtain a P\'olya type inequality with not a sharp constant. An example of a 2D domain is given for which numerical calculations suggest that the P\'olya inequality holds for it.