Connection between the Lieb--Thirring conjecture for Schroedinger operators and an isoperimetric problem for ovals on the plane

TL;DR

This paper explores a novel link between the Lieb--Thirring conjecture for Schrödinger operators and an isoperimetric problem involving plane ovals, aiming to understand sharp constants in spectral inequalities.

Contribution

It establishes a new connection between spectral inequalities and geometric isoperimetric problems, providing insights into the conjecture for specific exponents.

Findings

Connection between Lieb--Thirring conjecture and isoperimetric inequality for ovals

Potential implications for sharp constants in spectral inequalities

Insight into the case gamma=1 for Schrödinger operators

Abstract

To determine the sharp constants for the one dimensional Lieb--Thirring inequalities with exponent gamma in (1/2,3/2) is still an open problem. According to a conjecture by Lieb and Thirring the sharp constant for these exponents should be attained by potentials having only one bound state. Here we exhibit a connection between the Lieb--Thirring conjecture for gamma=1 and an isporimetric inequality for ovals in the plane.