Spectral estimates of the Dirichlet-Laplace operator in conformal regular domains

TL;DR

This paper develops spectral estimates for the Dirichlet-Laplace operator in conformal regular domains using geometric composition operator theory, providing bounds on eigenvalues and applications to quantum billiards.

Contribution

It introduces a novel approach using composition operators on Sobolev spaces to estimate eigenvalues in conformal domains, advancing spectral analysis techniques.

Findings

Derived lower bounds for the first eigenvalue of the Dirichlet-Laplace operator.

Established conformal estimates for the ground state energy in quantum billiards.

Linked geometric properties of domains to spectral characteristics.

Abstract

In this paper we consider conformal spectral estimates of the Dirichlet-Laplace operator in conformal regular domains $\Omega \subset \mathbb R^2$. This study is based on the geometric theory of composition operators on Sobolev spaces that permits us to estimate constants of the Poincar\'e-Sobolev inequalities. On this base we obtain lower estimates of the first eigenvalue of the Dirichlet-Laplace operator in a class of conformal regular domains. As a consequence we obtain conformal estimates of the ground state energy of quantum billiards.