Eigenvalue inequalities for relativistic Hamiltonians and fractional Laplacian

Abstract

Some eigenvalue inequalities for Klein-Gordon operators and fractional Laplacians restricted to a bounded domain are proved. Such operators became very popular recently as they arise in many problems ranging from mathematical finance to crystal dislocations, especially relativistic quantum mechanics and symmetric stable stochastic processes.

Many of the results obtained here are concerned with finding bounds for some functions of the spectrum of these operators. The subject, which is well developed for the Laplacian, is examined from the spectral theory perspective through some of the tools used to prove analogous results for the Laplacian. This work highlights some important results, sparking interest in constructing a similar theory for Klein-Gordon operators. For instance, the Weyl asymptotics and semiclassical bounds for the Klein-Gordon operator are developed. As a result, a Berezin-Li-Yau type inequality is derived and an improvement of the bound is proved in a separate chapter.

Other results involving some universal bounds for the Klein-Gordon Hamiltonian with an external interaction are also obtained.