Variational principle for Hamiltonians with degenerate bottom

TL;DR

This paper develops a variational approach to analyze Hamiltonians with degenerate minima, providing estimates for eigenvalues below the essential spectrum and demonstrating that negative potentials induce infinite discrete spectra.

Contribution

It introduces a variational principle tailored for Hamiltonians with degenerate bottoms and offers elementary proofs for the spectral effects of negative perturbations.

Findings

Negative potentials cause an infinite number of eigenvalues below the essential spectrum.

Provides variational estimates for eigenvalues in Hamiltonians with degenerate minima.

Applicable to models in spintronics, superconductivity, and superfluidity.

Abstract

We consider perturbations of Hamiltonians whose Fourier symbol attains its minimum along a hypersurface. Such operators arise in several domains, like spintronics, theory of supercondictivity, or theory of superfluidity. Variational estimates for the number of eigenvalues below the essential spectrum in terms of the perturbation potential are provided. In particular, we provide an elementary proof that negative potentials lead to an infinite discrete spectrum.