On the discrete spectrum of non-selfadjoint operators with applications to Schr\"odinger operators with complex potentials

TL;DR

This paper establishes bounds on the discrete eigenvalues of non-selfadjoint operators, extending classical results to complex potentials and providing new inequalities relevant to Schr"odinger operators.

Contribution

It introduces a novel connection between eigenvalue counts and the Birman--Schwinger operator in the non-selfadjoint setting, using tensor product techniques.

Findings

Derived upper bounds on eigenvalues in half-planes for non-selfadjoint operators.

Generalized Lieb--Thirring inequalities to complex potentials.

Analyzed the sharpness and optimality of the bounds.

Abstract

For relatively form-compact perturbations of non-negative selfadjoint operators, we obtain an upper bound on the number of discrete eigenvalues in half-planes separated from the positive real axis. The bound is given in terms of a partial trace of the real part of the Birman--Schwinger operator, or an appropriate rotation thereof. While eigenvalue counting estimates of this type are classical in the selfadjoint setting, no analogous connection between the number of discrete eigenvalues and the Birman--Schwinger operator has previously been established in the non-selfadjoint theory. The proof proceeds via techniques in antisymmetric tensor product spaces that serve as a non-selfadjoint replacement for the classical arguments. As an application to Schr\"odinger operators, we generalise the Cwikel--Lieb--Rozenblum inequality to complex potentials and derive new Lieb--Thirring type inequalities. We also analyse the sharpness of the obtained bounds and discuss their optimality within the considered framework.