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

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.