Estimating complex eigenvalues of non-self-adjoint Schr\"odinger operators via complex dilations

TL;DR

This paper investigates the spectral properties of non-self-adjoint Schrödinger operators with complex potentials, demonstrating how complex dilations can estimate eigenvalues and reveal hypo-coercivity effects in 1D semigroups.

Contribution

It introduces a method using complex dilations to estimate eigenvalues of Schrödinger operators with specific complex potentials, highlighting hypo-coercivity phenomena.

Findings

Eigenvalues' real parts grow as |b3|^{2/(b4+2)} for the given potential form

Complex dilation technique effectively estimates eigenvalues of non-self-adjoint operators

Hypo-coercivity manifests as increased contraction rates in the semigroup

Abstract

The phenomenon "hypo-coercivity," i.e., the increased rate of contraction for a semi-group upon adding a large skew-adjoint part to the generator, is considered for 1D semigroups generated by the Schr\"odinger operators $-\partial^2_x + x^2 + i{\gamma} f (x)$ with a complex potential. For $f$ of the special form$ f (x) = 1/(1 + |x|^\kappa)$, it is shown using complex dilations that the real part of eigenvalues of the operator are larger than a constant times $|\gamma|^{2/(\kappa+2)}$.