1D Schr\"odinger operators with complex potentials

TL;DR

This paper studies 1D Schrödinger operators with complex potentials, deriving trace formulas that include a new singular measure term and providing estimates relating eigenvalues and potential norms.

Contribution

It introduces a novel trace formula for complex potentials, incorporating a singular measure absent in real potential cases, and offers bounds on eigenvalues based on potential norms.

Findings

Derived new trace formulas with singular measure term

Estimated sum of eigenvalue imaginary parts using potential norms

Extended classical Hardy space results to complex potential analysis

Abstract

We consider a Schr\"odinger operator with complex-valued potentials on the line. The operator has essential spectrum on the half-line plus eigenvalues (counted with algebraic multiplicity) in the complex plane without the positive half-line. We determine series of trace formulas. Here we have the new term: a singular measure, which is absent for real potentials. Moreover, we estimate of sum of Im part of eigenvalues plus singular measure in terms of the norm of potentials. The proof is based on classical results about the Hardy spaces.