Resolvent estimates for one-dimensional Schr\"odinger operators with complex potentials

TL;DR

This paper provides precise asymptotic estimates for the resolvent norm of one-dimensional Schrödinger operators with complex potentials, revealing optimal bounds and their implications for spectral analysis.

Contribution

The paper derives explicit asymptotic estimates for the resolvent norm of Schrödinger operators with complex potentials, including leading order terms and remainders, extending previous results.

Findings

Resolvant norm estimates are optimal and explicit.

Asymptotic behavior differs for spectral and non-spectral parameters.

Extensions connect results to semigroup theory and examples.

Abstract

We study one-dimensional Schr\"odinger operators $\operatorname{H} = -\partial_x^2 + V$ with unbounded complex potentials $V$ and derive asymptotic estimates for the norm of the resolvent, $\Psi(\lambda) := \| (\operatorname{H} - \lambda)^{-1} \|$, as $|\lambda| \to +\infty$, separately considering $\lambda \in \operatorname{Ran} V$ and $\lambda \in \mathbb{R}_+$. In each case, our analysis yields an exact leading order term and an explicit remainder for $\Psi(\lambda)$ and we show these estimates to be optimal. We also discuss several extensions of the main results, their interrelation with some aspects of semigroup theory and illustrate them with examples.