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

Antonio Arnal; Petr Siegl

doi:10.1016/j.jfa.2023.109856

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arXiv · 2203.15938

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

Antonio Arnal·Petr Siegl

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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.

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BibTeX RIS

Antonio Arnal, Petr Siegl. 2022-03-29. Resolvent estimates for one-dimensional Schr\"odinger operators with complex potentials. https://doi.org/10.1016/j.jfa.2023.109856

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