Resolvent estimates for one-dimensional Schrödinger operators with complex potentials

Abstract

We study one-dimensional Schrödinger operators $H=-{\partial }_x^2+V$ with unbounded complex potentials $V$ and derive asymptotic estimates for the norm of the resolvent, $\Psi (\lambda ):=\Vert (H-\lambda {)}^{-1}\Vert$, as $|\lambda |\to +\infty$, separately considering $\lambda \in \mathrm{Ran}V$ 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.

Added a new section (5.1.2) on resolvent norm estimates along curves adjacent to the real axis. Minor additions/corrections throughout the text to enhance accuracy/clarity. Added some more references. Extended the notation section. Small corrections in the formulas in section 7.1

References in corpus (1)

• Pseudomodes for non-self-adjoint Dirac operators

Cited by in corpus (4)

• Spectral analysis of metamaterials in curved manifolds
• Resolvent estimates for the one-dimensional damped wave equation with unbounded damping
• The relativistic rotated harmonic oscillator
• Generalised Airy Operators