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Generalized Airy operators
Published online by Cambridge University Press: 01 August 2025
Abstract
We study the behaviour of the norm of the resolvent for non-self-adjoint operators of the form 
$A := -\partial_x + W(x)$, with 
$W(x) \ge 0$, defined in 
${L^2}({\mathbb{R}})$. We provide a sharp estimate for the norm of its resolvent operator, 
$\| (A - \lambda)^{-1} \|$, as the spectral parameter diverges 
$(\lambda \to +\infty)$. Furthermore, we describe the C0-semigroup generated by −A and determine its norm. Finally, we discuss the applications of the results to the asymptotic description of pseudospectra of Schrödinger and damped wave operators, and also the optimality of abstract resolvent bounds based on Carleman-type estimates.
Keywords
MSC classification
Primary:
34L05: General spectral theory
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- Type
- Research Article
- Information
- Copyright
- © The Author(s), 2025. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh
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