Generalised Airy Operators

Antonio Arnal; Petr Siegl

doi:10.1017/prm.2025.10056

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arXiv · 2208.14389

Generalised Airy Operators

Antonio Arnal·Petr Siegl

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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 $C_0$-semigroup generated by $-A$ and determine its norm. Finally, we discuss the applications of the results to the asymptotic description of pseudospectra of Schr\"odinger and damped wave operators and also the optimality of abstract resolvent bounds based on Carleman-type estimates.

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BibTeX RIS

Antonio Arnal, Petr Siegl. 2022-08-30. Generalised Airy Operators. https://doi.org/10.1017/prm.2025.10056

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