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Dynamics of Resonances for 0th Order Pseudodifferential Operators

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Abstract

We study the dynamics of resonances of analytic perturbations of 0th order pseudodifferential operators P(s). In particular, we prove a Fermi golden rule for resonances of P(s) at embedded eigenvalues of \(P=P(0)\). We answer the question on the generic absence of eigenvalues asked by Colin de Verdière (Anal PDE 13:1521–1537, 2020). We also study the dynamics of eigenvalues of \(P+it\Delta \) as the eigenvalues converge to simple eigenvalues of P. The 0th order pseudodifferential operators we consider satisfy natural dynamical assumptions and are used as microlocal models of internal waves.

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Acknowledgements

I would like to thank Maciej Zworski for suggesting this problem, for helpful advice and for his help in Matlab experiments. I would like the referees for careful reading of this paper and for their suggestions in improving the writing of the paper. Partial support by the National Science Fundation grant DMS-1952939 is also gratefully acknowledged.

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  1. Department of Mathematics, University of California, Berkeley, CA, 94720, USA

    Jian Wang

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  1. Jian Wang
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Correspondence to Jian Wang.

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Communicated by S. Dyatlov.

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Wang, J. Dynamics of Resonances for 0th Order Pseudodifferential Operators. Commun. Math. Phys. 391, 643–668 (2022). https://doi.org/10.1007/s00220-022-04327-8

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