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Bifurcations of Essential Spectra Generated by a Small Non-Hermitian Hole. II. Eigenvalues and Resonances

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Abstract

The paper focuses on bifurcations that occur in the essential spectrum of certain non-Hermitian operators. We consider an eigenvalue problem for an elliptic differential operator in a multidimensional tube-like domain which is infinite along one dimension and can be bounded or unbounded in other dimensions. This self-adjoint eigenvalue problem is perturbed by a small hole cut out of the domain. The boundary of the hole is described by a non-Hermitian Robin-type boundary condition. Our main result consists in sufficient conditions ensuring that this singular and non-Hermitian perturbation results in discrete eigenvalues or resonances bifurcating either from the edge or from certain internal points of the essential spectrum of the unperturbed problem. The location of the bifurcating eigenvalues and resonances is described in terms of asymptotic expansions with respect to the small linear size of the hole. Several illustrative examples are discussed.

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References

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Funding

The research is supported by the Russian Science Foundation (grant No. 20-11-19995).

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Authors and Affiliations

  1. Institute of Mathematics, Ufa Federal Research Center, Russian Academy of Sciences, Ufa, Russia

    D. I. Borisov & D. A. Zezyulin

  2. ITMO University, St.-Petersburg, 197101, Russia

    D. A. Zezyulin

Authors
  1. D. I. Borisov
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  2. D. A. Zezyulin
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Correspondence to D. I. Borisov or D. A. Zezyulin.

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Borisov, D.I., Zezyulin, D.A. Bifurcations of Essential Spectra Generated by a Small Non-Hermitian Hole. II. Eigenvalues and Resonances. Russ. J. Math. Phys. 29, 321–341 (2022). https://doi.org/10.1134/S1061920822030037

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  • DOI: https://doi.org/10.1134/S1061920822030037

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