Abstract
This paper focuses on bifurcations that occur in the essential spectrum of certain non-Hermitian operators. We consider the eigenvalue problem for a self-adjoint 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. The main result of the present paper states the existence and describes the properties of local meromorphic continuations of the resolvent of the operator in question through the essential spectrum. The continuations are constructed near the edge of the spectrum and in the vicinity of certain internal threshold points of the spectrum. Then we define the eigenvalues and resonances of the operator as the poles of these continuations and prove that both the edge and the internal thresholds bifurcate into eigenvalues and/or resonances. The total multiplicity of the eigenvalues and resonances bifurcating from internal thresholds can be up to twice larger than the multiplicity of the thresholds. In other words, the perturbation can increase the total multiplicity.
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References
C. M. Bender, “Making Sense of Non-Hermitian Hamiltonians”, Rep. Progr. Phys., 70 (2007), 947–1018.
R. El-Ganainy, K. G. Makris, M. Khajavikhan, Z. H. Musslimani, S. Rotter, and D. N. Christodoulides, “Non-Hermitian Physics and PT Symmetry”, Nature Physics, 14 (2018), 11–19.
R. El-Ganainy, M. Khajavikhan, D. N. Christodoulides, and S. K. Ozdemir, “The Dawn of Non-Hermitian Optics”, Communications Physics, 2 (2019), 37.
X. Zhu, H. Ramezani, C. Shi, J. Zhu, and X. Zhang, “\( {PT} \)-Symmetric Acoustics”, Phys. Rev. X, 4 (2014), 031042.
J. Z. Song, P. Bai, Z. H. Hang, and Y. Lai, “Acoustic Coherent Perfect Absorbers”, New J. Phys., 16 (2014), 033026.
R. Livi, R. Franzosi, and G.-L. Oppo, “Self-Localization of Bose–Einstein Condensates in Optical Lattices Via Boundary Dissipation”, Phys. Rev. Lett., 97 (2006), 060401.
A. Müllers, B. Santra, C. Baals, J. Jiang, J. Benary, R. Labouvie, D. A. Zezyulin, V. V. Konotop, and H. Ott, “Coherent Perfect Absorption of Nonlinear Matter Waves”, Sci. Adv., 4 (2018), eaat6539.
V. V. Konotop, J. Yang, and D. A. Zezyulin, “Nonlinear Waves in \( {PT} \)-Symmetric Systems”, Rev. Mod. Phys., 88 (2016), 035002.
W. D. Heiss, “The Physics of Exceptional Points”, J. Phys. A: Math. Theor., 45 (2011), 444016.
A. Mostafazadeh, Physics of Spectral Singularities, in Geometric Methods in Physics (Birkhäuser, Cham), 2015.
J. Yang, “Classes of Non-Parity-Time-Symmetric Optical Potentials with Exceptional-Point-Free Phase Transitions”, Opt. Lett., 42 (2017), 4067–4070.
V. V. Konotop and D. A. Zezyulin, “Phase Transition through the Splitting of Self-Dual Spectral Singularity in Optical Potentials”, Opt. Lett., 42 (2017), 5206–5209.
D. I. Borisov, D. A. Zezyulin, and M. Znojil, “Bifurcations of Thresholds in Essential Spectra of Elliptic Operators under Localized Non-Hermitian Perturbations”, Stud. Appl. Math., 146 (2021), 834–880.
Y. V. Kartashov, C. Milián, V. V. Konotop, and L. Torner, “Bound States in the Continuum in a Two-Dimensional PT-Symmetric System”, Opt. Lett., 43 (2018), 575–578.
B. Midya and V. V. Konotop, “Waveguides with Absorbing Boundaries: Nonlinearity Controlled by an Exceptional Point and Solitons”, Phys. Rev. Lett., 119 (2017), 033905.
T. B. A. Senior, “Impedance Boundary Conditions for Imperfectly Conducting Surfaces”, Appl. Sci. Res., 8 (1960), 418.
L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media, Pergamon Press Ltd, New York, 1963.
T. B. A. Senior and J. L. Volakis, Approximate Boundary Conditions in Electromagnetics, IEE Press, 1995.
D. A. Hill, Electromagnetic Fields in Cavities. Deterministic and Statistical Theories, John Wiley & Sons, Inc., New York, 2009.
D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory. Fourth Edition, Springer, 2019.
H.-C. Kaiser, H. Neidhardt, and J. Rehberg, “Macroscopic Current Induced Boundary Conditions for Schrödinger-Type Operators”, Integral. Equations Operator Theory, 45 (2003), 39–63.
H. Ammari, A. Dabrowski, B. Fitzpatrick, and P. Millien, “Perturbation of the Scattering Resonances of an Open Cavity by Small Particles. Part I: the Transverse Magnetic Polarization Case”, Z. Angew. Math. Phys., 71 (2020), 102.
S. Longhi, “Bound States in the Continuum in \( {PT} \)-Symmetric Optical Lattices”, Opt. Lett., 39 (2014), 1697–1700.
A. M. Il’in, Matching of Asymptotic Expansions of Solutions of Boundary Value Problems, Amer. Math. Soc., Providence, RI, 1992.
V. G. Maz’ya, S. A. Nazarov, and B. A. Plamenevskii, “Asymptotic Expansions of the Eigenvalues of Boundary Value Problems for the Laplace Operator in Domains with Small Holes”, Math. USSR-Izv., 24 (1985), 321–345.
V. G. Maz’ya, S. A. Nazarov, and B. A. Plamenevskii, Asymptotic Theory of Elliptic Boundary Value Problems in Singularly Perturbed Domains, Birkhäuser, Basel, 2000.
D. Borisov, “On a \({PT}\)-Symmetric Waveguide with a Pair of Small Holes”, Proceedings of Steklov Institute of Mathematics, 281:1 (2013), 5–21.
R. R. Gadyl’shin, “On Regular and Singular Perturbations of Acoustic and Quantum Waveguides”, C.R. Méch., 332 (2004), 647–652.
S. A. Nazarov, “Variational and Asymptotic Methods for Finding Eigenvalues below the Continuous Spectrum Threshold”, Siberian Math. J., 51 (2010), 866–878.
S. A. Nazarov, “Asymptotics of Trapped Modes and Eigenvalues below the Continuous Spectrum of a Waveguide with a Thin Shielding Obstacle”, St. Petersburg Math. J., 23 (2012), 571–601.
D. Borisov, “Asymptotic Behaviour of the Spectrum of a Waveguide with Distant Perturbation”, Math. Phys. Anal. Geom., 10 (2007), 155–196.
D. I. Borisov, “Distant Perturbations of the Laplacian in a Multi-Dimensional Space”, Ann. Henri Poincaré, 8 (2007), 1371–1399.
D. Borisov and D. Krejčiřík, “The Effective Hamiltonian for Thin Layers with Non-Hermitian Robin-Type Boundary Conditions”, Asymptot. Anal., 76 (2012), 49–59.
D. Borisov, “Discrete Spectrum of Thin \({PT}\)-Symmetric Waveguide”, Ufa Math. J., 6 (2014), 29–55.
D. Borisov, “Emergence of Eigenvalue for \({PT}\)-Symmetric Operator in Thin Strip”, Math. Notes, 98 (2015), 827–883.
D. I. Borisov and M. Znojil, “On Eigenvalues of a \({PT}\)-Symmetric Operator in a Thin Layer”, Sb. Math., 208 (2017), 173–199.
D. I. Borisov and A. I. Mukhametrakhimova, “On Norm Resolvent Convergence for Elliptic Operators in Multi-Dimensional Domains with Small Holes”, J. Math. Sci., 232 (2018), 283–298.
D. Borisov, G. Cardone, and T. Durante, “Homogenization and Uniform Resolvent Convergence for Elliptic Operators in a Strip Perforated along a Curve”, Proc. Roy. Soc. Edinburgh Sec. A Math., 146 (2016), 1115–1158.
A. Majda, “Outgoing Solutions for Perturbation of \(- \Delta \) with Applications to Spectral and Scattering Theory”, J. Diff. Equat., 16 (1974), 515–547.
D. Borisov, “Discrete Spectrum of a Pair of Non-Symmetric Waveguides Coupled by a Window”, Sb. Math., 197 (2006), 475–504.
D. Borisov, “On Spectrum of Two-Dimensional Periodic Operator with Small Localized Perturbation”, Izv. Math., 75 (2011), 471–505.
D. I. Borisov and D. A. Zezyulin, “Bifurcations of Essential Spectra Generated by a Small Non-Hermitian Hole. II. Eigenvalues and Resonances”, Russ. J. Math. Phys., :(submitted).
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The research is supported by the Russian Science Foundation (grant No. 20-11-19995).
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Borisov, D.I., Zezyulin, D.A. Bifurcations of Essential Spectra Generated by a Small Non-Hermitian Hole. I. Meromorphic Continuations. Russ. J. Math. Phys. 28, 416–433 (2021). https://doi.org/10.1134/S1061920821040026
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DOI: https://doi.org/10.1134/S1061920821040026