Spectra of definite type in waveguide models
HTML articles powered by AMS MathViewer
- by Vladimir Lotoreichik and Petr Siegl PDF
- Proc. Amer. Math. Soc. 145 (2017), 1231-1246 Request permission
Abstract:
We develop an abstract method to identify spectral points of definite type in the spectrum of the operator $T_1\otimes I_2 + I_1\otimes T_2$, applicable in particular for non-self-adjoint waveguide type operators with symmetries. Using the remarkable properties of the spectral points of definite type, we obtain new results on realness of weakly coupled bound states and of low lying essential spectrum in the $\mathcal {P}\mathcal {T}$-symmetric waveguide. Moreover, we show that the pseudospectrum has a tame behavior near the low lying essential spectrum and exclude the accumulation of non-real eigenvalues from this part of the essential spectrum. The advantage of our approach is particularly visible when the resolvent of the unperturbed operator cannot be explicitly expressed and most of the mentioned conclusions are extremely hard to prove by direct methods.References
- T. Ya. Azizov and I. S. Iokhvidov, Linear operators in spaces with an indefinite metric, Pure and Applied Mathematics (New York), John Wiley & Sons, Ltd., Chichester, 1989. Translated from the Russian by E. R. Dawson; A Wiley-Interscience Publication. MR 1033489
- T. Ya. Azizov, J. Behrndt, P. Jonas, and C. Trunk, Spectral points of definite type and type $\pi$ for linear operators and relations in Krein spaces, J. Lond. Math. Soc. (2) 83 (2011), no. 3, 768–788. MR 2802510, DOI 10.1112/jlms/jdq098
- T. Ya. Azizov, M. Denisov, and F. Philipp, Spectral functions of products of selfadjoint operators, Math. Nachr. 285 (2012), no. 14-15, 1711–1728. MR 2988002, DOI 10.1002/mana.201200169
- Tomas Ya. Azizov, Peter Jonas, and Carsten Trunk, Spectral points of type $\pi _+$ and $\pi _-$ of self-adjoint operators in Krein spaces, J. Funct. Anal. 226 (2005), no. 1, 114–137. MR 2158177, DOI 10.1016/j.jfa.2005.03.009
- Jiří Blank, Pavel Exner, and Miloslav Havlíček, Hilbert space operators in quantum physics, 2nd ed., Theoretical and Mathematical Physics, Springer, New York; AIP Press, New York, 2008. MR 2458485
- Denis Borisov and David Krejčiřík, The effective Hamiltonian for thin layers with non-Hermitian Robin-type boundary conditions, Asymptot. Anal. 76 (2012), no. 1, 49–59. MR 2918878, DOI 10.3233/ASY-2011-1061
- Denis Borisov and David Krejčiřík, $\scr P\scr T$-symmetric waveguides, Integral Equations Operator Theory 62 (2008), no. 4, 489–515. MR 2470121, DOI 10.1007/s00020-008-1634-1
- D. E. Edmunds and W. D. Evans, Spectral theory and differential operators, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1987. Oxford Science Publications. MR 929030
- Peter Jonas, On locally definite operators in Krein spaces, Spectral analysis and its applications, Theta Ser. Adv. Math., vol. 2, Theta, Bucharest, 2003, pp. 95–127. MR 2082429
- Tosio Kato, Perturbation theory for linear operators, Classics in Mathematics, Springer-Verlag, Berlin, 1995. Reprint of the 1980 edition. MR 1335452, DOI 10.1007/978-3-642-66282-9
- D. Krejčiřík, H. Bíla, and M. Znojil, Closed formula for the metric in the Hilbert space of a $\scr {PT}$-symmetric model, J. Phys. A 39 (2006), no. 32, 10143–10153. MR 2252706, DOI 10.1088/0305-4470/39/32/S15
- David Krejčiřík and Petr Siegl, $\scr {PT}$-symmetric models in curved manifolds, J. Phys. A 43 (2010), no. 48, 485204, 30. MR 2738140, DOI 10.1088/1751-8113/43/48/485204
- David Krejčiřík, Petr Siegl, and Jakub Železný, On the similarity of Sturm-Liouville operators with non-Hermitian boundary conditions to self-adjoint and normal operators, Complex Anal. Oper. Theory 8 (2014), no. 1, 255–281. MR 3147722, DOI 10.1007/s11785-013-0301-y
- D. Krejčiřík and M. Tater, Non-Hermitian spectral effects in a $\scr {PT}$-symmetric waveguide, J. Phys. A 41 (2008), no. 24, 244013, 14. With online multimedia enhancements. MR 2455811, DOI 10.1088/1751-8113/41/24/244013
- P. Lancaster, A. S. Markus, and V. I. Matsaev, Definitizable operators and quasihyperbolic operator polynomials, J. Funct. Anal. 131 (1995), no. 1, 1–28. MR 1343157, DOI 10.1006/jfan.1995.1080
- H. Langer, A. Markus, and V. Matsaev, Locally definite operators in indefinite inner product spaces, Math. Ann. 308 (1997), no. 3, 405–424. MR 1457739, DOI 10.1007/s002080050082
- Vladimir Lotoreichik and Jonathan Rohleder, Schatten-von Neumann estimates for resolvent differences of Robin Laplacians on a half-space, Spectral theory, mathematical system theory, evolution equations, differential and difference equations, Oper. Theory Adv. Appl., vol. 221, Birkhäuser/Springer Basel AG, Basel, 2012, pp. 453–468. MR 2954082, DOI 10.1007/978-3-0348-0297-0_{2}6
- William McLean, Strongly elliptic systems and boundary integral equations, Cambridge University Press, Cambridge, 2000. MR 1742312
- Radek Novák, Bound states in waveguides with complex Robin boundary conditions, Asymptot. Anal. 96 (2016), no. 3-4, 251–281. MR 3466467, DOI 10.3233/ASY-151338
- Friedrich Philipp and Carsten Trunk, Spectral points of type $\pi _+$ and type $\pi _-$ of closed operators in indefinite inner product spaces, Oper. Matrices 9 (2015), no. 3, 481–506. MR 3399333, DOI 10.7153/oam-09-30
- Michael Reed and Barry Simon, Methods of modern mathematical physics. IV. Analysis of operators, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR 0493421
- Konrad Schmüdgen, Unbounded self-adjoint operators on Hilbert space, Graduate Texts in Mathematics, vol. 265, Springer, Dordrecht, 2012. MR 2953553, DOI 10.1007/978-94-007-4753-1
- C. Trunk, Operator Theory, ch. “Locally Definitizable Operators: The Local Structure of the Spectrum”, Springer Basel, 2015, pp. 1–18.
Additional Information
- Vladimir Lotoreichik
- Affiliation: Nuclear Physics Institute CAS, 25068 Řež, Czech Republic
- MR Author ID: 904474
- Email: lotoreichik@ujf.cas.cz
- Petr Siegl
- Affiliation: Mathematisches Institut, Universität Bern, Alpeneggstr. 22, 3012 Bern, Switzerland (On leave from Nuclear Physics Institute CAS, 25068 Řež, Czech Republic)
- MR Author ID: 851879
- Email: petr.siegl@math.unibe.ch
- Received by editor(s): March 4, 2016
- Received by editor(s) in revised form: May 19, 2016
- Published electronically: November 21, 2016
- Additional Notes: The first author was supported by the Austrian Science Fund (FWF): Project P 25162-N26 and the Czech Science Foundation: Project 14-06818S
The second author was supported by SNSF Ambizione project PZ00P2_154786
Both the authors acknowledge the support by the Austria-Czech Republic co-operation grant CZ01/2013 - Communicated by: Michael Hitrik
- © Copyright 2016 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 145 (2017), 1231-1246
- MSC (2010): Primary 47A55, 47B50, 81Q12
- DOI: https://doi.org/10.1090/proc/13316
- MathSciNet review: 3589322