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Spectra of definite type in waveguide models


Authors: Vladimir Lotoreichik and Petr Siegl
Journal: Proc. Amer. Math. Soc. 145 (2017), 1231-1246
MSC (2010): Primary 47A55, 47B50, 81Q12
DOI: https://doi.org/10.1090/proc/13316
Published electronically: November 21, 2016
MathSciNet review: 3589322
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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.


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Additional Information

Vladimir Lotoreichik
Affiliation: Nuclear Physics Institute CAS, 25068 Řež, Czech Republic
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)
Email: petr.siegl@math.unibe.ch

DOI: https://doi.org/10.1090/proc/13316
Keywords: Spectral points of definite and of type $\pi$, weakly coupled bound states, perturbations of essential spectrum, pseudospectrum, $\mathcal{PT}$-symmetric waveguide
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
Article copyright: © Copyright 2016 American Mathematical Society