Spectra of definite type in waveguide models

Lotoreichik, Vladimir; Siegl, Petr

doi:10.1090/proc/13316

Cited by 4 publications

(2 citation statements)

References 22 publications

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“…Non-self-adjoint elliptic operators with Robin boundary conditions on such domains have been investigated recently in connection with non-Hermitian quantum waveguides and layers; see, e.g. [34,35,36,113]. Further, let…”

Section: Elliptic Operators With Non-local Robin Boundary Conditionsmentioning

confidence: 99%

Spectral enclosures for non-self-adjoint extensions of symmetric operators

Behrndt

Langer²,

Lotoreichik

et al. 2018

Journal of Functional Analysis

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The spectral properties of non-self-adjoint extensions A [B] of a symmetric operator in a Hilbert space are studied with the help of ordinary and quasi boundary triples and the corresponding Weyl functions. These extensions are given in terms of abstract boundary conditions involving an (in general non-symmetric) boundary operator B. In the abstract part of this paper, sufficient conditions for sectoriality and m-sectoriality as well as sufficient conditions for A [B] to have a non-empty resolvent set are provided in terms of the parameter B and the Weyl function. Special attention is paid to Weyl functions that decay along the negative real line or inside some sector in the complex plane, and spectral enclosures for A [B] are proved in this situation. The abstract results are applied to elliptic differential operators with local and non-local Robin boundary conditions on unbounded domains, to Schrödinger operators with δ-potentials of complex strengths supported on unbounded hypersurfaces or infinitely many points on the real line, and to quantum graphs with non-self-adjoint vertex couplings.

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Section: Elliptic Operators With Non-local Robin Boundary Conditionsmentioning

confidence: 99%

Spectral enclosures for non-self-adjoint extensions of symmetric operators

Behrndt

Langer²,

Lotoreichik

et al. 2018

Journal of Functional Analysis

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“…The essential spectrum for (44) cf. [34,39]. A numerical analysis of eigenvalues of (44) can be found in [8].…”

Section: 2mentioning

confidence: 99%

Bifurcation of eigenvalues in nonlinear problems with antilinear symmetry

Dohnal

Siegl

2016

Journal of Mathematical Physics

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Abstract. Many physical systems can be described by nonlinear eigenvalues and bifurcation problems with a linear part that is non-selfadjoint e.g. due to the presence of loss and gain. The balance of these effects is reflected in an antilinear symmetry, like e.g. the PT -symmetry, of the problem. Under this condition we show that the nonlinear eigenvalues bifurcating from real linear eigenvalues remain real and the corresponding nonlinear eigenfunctions remain symmetric. The abstract results are applied in a number of physical models of Bose-Einstein condensation, nonlinear optics and superconductivity, and further numerical analysis is performed.

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Hermitian-to-quasi-Hermitian quantum phase transitions

Znojil

2018

Phys. Rev. A

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The phenomenon of quantum phase transition is considered in the special case in which the evolution laws remain unitary and in which the bound-state energies remain observable. The conventional Hermiticity of observables is lost at the interface, replaced by the so called quasi-Hermiticity. Several features of the passage of the system through the interface are discussed and illustrated by elementary illustrative PT −symmetric examples.

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Spectra of definite type in waveguide models

Cited by 4 publications

References 22 publications

Spectral enclosures for non-self-adjoint extensions of symmetric operators

Spectral enclosures for non-self-adjoint extensions of symmetric operators

Bifurcation of eigenvalues in nonlinear problems with antilinear symmetry

Hermitian-to-quasi-Hermitian quantum phase transitions

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