Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 7 (2011), 111, 19 pages      arXiv:1107.5911
Contribution to the Proceedings of the Workshop “Supersymmetric Quantum Mechanics and Spectral Design”

Resolutions of Identity for Some Non-Hermitian Hamiltonians. I. Exceptional Point in Continuous Spectrum

Alexander A. Andrianov a, b and Andrey V. Sokolov a
a) V.A. Fock Department of Theoretical Physics, Sankt-Petersburg State University, 198504 St. Petersburg, Russia
b) ICCUB, Universitat de Barcelona, 08028 Barcelona, Spain

Received August 06, 2011, in final form November 25, 2011; Published online December 05, 2011

Resolutions of identity for certain non-Hermitian Hamiltonians constructed from biorthogonal sets of their eigen- and associated functions are given for the spectral problem defined on entire axis. Non-Hermitian Hamiltonians under consideration possess the continuous spectrum and the following peculiarities are investigated: (1) the case when there is an exceptional point of arbitrary multiplicity situated on a boundary of continuous spectrum; (2) the case when there is an exceptional point situated inside of continuous spectrum. The reductions of the derived resolutions of identity under narrowing of the classes of employed test functions are revealed. It is shown that in the case (1) some of associated functions included into the resolution of identity are normalizable and some of them may be not and in the case (2) the bounded associated function corresponding to the exceptional point does not belong to the physical state space. Spectral properties of a SUSY partner Hamiltonian for the Hamiltonian with an exceptional point are examined.

Key words: non-Hermitian quantum mechanics; supersymmetry; exceptional points; resolution of identity.

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