August 2011
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Day August 1, 2011

Resolutions of identity for some non-Hermitian Hamiltonians II: proofs

A.V. Sokolov

This part is a continuation of the Part I where we built resolutions of identity for certain non-Hermitian Hamiltonians constructed from biorthogonal sets of their eigen- and associated functions for the spectral problem defined on entire axis. Non-Hermitian Hamiltonians under consideration were taken with continuous spectrum and the following cases were examined: an exceptional point situated on a boundary of continuous spectrum and an exceptional point situated inside of continuous spectrum. In the present work the rigorous proofs are given for the resolutions of identity in both cases.

http://arxiv.org/abs/1107.5916
Mathematical Physics (math-ph); High Energy Physics – Theory (hep-th); Nuclear Theory (nucl-th); Quantum Physics (quant-ph)

Resolutions of identity for some non-Hermitian Hamiltonians I: exceptional point in continuous spectrum

A.A. Andrianov, A.V. Sokolov

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 are taken with continuous spectrum and the following peculiarities are investigated: (1) the case when there is an exceptional point 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.

http://arxiv.org/abs/1107.5911
Mathematical Physics (math-ph); High Energy Physics – Theory (hep-th); Nuclear Theory (nucl-th); Quantum Physics (quant-ph)