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Resolutions of Identity for Some Non-Hermitian Hamiltonians. I. Exceptional Point in Continuous Spectrum
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- Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)
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- Symmetry, Integrability and Geometry: Methods and Applications, 2011, том 7
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Resolutions of Identity for Some Non-Hermitian Hamiltonians. I. Exceptional Point in Continuous Spectrum
Інший ID:
2010 Mathematics Subject Classification: 81Q60; 81R15; 47B15
DOI: http://dx.doi.org/10.3842/SIGMA.2011.111
DOI: http://dx.doi.org/10.3842/SIGMA.2011.111
Посилання:
Resolutions of Identity for Some Non-Hermitian Hamiltonians. I. Exceptional Point in Continuous Spectrum / A.A. Andrianov, A.V. Sokolov // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 29 назв. — англ.
Підтримка:
This paper is a contribution to the Proceedings of the Workshop “Supersymmetric Quantum Mechanics and Spectral Design” (July 18–30, 2010, Benasque, Spain). The full collection is available at http://www.emis.de/journals/SIGMA/SUSYQM2010.html.
This work was supported by Grant RFBR 09-01-00145-a and by the SPbSU project 11.0.64.2010. The work of A.A. was also supported by grants 2009SGR502, FPA2007-66665 and by the Consolider-Ingenio 2010 Program CPAN (CSD2007-00042).
Дата:
2011
Переглядів:
472
Завантажень:
227
Анотація:
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.
