Resolutions of Identity for Some Non-Hermitian Hamiltonians. I. Exceptional Point in Continuous Spectrum
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 pecul...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2011 |
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| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2011
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/148088 |
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| Cite this: | 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. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862703859120472064 |
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| author | Andrianov, A.A. Sokolov, A.V. |
| author_facet | Andrianov, A.A. Sokolov, A.V. |
| citation_txt | 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. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | 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.
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| first_indexed | 2025-12-07T16:49:45Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-148088 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T16:49:45Z |
| publishDate | 2011 |
| publisher | Інститут математики НАН України |
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| spelling | Andrianov, A.A. Sokolov, A.V. 2019-02-16T20:50:51Z 2019-02-16T20:50:51Z 2011 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. 1815-0659 2010 Mathematics Subject Classification: 81Q60; 81R15; 47B15 DOI: http://dx.doi.org/10.3842/SIGMA.2011.111 https://nasplib.isofts.kiev.ua/handle/123456789/148088 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. 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). en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Resolutions of Identity for Some Non-Hermitian Hamiltonians. I. Exceptional Point in Continuous Spectrum Article published earlier |
| spellingShingle | Resolutions of Identity for Some Non-Hermitian Hamiltonians. I. Exceptional Point in Continuous Spectrum Andrianov, A.A. Sokolov, A.V. |
| title | Resolutions of Identity for Some Non-Hermitian Hamiltonians. I. Exceptional Point in Continuous Spectrum |
| title_full | Resolutions of Identity for Some Non-Hermitian Hamiltonians. I. Exceptional Point in Continuous Spectrum |
| title_fullStr | Resolutions of Identity for Some Non-Hermitian Hamiltonians. I. Exceptional Point in Continuous Spectrum |
| title_full_unstemmed | Resolutions of Identity for Some Non-Hermitian Hamiltonians. I. Exceptional Point in Continuous Spectrum |
| title_short | Resolutions of Identity for Some Non-Hermitian Hamiltonians. I. Exceptional Point in Continuous Spectrum |
| title_sort | resolutions of identity for some non-hermitian hamiltonians. i. exceptional point in continuous spectrum |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/148088 |
| work_keys_str_mv | AT andrianovaa resolutionsofidentityforsomenonhermitianhamiltoniansiexceptionalpointincontinuousspectrum AT sokolovav resolutionsofidentityforsomenonhermitianhamiltoniansiexceptionalpointincontinuousspectrum |