Singular Eigenfunctions of Calogero-Sutherland Type Systems and How to Transform Them into Regular Ones
There exists a large class of quantum many-body systems of Calogero-Sutherland type where all particles can have different masses and coupling constants and which nevertheless are such that one can construct a complete (in a certain sense) set of exact eigenfunctions and corresponding eigenvalues, e...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2007 |
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| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2007
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/147783 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Singular Eigenfunctions of Calogero-Sutherland Type Systems and How to Transform Them into Regular Ones / E. Langmann // Symmetry, Integrability and Geometry: Methods and Applications. — 2007. — Т. 3. — Бібліогр.: 26 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862710108467757056 |
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| author | Langmann, E. |
| author_facet | Langmann, E. |
| citation_txt | Singular Eigenfunctions of Calogero-Sutherland Type Systems and How to Transform Them into Regular Ones / E. Langmann // Symmetry, Integrability and Geometry: Methods and Applications. — 2007. — Т. 3. — Бібліогр.: 26 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | There exists a large class of quantum many-body systems of Calogero-Sutherland type where all particles can have different masses and coupling constants and which nevertheless are such that one can construct a complete (in a certain sense) set of exact eigenfunctions and corresponding eigenvalues, explicitly. Of course there is a catch to this result: if one insists on these eigenfunctions to be square integrable then the corresponding Hamiltonian is necessarily non-hermitean (and thus provides an example of an exactly solvable PT-symmetric quantum-many body system), and if one insists on the Hamiltonian to be hermitean then the eigenfunctions are singular and thus not acceptable as quantum mechanical eigenfunctions. The standard Calogero-Sutherland Hamiltonian is special due to the existence of an integral operator which allows to transform these singular eigenfunctions into regular ones.
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| first_indexed | 2025-12-07T17:22:12Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-147783 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T17:22:12Z |
| publishDate | 2007 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Langmann, E. 2019-02-16T08:08:16Z 2019-02-16T08:08:16Z 2007 Singular Eigenfunctions of Calogero-Sutherland Type Systems and How to Transform Them into Regular Ones / E. Langmann // Symmetry, Integrability and Geometry: Methods and Applications. — 2007. — Т. 3. — Бібліогр.: 26 назв. — англ. 1815-0659 2000 Mathematics Subject Classification: 81U15; 33C50; 05E05 https://nasplib.isofts.kiev.ua/handle/123456789/147783 There exists a large class of quantum many-body systems of Calogero-Sutherland type where all particles can have different masses and coupling constants and which nevertheless are such that one can construct a complete (in a certain sense) set of exact eigenfunctions and corresponding eigenvalues, explicitly. Of course there is a catch to this result: if one insists on these eigenfunctions to be square integrable then the corresponding Hamiltonian is necessarily non-hermitean (and thus provides an example of an exactly solvable PT-symmetric quantum-many body system), and if one insists on the Hamiltonian to be hermitean then the eigenfunctions are singular and thus not acceptable as quantum mechanical eigenfunctions. The standard Calogero-Sutherland Hamiltonian is special due to the existence of an integral operator which allows to transform these singular eigenfunctions into regular ones. This paper is a contribution to the Vadim Kuznetsov Memorial Issue ‘Integrable Systems and Related Topics’. I would like to thank Orlando Ragnisco for organizing and inviting me to several inspiring meetings in Rome and for showing me Ref. [26]. I thank Martin Halln¨as for useful comments on the manuscript and Alexander Veselov for reading the paper and several helpful comments. This work was supported by the Swedish Science Research Council (VR) and the European Union through the FP6 Marie Curie RTN ENIGMA (Contract number MRTN-CT-2004-5652). en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Singular Eigenfunctions of Calogero-Sutherland Type Systems and How to Transform Them into Regular Ones Article published earlier |
| spellingShingle | Singular Eigenfunctions of Calogero-Sutherland Type Systems and How to Transform Them into Regular Ones Langmann, E. |
| title | Singular Eigenfunctions of Calogero-Sutherland Type Systems and How to Transform Them into Regular Ones |
| title_full | Singular Eigenfunctions of Calogero-Sutherland Type Systems and How to Transform Them into Regular Ones |
| title_fullStr | Singular Eigenfunctions of Calogero-Sutherland Type Systems and How to Transform Them into Regular Ones |
| title_full_unstemmed | Singular Eigenfunctions of Calogero-Sutherland Type Systems and How to Transform Them into Regular Ones |
| title_short | Singular Eigenfunctions of Calogero-Sutherland Type Systems and How to Transform Them into Regular Ones |
| title_sort | singular eigenfunctions of calogero-sutherland type systems and how to transform them into regular ones |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/147783 |
| work_keys_str_mv | AT langmanne singulareigenfunctionsofcalogerosutherlandtypesystemsandhowtotransformthemintoregularones |