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Singular Eigenfunctions of Calogero-Sutherland Type Systems and How to Transform Them into Regular Ones
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- Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)
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- Symmetry, Integrability and Geometry: Methods and Applications, 2007, том 3
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Singular Eigenfunctions of Calogero-Sutherland Type Systems and How to Transform Them into Regular Ones
Інший ID:
2000 Mathematics Subject Classification: 81U15; 33C50; 05E05
Посилання:
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.
Підтримка:
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).
Дата:
2007
Переглядів:
401
Завантажень:
216
Анотація:
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.
