Singular Eigenfunctions of Calogero-Sutherland Type Systems and How to Transform Them into Regular Ones

• Published in: Symmetry, Integrability and Geometry: Methods and Applications
• Date: 2007
• Main Author: Langmann, E.
• Format: Article
• Language: English
• Published: Інститут математики НАН України 2007
• Online Access: https://nasplib.isofts.kiev.ua/handle/123456789/147783
• 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.

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.