Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 3 (2007), 031, 18 pages      math-ph/0702089
Contribution to the Vadim Kuznetsov Memorial Issue

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

Edwin Langmann
Theoretical Physics, KTH Physics, AlbaNova, SE-106 91 Stockholm, Sweden

Received November 02, 2006, in final form January 29, 2007; Published online February 26, 2007

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.

Key words: quantum integrable systems; orthogonal polynomials; singular eigenfunctions.

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