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SIGMA 3 (2007), 031, 18 pages math-ph/0702089
https://doi.org/10.3842/SIGMA.2007.031
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
Abstract
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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