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

Edwin Langmann
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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.

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Edwin Langmann. 2007-02-26. Singular Eigenfunctions of Calogero-Sutherland Type Systems and How to Transform Them into Regular Ones. https://doi.org/10.3842/sigma.2007.031

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