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

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 Added by Edwin Langmann
 Publication date 2007
  fields Physics
and research's language is English




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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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An interesting observation was reported by Corrigan-Sasaki that all the frequencies of small oscillations around equilibrium are quantised for Calogero and Sutherland (C-S) systems, typical integrable multi-particle dynamics. We present an analytic proof by applying recent results of Loris-Sasaki. Explicit forms of `classical and quantum eigenfunctions are presented for C-S systems based on any root systems.
Complete description of the singular sectors of the 1-layer Benney system (classical long wave equation) and dToda system is presented. Associated Euler-Poisson-Darboux equations E(1/2,1/2) and E(-1/2,-1/2) are the main tool in the analysis. A complete list of solutions of the 1-layer Benney system depending on two parameters and belonging to the singular sector is given. Relation between Euler-Poisson-Darboux equations E(a,a) with opposite sign of a is discussed.
323 - V. Prokofev , A. Zabrodin 2019
We consider solutions of the matrix KP hierarchy that are trigonometric functions of the first hierarchical time $t_1=x$ and establish the correspondence with the spin generalization of the trigonometric Calogero-Moser system on the level of hierarchies. Namely, the evolution of poles $x_i$ and matrix residues at the poles $a_i^{alpha}b_i^{beta}$ of the solutions with respect to the $k$-th hierarchical time of the matrix KP hierarchy is shown to be given by the Hamiltonian flow with the Hamiltonian which is a linear combination of the first $k$ higher Hamiltonians of the spin trigonometric Calogero-Moser system with coordinates $x_i$ and with spin degrees of freedom $a_i^{alpha}, , b_i^{beta}$. By considering evolution of poles according to the discrete time matrix KP hierarchy we also introduce the integrable discrete time version of the trigonometric spin Calogero-Moser system.
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