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SCGP Weekly Talk: Alexander Turbiner, Nuclear Science Institute, National Autonomous University of Mexico

Title: Solvable Schroedinger equations and representation theory
Speaker: Alexander Turbiner, Nuclear Science Institute, National Autonomous University of Mexico
Date: Tuesday, May 12, 2015
Time: 01:00pm – 02:00pm
Place: Lecture Hall 102, Simons Center

 

 

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Abstract: For any quantum Calogero-Sutherland (CS) Hamiltonian (rational, trigonometric, elliptic) for any root system:

(I) one can indicate a change of variables in which its potential is a rational function and contravariant (flat) metric has polynomial entries;

(II) there exists a similarity transformation (gauge rotation) of the Hamiltonian after which it becomes the 2nd order differential, algebraic operator with polynomial coefficient functions;

(III) the algebraic operator has a finite-dimensional invariant subspace in polynomials which is a representation space of an algebra of differential operators (for $A_n$ and $BC_n$ cases this is the $gl(n+1)$-algebra)

In one-dimensional case ($A_1$ and $BC_1$ CS models) the algebraic operator is the (restricted) Heun operator (for elliptic) and the Riemann operator (for rational and trigonometric).