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Algebraic integrability for the Schrödinger equation and finite reflection groups. (English. Russian original) Zbl 0805.47070
Theor. Math. Phys. 94, No. 2, 182-197 (1993); translation from Teor. Mat. Fiz. 94, No. 2, 253-275 (1993).
Summary: Algebraic integrability of an $$n$$-dimensional Schrödinger equation means that it has more than $$n$$ independent quantum integrals. For $$n=1$$, the problem of describing such equations arose in the theory of finite- gap potentials. The present paper gives a construction which associates finite reflection groups (in particular, Weyl groups of simple Lie algebras) with algebraically integrable multidimensional Schrödinger equations. These equations correspond to special values of the parameters in the generalization of the Calogero-Sutherland system proposed by Olshanetsky and Perelomov. The analytic properties of a joint eigenfunction of the corresponding commutative rings of differential operators are described. Explicit expressions are obtained for the solution of the quantum Calogero-Sutherland problem for a special value of the coupling constant.

##### MSC:
 47N50 Applications of operator theory in the physical sciences 47F05 General theory of partial differential operators (should also be assigned at least one other classification number in Section 47-XX) 81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
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