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Integrability in the theory of Schrödinger operator and harmonic analysis. (English) Zbl 0767.35066
Summary: The algebraic integrability for the Schrödinger equation in \(\mathbb{R}^ n\) and the role of the quantum Calogero-Sutherland problem and root systems in this context are discussed. For the special values of the parameters in the potential the explicit formula for the eigenfunction of the corresponding Sutherland operator is found. As an application the explicit formula for the zonal spherical functions on the symmetric spaces \(SU_{2n}^*/Sp_ n\) (type A II in Cartan notations) is presented.

35Q40 PDEs in connection with quantum mechanics
43A90 Harmonic analysis and spherical functions
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
Full Text: DOI
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