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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.

MSC:
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.)
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[1] Dubrovin, B.A., Matveev, V.B., Novikov, S.P.: Nonlinear equation of KdV type, finite-zone linear operators and abelian varieties. Russ. Math. Surv.31, 51–125 (1976) · Zbl 0346.35025 · doi:10.1070/RM1976v031n01ABEH001446
[2] Chalykh, O.A., Veselov, A.P.: Commutative rings of partial differential operators and Lie algebras. Preprint of FIM (ETH, Zürich), 1988; Commun. Math. Phys.126, 597–611 (1990) · Zbl 0746.47025
[3] Krichever, I.M.: Methods of algebraic geometry in the theory of nonlinear equations. Russ. Math. Surv.32 (1977) · Zbl 0372.35002
[4] Feldman, J., Knörrer, H., Trubowitz, E.: There is no two-dimensional analogue of Lamé’s equation. Preprint of FIM (ETH, Zürich) 1991; Math. Ann.294, 295–324 (1992) · Zbl 0763.58027
[5] Sutherland, B.: Exact results for a quantum many-body problem in one-dimension. Phys. Rev.A4, 2019–2021 (1976); Phys. Rev.A5, 1372–1376 (1972) · doi:10.1103/PhysRevA.4.2019
[6] Calogero, F.: Solution of the one-dimensionaln-body problem with quadratic and/or inversely quadratic pair potentials. J. Math. Phys.12, 419–436 (1971) · doi:10.1063/1.1665604
[7] Moser, J.: Three integrable Hamiltonian systems, connected with isospectral deformations. Adv. Math.16, 1–23 (1978)
[8] Olshanetsky, M.A., Perelomov, A.M.: Completely integrable Hamiltonian systems associated with semisimple Lie algebra. Invent. Math.37, 93–108 (1976) · Zbl 0342.58017 · doi:10.1007/BF01418964
[9] Olshanetsky, M.A., Perelomov, A.M.: Quantum completely integrable systems connected with semisimple Lie algebras. Lett. Math. Phys.2, 7–13 (1977) · Zbl 0366.58005 · doi:10.1007/BF00420664
[10] Olshanetsky, M.A., Perelomov, A.M.: Quantum integrable systems related to Lie algebras. Phys. Rep.94, 313–404 (1983) · doi:10.1016/0370-1573(83)90018-2
[11] Berezin, F.S., Pokhil, G.P., Finkelberg, V.M.: Schrödinger equation for the system of onedimensional particles with point interaction. Vestnik Mosk. Univ.21–28 (1964) (in Russian)
[12] Olshanetsky, M.A., Perelomov, A.M.: Quantum systems related to root systems and radial parts of Laplace operators. Funct. Anal. and its Appl.12, 57–65 (1978)
[13] Helgason, S.: Differential geometry, Lie groups and symmetric spaces. New York: Academic Press 1978 · Zbl 0451.53038
[14] Berezin, F.A.: Laplace operators on semisimple Lie groups. Proc Mosc. Math. Soc.6, 371–463 (1957) · Zbl 0091.28201
[15] Vretare, L.: Formulas for elementary spherical functions and generalized Jacobi polynomials. SIAM J. Math. Ann.15 (4), 805–833 (1984) · Zbl 0549.43006 · doi:10.1137/0515062
[16] Beerends, R.: On the Abel transformation and its inversion. Comp. Math.66, 145–197 (1988) · Zbl 0661.43006
[17] Heckman, G.J., Opdam, E.M.: Root systems and hypergeometric functions I. Comp. Math.64, 329–352 (1987) · Zbl 0656.17006
[18] Heckman, G.J.: Root system and hypergeometric function II. Comp. Math.64, 353–373 (1987) · Zbl 0656.17007
[19] Opdam, E.M.: Root systems and hypergeometric functions III, IV. Comp. Math.67, 21–49, 191–209 (1988) · Zbl 0669.33007
[20] Opdam, E.M.: Some applications of hypergeometric shift operators. Invent. Math.98, 1–18 (1989) · Zbl 0696.33006 · doi:10.1007/BF01388841
[21] Heckman, G.J.: An elementary approach to the hypergeometric shift operators of Opdam. Invent. Math.103, 341–350 (1991) · Zbl 0721.33009 · doi:10.1007/BF01239517
[22] Chalykh, O.A.: On one construction of the commutative rings of partial differential operators. (To appear in Math. Notes) · Zbl 0813.47058
[23] Helgason, S.: Groups and geometry analysis. New York: Academic Press 1984 · Zbl 0543.58001
[24] Gelfand, I.M.: Spherical functions on the symmetric Riemannian spaces. Dokl. AN SSSR,70/1, 5–8 (1950)
[25] Styrkas, K.: Commutative rings of differential operators, Lie algebras and groups, generated by reflections. Diplom work, Moscow State University 1992 (to be published in Math. Notes)
[26] Veselov, A.P., Chalykh, O.A.: Explicit formulas for spherical functions on symmetric spaces of type AII. Funct. Anal. and its Appl.26/1, 59–61 (1992) · Zbl 0760.43004 · doi:10.1007/BF01077080
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