×

Intertwining operators for solving differential equations, with applications to symmetric spaces. (English) Zbl 0704.58025

There are considered general properties of intertwining operators and their application to solving ordinary and partial differential equations. For example, the heat kernels on the rank-one and rank-two symmetric spaces are constructed.
Reviewer: J.Danesova

MSC:

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.)
58J35 Heat and other parabolic equation methods for PDEs on manifolds
53C35 Differential geometry of symmetric spaces
35K05 Heat equation
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Koornwinder, T. H.: Orthogonal polynomials in two variables which are eigenfunctions of two algebraically independent partial differential operators, I, II, III, IV. Proc. Kon. Ned. Akad. Wetensch. A77 =Indag. Math.36, 48, 59, 357, 370 (1974) · Zbl 0263.33011
[2] Sprinkhuizen-Kuyper, I. G.: Orthogonal polynomials in two variables. A further analysis of the polynomials orthogonal over a region bounded by two lines and a parabola. SIAM. J. Math. Anal.7, 501 (1976) · Zbl 0332.33011
[3] Dowker, J. S.: Quantum mechanics on group space and Huygens’ principle. Ann. Phys. (NY)62, 361 (1971)
[4] Anderson, A.: Operator method for finding new propagators from old. Phys. Rev. D37, 536 (1988)
[5] Segal, G., Wilson, G.: Loop groups and equations of KdV type. Publ. IHES61, 5 (1985) · Zbl 0592.35112
[6] Olshanetsky, M.A., Perelomov, A.M.: Quantum integrable systems related to Lie algebras. Phys. Rep.94, 313 (1983); Classical integrable finite-dimensional systems related to Lie algebras. Phys. Rep.71, 313 (1981)
[7] Anker, J.-Ph.: Le noyau de la chaleur sur les espaces symetriquesU(p,q)/U(p){\(\times\)}U(q). In: Harmonic Analysis, Lecture Notes in Mathematics vol.1359 Eymard, P., Pier, J.-P. (eds.) p. 60. Berlin, Heidelberg, New York: Springer 1988 (in French) · Zbl 0669.43009
[8] Debiard, A., Gaveau, B.: Noyaux de la chaleur pour certaines equations hypergéométriques et application aux espaces symétriques de rang 1. C.R. Acad. Sci. Paris303- serie I, 849 (1986) (in French) · Zbl 0623.35017
[9] Debiard, A., Gaveau, B.: Noyaux de la chaleur, systèmes de racines, et espaces symétriques de typeB 2. C.R. Acad. Sci. Paris303- serie I, 951 (1986) (in French) · Zbl 0637.35040
[10] Debiard, A., Gaveau, B.: Noyaux de la chaleur de certaines equations aux dérivées partielles à singularité régulière et applications à certains espaces symétriques. C.R. Acad. Sci. Paris304- serie I, 131 (1987) (in French)
[11] Infeld, L., Hull, T. E.: The factorization method. Rev. Mod. Phys.23, 21 (1951) · Zbl 0043.38602
[12] Helgason, S.: Differential geometry, Lie groups, and symmetric spaces. New York: Academic Press 1978 · Zbl 0451.53038
[13] Helgason, S.: Groups and Geometric Analysis. New York: Academic Press 1984 · Zbl 0543.58001
[14] Oldham, K. B., Spanier, J.: The Fractional Calculus. New York: Academic Press 1974 · Zbl 0292.26011
[15] Osler, T. J.: Leibniz rule for fractional derivatives generalized and an application to infinite series. SIAM J. Appl. Math.18, 658 (1970) · Zbl 0201.44102
[16] Gindikin, S. G., Karpelevic, F. I.: Plancherel measure for Riemannian symmetric spaces of nonpositive curvature. Sov. Math. Dokl.3, 962 (1962) (in Russian) · Zbl 0156.03201
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.