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Lie theory and separation of variables. 10. Nonorthogonal R-separable solutions of the wave equation [partial-derivative]ttpsi=Delta2psi
We classify and discuss the possible nonorthogonal coordinate systems which lead to R-separable solutions of the wave equation. Each system is associated with a pair of commuting operators in the symm...
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Lie theory and separation of variables. 11. The EPD equation

J. Math. Phys. 17, 369 (1976); doi:10.1063/1.522902

Issue Date: March 1976

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E. G. Kalnins
Centre de Recherches Mathématiques, Université de Montréal, Montréal 101, P.Q., Canada

Willard Miller, Jr.
School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455
We show that the Euler–Poisson–Darboux equation {[partial-derivative]tt−[partial-derivative]rr−[(2m+1)/r][partial-derivative]r}Theta=0 separates in exactly nine coordinate systems corresponding to nine orbits of symmetric second-order operators in the enveloping algebra of SL(2,R), the symmetry group of this equation. We employ techniques developed in earlier papers from this series and use the representation theory of SL(2,R) to derive special function identities relating the separated solutions. We also show that the complex EPD equation separates in exactly five coordinate systems corresponding to five orbits of symmetric second-order operators in the enveloping algebra of SL(2,C). Journal of Mathematical Physics is copyrighted by The American Institute of Physics.
Permalink: http://link.aip.org/link/?JMAPAQ/17/369/1
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PACS

  • 02.20.Qs
    Mathematical methods in physics Group theory General properties, structure, and representation of Lie groups
  • 02.30.Hq
    Mathematical methods in physics Function theory, analysis Ordinary differential equations
  • YEAR: 1976

PUBLICATION DATA

ISSN:
0022-2488 (print)   1089-7658 (online)
Publisher:
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REFERENCES (14)
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  1. E. G. Kalnins and W. Miller, Jr., J. Math. Phys. 17, xxx, xxx, xxx (1976), Papers 8–10.
  2. B. Viswanathan, Can. J. Math. 20, 120 (1968).
  3. L. Weisner, Pacific J. Math. 5, 1033 (1955).
  4. W. Miller, Jr., Lie Theory and Special Functions (Academic, New York, 1968).
  5. W. Miller, Jr., SIAM J. Math. Anal. 4, 314 (1973).
  6. A. Erdélyi et al., High Transcendental Functions (McGraw-Hill, New York, 1953), Vols. I and II.
  7. P. Sally, Analytic Continuation of the Irreducible Unitary Representations of the Universal Covering Group of SL (2,R), AMS Mem., No. 69 (Am. Math. Soc., Providence, R.I., 1967).
  8. W. Montgomery and L. O'Raifeartaigh, J. Math. Phys. 15, 380 (1974).
  9. V. Bargmann, Ann. Math. 48, 568 (1947).
  10. E. C. Titchmarsh, Eigenfunction Expansions (Oxford U.P., Oxford, 1962), Part One, 2nd ed.
  11. E. G. Kalnins and W. Miller, Jr., J. Math. Phys. 15, 1263 (1974).
  12. W. Miller Jr., SIAM J. Math. Anal. 5, 822 (1974).
  13. W. Miller Jr., SIAM J. Math. Anal. 5, 822 (1974).
  14. L. J. Slater, Generalized Hypergeometric Functions (Cambridge U.P., London, 1966).

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