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Splitting of the spectrum of the laplace operator on an ellipse with cuts

Abstract

The exponential asymptotics with a factor of the semiclassical form multiplying the exponent is proved for the splitting of the eigenvalues of the the Laplace operator with the Dirichlet boundary condition in the case of an ellipse with cuts. The proof is based on the method of separation of variables.

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Literature cited

  1. 1.

    V. F. Lazutkin, “The complex billiard,” Probl. Mat. Fiz.,11, 138–164 (1986).

  2. 2.

    A. G. Alenitsyn, “Spectrum splitting generated by a potential barrier in problems with a symmetric potential,” Differents. Uravn.,18, No. 11, 1971–1975 (1982).

  3. 3.

    L. D. Landau and E. M. Lifshitz, Quantum Mechanics, Nonrelativistic Theory, Pergamon Press, Oxford (1965).

  4. 4.

    M. V. Fedoryuk, “Discrete spectrum asymptotics of the operatorω"(x)-λ 2 p(x)ω(x),” Mat. Sb.,68, No. 1, 81–110 (1965).

  5. 5.

    E. M. Harrell, “Double wells,” Comm. Math. Phys.,75, 239–261 (1980).

  6. 6.

    V. S. Buldyrev and S. Yu. Slavyanov, “Regularization of phase integrals near the top of a barrier,” Probl. Mat. Fiz.,10, 50–70 (1982).

  7. 7.

    V. I. Smirnov, Course in Higher Mathematics, Vol. 3, Part 2 [in Russian], Nauka (1974).

  8. 8.

    F. W. J. Olver, Introduction to Asymptotics and Special Functions, Academic Press, New York-London (1974).

  9. 9.

    V. A. Fok, Tables of Airy Functions [in Russian], GITTL (1946).

  10. 10.

    V. M. Babich and V. S. Buldyrev, Asymptotic Methods in Problems of Short Wave Diffraction [in Russian], Nauka (1972).

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Additional information

Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova, Vol. 179, pp. 173–178, 1989.

The author wishes to express his gratitude to V. F. Lazutkin for stimulating influence, interest in this work, and valuable comments.

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Terman, D.Y. Splitting of the spectrum of the laplace operator on an ellipse with cuts. J Math Sci 57, 3176–3180 (1991). https://doi.org/10.1007/BF01098988

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Keywords

  • Boundary Condition
  • Dirichlet Boundary
  • Laplace Operator
  • Dirichlet Boundary Condition
  • Exponential Asymptotics