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Inverse Spectral Problem for Analytic Domains I: Balian-Bloch Trace Formula

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Abstract

This is the first in a series of papers [Z3, Z4] on inverse spectral/resonance problems for analytic plane domains Ω. In this paper, we present a rigorous version of the Balian-Bloch trace formula [BB1, BB2]. It is an asymptotic formula for the trace Tr1ΩR ρ (k+iτ log k) of the regularized resolvent of the Dirichlet or Neumann Laplacian of Ω as k→∞ with τ>0. When the support of contains the length L γ of precisely one periodic reflecting ray γ, then the asymptotic expansion of Tr1ΩR ρ (k+iτ log k) is essentially the same as the wave trace expansion at γ. The raison d’ètre for this approach is that it leads to relatively simple explicit formulae for wave invariants. For example, we give the first formulae for wave invariants of bouncing ball orbits of plane domains (the details will appear in [Z3]). Although we only present details in dimension 2, the methods and results extend with few modifications to all dimensions.

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References

  1. Abramowitz, M., Stegun, I.: Handbook of Mathematical Functions. New York: Dover, 1965

  2. Alonso, D., Gaspard, P.: ℏ expansion for the periodic orbit quantization of chaotic systems. Chaos 3(4), 601–612 (1993)

    Article  MATH  Google Scholar 

  3. Andersson, K.G., Melrose, R.B.: The propagation of singularities along gliding rays. Invent. Math. 41(3), 197–232 (1977)

    MATH  Google Scholar 

  4. Balian, R., Bloch, C.: Distribution of eigenfrequencies for the wave equation in a finite domain I: Three-dimensional problem with smooth boundary surface. Ann. Phys. 60, 401–447 (1970)

    MATH  Google Scholar 

  5. Balian, R., Bloch, C.: Distribution of eigenfrequencies for the wave equation in a finite domain. III. Eigenfrequency density oscillations. Ann. Phys. 69, 76–160 (1972)

    MATH  Google Scholar 

  6. Burmeister, B.: Korrekturen zur Gutzwillerschen Spurformel fuer Quantenbillards, Diplomarbeit am II. Hamburg: Institut fuer Theoretische Physik der Universität Hamburg, 1995

  7. Chazarain, J.: Construction de la paramétrix du problème mixte hyperbolique pour l’equation des ondes. C. R. Acad. Sci. Paris Ser. A-B 276, A1213–A1215 (1973)

  8. Colin de Verdière, Y.: Sur les longueurs des trajectoires périodiques d’un billard. In: P. Dazord and N. Desolneux-Moulis (eds.) Géométrie Symplectique et de Contact: Autour du Theoreme de Poincaré-Birkhoff. Travaux en Cours, Sem. Sud-Rhodanien de Géométrie III Paris: Herman, 1984, pp. 122–139

  9. Colin de Verdière, Y.: Spectre du laplacien et longueurs des gèodèsiques pèriodiques. I, II. Compositio Math. 27, 83–106 (1973); ibid. 27, 159–184 (1973)

    Google Scholar 

  10. Duistermaat, J.J., Guillemin, V.: The spectrum of positive elliptic operators and periodic bicharacteristics. Invent. Math. 29, 39–79 (1975)

    MATH  Google Scholar 

  11. Eckmann, J.-P., Pillet, C.-A.: Zeta functions with Dirichlet and Neumann boundary conditions for exterior domains. Papers honouring the 60th birthday of Klaus Hepp and of Walter Hunziker, Part II (Zürich, 1995). Helv. Phys. Acta 70(1–2), 44–65 (1997)

  12. Folland, G.: Introduction to Partial Differential Equations. Princeton Math. Notes, Princeton, NJ: Princeton U. Press, 1976

  13. Gonzalez-Barrios, J.M., Dudley, R.M.: Metric entropy conditions for an operator to be of trace class. Proc. AMS 118, 175–180 (1993)

    MathSciNet  MATH  Google Scholar 

  14. Guillemin, V., Melrose, R.B.: The Poisson summation formula for manifolds with boundary. Adv. Math. 32, 204–232 (1979)

    MathSciNet  MATH  Google Scholar 

  15. Hassell, A., Zelditch, S.: Quantum ergodicity of boundary values of eigenfunctions. to appear in Commun. Math. Phys. (preprint ar**v: math.SP/0211140)

  16. Hörmander, L.: The Analysis of Linear Partial Differential Operators, Volumes I–IV. Berlin Heidelberg: Springer-Verlag, 1983

  17. Iantchenko, A., Sjöstrand, J., Birkhoff, J.: normal forms for Fourier integral operators. II. Am. J. Math. 124(4), 817–850 (2002)

    MATH  Google Scholar 

  18. Iantchenko, A., Sjöstrand, J., Zworski, M.: Birkhoff normal forms in semi-classical inverse problems. Math. Res. Lett. 9(2–3), 337–362 (2002) (ar**v.org/abs/math.SP/0201191)

    Google Scholar 

  19. Kozlov, V.V., Treshchev, D.V.: Billiards: A Genetic Introduction to the Dynamics of Systems with Impacts. Translations of Math. Monographs 89, Providence, RI: AMS publications, 1991

  20. Lions, J.-L. Magenes, E.: Non-homogeneous Boundary Value Problems and Applications. Volume I, New York: Springer-Verlag, 1972

  21. Melrose, R.B., Sjöstrand, J.: Singularities of boundary value problems. I. Comm. Pure Appl. Math. 31(5), 593–617 (1978)

    MATH  Google Scholar 

  22. Miller, L.: Escape function conditions for the observation, control and stabilization of the wave equation. SIAM J. Control Optim 41, 1554–1566 (2003)

    Article  MATH  Google Scholar 

  23. Petkov, V.M., Stoyanov, L.N.: Geometry of reflecting rays and inverse spectral problems. Pure and Applied Mathematics. Chichester: John Wiley & Sons, Ltd., 1992

  24. Oberhettinger, F.: Tables of Fourier transforms and Fourier transforms of distributions. Berlin: Springer-Verlag, 1990

  25. Seeley, R.T.: Analytic extension of the trace associated with elliptic boundary problems. Am. J. Math. 91, 963–983 (1969)

    MATH  Google Scholar 

  26. Seeley, R.T.: Norms and domains of the complex powers A B z. Am. J. Math. 93, 299–309 (1971)

    MATH  Google Scholar 

  27. Seeley, R.T.: Complex powers of an elliptic operator. In: Singular Integrals, (Proc. Sympos. Pure Math., Chicago, Ill., 1966), Providence, RI: Amer. Math. Soc., pp. 288–307

  28. Sogge, C.D.: Fourier integrals in classical analysis. Cambridge Tracts in Mathematics, 105. Cambridge: Cambridge University Press, 1993

  29. Sjöstrand, J., Zworski, M.: Quantum monodromy and semi-classical trace formulae, J. Math. Pures Appl. (9) 81(1), 1–33 (2002)

    Google Scholar 

  30. Taylor, M.E.: Partial Differential Equations, I- II. Appl. Math. Sci., Berlin-Heidelberg-New York: Springer-Verlag, 1996, pp. 115–116

  31. Zelditch, S.: Spectral determination of analytic bi-axisymmetric plane domains (announcement). Math. Res. Lett. 6, 457–464 (1999)

    MathSciNet  MATH  Google Scholar 

  32. Zelditch, S.: Spectral determination of analytic bi-axisymmetric plane domains. Geom. Funct. Anal. 10(3), 628–677 (2000)

    MATH  Google Scholar 

  33. Zelditch, S.: Inverse spectral problem for analytic domains II: Domains with one symmetry (ar**v preprint, math.SP/0111078)

  34. Zelditch, S.: Inverse resonance problem for ℤ2 symmetric analytic obstacles in the plane. IMA Volume 137: Geometric Methods in Inverse Problems and PDE Control. C.B. Croke, I. Lasiecka, G. Uhlmann, M. S.Vogelius, eds. (2004)

  35. Zelditch, S.: Norm estimates in potential theory. Unpublished notes

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Authors and Affiliations

  1. Department of Mathematics, Johns Hopkins University, Baltimore, MD 21218, USA

    Steve Zelditch

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  1. Steve Zelditch
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Correspondence to Steve Zelditch.

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Communicated by P. Sarnak

Research partially supported by NSF grants #DMS-0071358 and #DMS-0302518.

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Zelditch, S. Inverse Spectral Problem for Analytic Domains I: Balian-Bloch Trace Formula. Commun. Math. Phys. 248, 357–407 (2004). https://doi.org/10.1007/s00220-004-1074-y

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