Inverse Resonance Problem for ℤ2-Symmetric Analytic Obstacles in the Plane

  • Steve Zelditch
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 137)


We given an exposition of a proof that a mirror symmetric configuration of two convex analytic obstacles in ℝ2determined by its Dirichlet resonance poles. It is the analogue for exterior domains of the proof that a mirror symmetric bounded simply connected analytic plane domain is determined by its Dirichlet eigenvalues. The proof uses ‘interior/exterior duality’ to simplify the argument.


Open Vertex Exterior Domain Fredholm Determinant Inverse Spectral Problem Length Spectrum 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [AG]
    D. Alonso and P. Gaspard, ħ expansion for the periodic orbit quantization of chaotic systems. Chaos 3 (1993), No. 4, 601–612.MathSciNetCrossRefGoogle Scholar
  2. [AM]
    K. G. Andersson and R. B. Melrose, The propagation of singularities along gliding rays. Invent. Math. 41 (1977), No. 3, 197–232.MathSciNetMATHCrossRefGoogle Scholar
  3. [B]
    R. Bacher (unpublished note, 2002).Google Scholar
  4. [BB1]
    R. Balian and C. Bloch, Distribution of eigenfrequencies for the wave equation in a finite domain I: three-dimensional problem with smooth boundary surface, Ann. Phys. 60 (1970), 401–447.MathSciNetMATHCrossRefGoogle Scholar
  5. [BB2]
    R. Balian andC. Bloch, Distribution of eigenfrequencies for the wave equation in a finite domain. III. Eigenfrequency density oscillations. Ann. Physics 69 (1972), 76–160. MathSciNetMATHCrossRefGoogle Scholar
  6. [BGR]
    C. Bardos, J. C. Guillot and J. Ralston, La relation de Poisson pour I’quation des ondes dans un ouvert non born. Application la thorie de la diffusion. Comm. Partial Differential Equations 7 (1982), No. 8, 905–958.MathSciNetMATHCrossRefGoogle Scholar
  7. [CdV]
    Y. Colin de Verdirb, Sur les longueurs des trajectoires priodiques d’un bil- lard, South Rhone seminar on geometry, III (Lyon, 1983), 122–139, Travaux en Cours, Hermann, Paris, 1984.Google Scholar
  8. [EP]
    J.P. Eckmann and C.A. Pillet, Zeta functions with Dirichlet and Neumann boundary conditions for exterior problems, Helv. Phys. Acta 70 (1997), 44–65.MathSciNetMATHGoogle Scholar
  9. [EP2]
    J.P. Eckmann and C.A. Pillet, Scattering phases and density of states for exterior domains. Ann. Inst. H. Poincar Phys. Thor. 62 (1995), No. 4, 383–399.MathSciNetMATHGoogle Scholar
  10. [E]
    G.B. Folland, Introduction to Partial Differential Equations, Math. Notes. 17, Princeton U. Press, Princeton (1976).Google Scholar
  11. [GP]
    B. Georgeot and R.E. Prange, Exact and quasiclassical Predholm solutions of quantum billiards. Phys. Rev. Lett. 74 (1995), No. 15, 2851–2854.MathSciNetMATHCrossRefGoogle Scholar
  12. [G]
    V. Guillemin, Wave-trace invariants. Duke Math. J. 83 (1996), No. 2, 287–352.MathSciNetMATHCrossRefGoogle Scholar
  13. [GM]
    V. Guillemin and R.B. Melrose, The Poisson summation formula for manifolds with boundary. Adv. in Math. 32 (1979), No. 3, 204–232.MathSciNetMATHCrossRefGoogle Scholar
  14. [HZ]
    A. Hassell and S. Zelditch, Quantum ergodicity of boundary values of eigen- functions (ariv preprint math. SP/0211140).Google Scholar
  15. [ISZ]
    A. Iantchenko, J.Sjostrand and M. Zworski, Birkhoff normal forms in semi-classical inverse problems, ( Scholar
  16. [KT]
    V.V. Kozlov and D.V. Treshchev, Billiards: A Genetic Introduction to the Dynamics of Systems with Impacts, Translations of Math. Monographs 89, AMS publications. Providence, R.L (1991).Google Scholar
  17. [M]
    R.B. Melrose, Polynomial bounds on the distribution of poles in scattering by an obstacle, Journees “Equations aux Derivee Partielles”, Saint-Jean de Monts (1984).MATHGoogle Scholar
  18. [MS]
    R. B. Melrose and J. Sjostrand, Singularities of boundary value problems. I. Comm. Pure Appl. Math. 31 (1978), No. 5, 593–617.MathSciNetMATHCrossRefGoogle Scholar
  19. [PS]
    V.M. Petkov and L.N. Stoyanov, Geometry of Reflecting Rays and Inverse Spectral Problems, John Wiley and Sons, N.Y. (1992).MATHGoogle Scholar
  20. [THS]
    T. Harayama, A. Shudo,and S. Tasaki, Semiclassical Predholm determinant for strongly chaotic billiards. Nonlinearity 12 (1999), No. 4, 1113–1149.MathSciNetMATHCrossRefGoogle Scholar
  21. [THS2]
    T. Harayama, A. Shudo, and S. Tasaki, Interior Dirichlet eigenvalue problem, exterior Neumann scattering problem, and boundary element method for quantum billiards. Phys. Rev. E (3) 56 (1997), No. 1, part A, R13-R16.MathSciNetGoogle Scholar
  22. [P]
    A. Pleuel, A study of certain Green’s functions with applications in the theory of vibrating membranes, Arkiv for Math 2 (1952), 553–569.Google Scholar
  23. [S]
    B. Simon, Trace ideals and Their Applications, London Math. Soc. Lecture Note Series 35, Cambridge Univ. Press, Cambridge (1979).Google Scholar
  24. [TI]
    M.E. Taylor, Partial differential equations. L Basic theory. Applied Mathematical Sciences, 115. Springer-Ver lag, New York, 1996.Google Scholar
  25. [TII]
    M.E. Taylor, Partial differential equations. II. Qualitative studies of linear equations. Applied Mathematical Sciences, 116. Springer-Verlag, New York, 1996.Google Scholar
  26. [Z1]
    S. Zelditch, Inverse spectral problem for analytic plane domains I: Balian- Bloch trace formula (preprint, 2001).Google Scholar
  27. [Z2]
    S. Zelditch, Inverse spectral problem for analytic plane domains II: domains with symmetry (preprint, 2001).Google Scholar
  28. [Z3]
    S. Zelditch, Spectral determination of analytic bi-axisymmetric plane domains. Geom. Funct. Anal. 10 (2000), No. 3, 628–677.MathSciNetMATHCrossRefGoogle Scholar
  29. [Zw]
    M. Zworski, Poisson formula for resonances in even dimensions. Asian J. Math. 2 (1998), No. 3, 609–617.MathSciNetMATHGoogle Scholar

Copyright information

© Springer-Verlag New York, Inc. 2004

Authors and Affiliations

  • Steve Zelditch
    • 1
  1. 1.Department of MathematicsJohns Hopkins UniversityBaltimoreUSA

Personalised recommendations