Skip to main content
Log in

The Neumann Isospectral Problem for Trapezoids

  • Published:
Annales Henri Poincaré Aims and scope Submit manuscript

Abstract

We show that non-obtuse trapezoids with identical Neumann spectra are congruent up to rigid motions of the plane. The proof is based on heat trace invariants and some new wave trace invariants associated to certain diffractive billiard trajectories. We use the method of reflections to express the Dirichlet and Neumann wave kernels in terms of the wave kernel of the double polygon. Using Hillairet’s trace formulas for isolated diffractive geodesics and one-parameter families of regular geodesics with geometrically diffractive boundaries for Euclidean surfaces with conical singularities (Hillairet in J Funct Anal 226(1):48–89, 2005), we obtain the new wave trace invariants for trapezoids. To handle the reflected term, we use another result of Hillairet (J Funct Anal 226(1):48–89, 2005), which gives a Fourier integral operator representation for the Keller and Friedlander parametrix (Keller in Proc Symp Appl Math 8:27–52, 1958; Friedlander in Math Proc Camb Philos Soc 90(2):335–341, 1981) of the wave propagator near regular diffractive geodesics. The reason we can only treat the Neumann case is that the wave trace is “more singular” for the Neumann case compared to the Dirichlet case. This is a new observation which is of independent interest.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

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

    Article  ADS  MATH  MathSciNet  Google Scholar 

  2. Bérard, Pierre H.: On the wave equation on a compact Riemannian manifold without conjugate points. Math. Z. 155(3), 249–276 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  3. van den Berg, M., Srisatkunarajah, S.: Heat equation for a region in \({ R}^2\) with a polygonal boundary. J. Lond. Math. Soc. 37(1), 119–127 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  4. Berry, M.V., Tabor, M.: Level clustering in the regular spectrum. Proc. R. Soc. Lond. Ser. A 356(1656), 375–394 (1977)

    Article  ADS  MATH  Google Scholar 

  5. Bogomolny, E., Pavloff, N., Schmit, C.: Diffractive corrections in the trace formula for polygonal billiards. Phys. Rev. E 61(4), 3689–3711 (2000)

    Article  ADS  MathSciNet  Google Scholar 

  6. Bohigas, O., Giannoni, M.-J., Schmit, C.: Characterization of chaotic quantum spectra and universality of level fluctuation laws. Phys. Rev. Lett. 52(1), 1–4 (1984)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  7. Buser, P.: Isospectral Riemann surfaces. Ann. Inst. Fourier (Grenoble) 36(2), 167–192 (1986). (English, with French summary)

    Article  MATH  MathSciNet  Google Scholar 

  8. Chapman, S.J.: Drums that sound the same. Am. Math. Mon. 102, 124–138 (1995)

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  10. Chazarain, J.: Formule de Poisson pour les variétés riemanniennes. Invent. Math. 24, 65–82 (1974). (French)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  11. Cheeger, J., Taylor, M.: On the diffraction of waves by conical singularities I. Commun. Pure Appl. Math. 35(3), 275–331 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  12. Cheeger, J., Taylor, M.: On the diffraction of waves by conical singularities II. Commun. Pure Appl. Math. 35(4), 487–529 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  13. Colin Verdière, Y.: Spectre du laplacien et longueurs des géodésiques périodiques I, II. Compos. Math. 27, 159–184 (1973)

    MATH  Google Scholar 

  14. Colin de Verdière, Y.: Sur les longueurs des trajectoires périodiques d’un billard. In: South Rhone Seminar on Geometry, III (Lyon, 1983). Travaux en Cours, Hermann, Paris, pp. 122–139 (1984)

  15. Datchev, K., Hezari, H.: Inverse problems in spectral geometry. Inverse problems and applications: inside out. II. Math. Sci. Res. Inst. Publ. 60, 455–485 (2013). Cambridge Univ. Press, Cambridge

    MATH  Google Scholar 

  16. De Simoi, J., Kaloshin, V., Wei, Q.: Dynamical spectral rigidity among \({\mathbb{Z}}_2\)-symmetric strictly convex domains close to a circle, with a joint appendix by H. Hezari. Ann. Math. (2017). arXiv:1606.00230

  17. Duistermaat, J.J., Guillemin, V.W.: The spectrum of positive elliptic operators and periodic bicharacteristics. Invent. Math. 29(1), 39–79 (1975)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  18. Duistermaat, J.J.: Fourier integral operators. In: Modern Birkhäuser Classics. Reprint of the 1996 edition [MR1362544], based on the original lecture notes published in 1973 [MR0451313], Springer, New York (2011)

  19. Durso, C.: On the inverse spectral problem for polygonal domains. ProQuest LLC, Ann Arbor, MI, (1988). Thesis (Ph.D.)–Massachusetts Institute of Technology

  20. Ford, A., Hassell, A., Hillairet, L.: Wave propagation on Euclidean surfaces with conical singularities. I: geometric diffraction. J. Spectr. Theory (2015). Arxiv:1505.01043

  21. Ford, A., Wunsch, J.: The diffractive wave trace on manifolds with conic singularities. Adv. Math. 304, 1330–1385 (2017)

    Article  MATH  MathSciNet  Google Scholar 

  22. Friedlander, F.G.: Multivalued solutions of the wave equation. Math. Proc. Camb. Philos. Soc. 90(2), 335–341 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  23. Friedlander, F.G.: On the wave equation in plane regions with polygonal boundary. Adv. Microlocal Anal. 135–150 (1986)

  24. Fursaev, D.V.: The heat-kernel expansion on a cone and quantum fields near cosmic strings. Class. Quantum Gravity 11(6), 1431–1443 (1994)

    Article  ADS  MathSciNet  Google Scholar 

  25. Gordon, C., Webb, D.L., Wolpert, S.: One cannot hear the shape of a drum. Bull. Am. Math. Soc. (N.S.) 27(1), 134–138 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  26. Gordon, C., Webb, D.L., Wolpert, S.: Isospectral plane domains and surfaces via Riemannian orbifolds. Invent. Math. 110(1), 1–22 (1992)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  27. Grieser, D., Maronna, S.: Hearing the shape of a triangle. Not. Am. Math. Soc. 60(11), 1440–1447 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  28. Guillemin, V., Melrose, R.: The Poisson summation formula for manifolds with boundary. Adv. Math. 32(3), 204–232 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  29. Hezari, H., Zelditch, S.: \(C^\infty \) spectral rigidity of the ellipse. Anal. PDE 5(5), 1105–1132 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  30. Hillairet, L.: Formule de trace sur une surface euclidienne à singularités coniques. C. R. Math. Acad. Sci. Paris 335(12), 1047–1052 (2002). (French, with English and French summaries)

    Article  MATH  MathSciNet  Google Scholar 

  31. Hillairet, L.: Contribution of periodic diffractive geodesics. J. Funct. Anal. 226(1), 48–89 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  32. Hillairet, L.: Diffractive geodesics of a polygonal billiard. Proc. Edinb. Math. Soc. 49(1), 71–86 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  33. Kac, M.: Can one hear the shape of a drum? Am. Math. Mon. 73(4), 1–23 (1966)

    Article  MATH  MathSciNet  Google Scholar 

  34. Hörmander, L.: The analysis of linear partial differential operators. I. In: Distribution Theory and Fourier Analysis MR1065136, 2nd edn. Springer, Berlin (1990)

  35. Hörmander, L.: The analysis of linear partial differential operators. III. Classics in mathematics. In: Pseudo-Differential Operators, Springer, Berlin (2007) (Reprint of the 1994 edition)

  36. Hörmander, L.: The analysis of linear partial differential operators. IV. Classics in mathematics. In: Fourier Integral Operators, Springer, Berlin (2009) (Reprint of the 1994 edition)

  37. Hörmander, L.: Fourier integral operators. I. Acta Math. 127(1–2), 79–183 (1971)

    Article  MATH  MathSciNet  Google Scholar 

  38. Ivrii, V.: Microlocal analysis and precise spectral asymptotics. In: Springer Monographs in Mathematics MR1631419, Springer, Berlin (1998)

  39. Keller, J.B.: A geometrical theory of diffraction. Calculus of variations and its applications. Proc. Symp. Appl. Math. 8, 27–52 (1958)

    Article  Google Scholar 

  40. Kokotov, A.: On the spectral theory of the Laplacian on compact polyhedral surfaces of arbitrary genus. Comput. Approach Riemann Surf. Lect. Notes Math. 2013, 227–253 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  41. Lu, Z., Rowlett, J.: The sound of symmetry. Am. Math. Mon. 122(9), 815–835 (2015)

    Article  MATH  MathSciNet  Google Scholar 

  42. Lu, Z., Rowlett, J.: One can hear the corners of a drum. Bull. Lond. Math. Soc. 48(1), 85–93 (2016)

    Article  MATH  MathSciNet  Google Scholar 

  43. Netrusov, Y., Safarov, Y.: Weyl asymptotic formula for the Laplacian on domains with rough boundaries. Commun. Math. Phys. 253(2), 481–509 (2005). https://doi.org/10.1007/s00220-004-1158-8. MR2140257

    Article  ADS  MATH  MathSciNet  Google Scholar 

  44. McKean Jr., H.P., Singer, I.M.: Curvature and the eigenvalues of the Laplacian. J. Differ. Geometry 1(1), 43–69 (1967)

    Article  MATH  MathSciNet  Google Scholar 

  45. Melrose, R.: The inverse spectral problem for planar domains. In: Instructional Workshop on Analysis and Geometry, Part I (Canberra, 1995). Proceedings of Centre for Mathematics and Its Applications, Australian National University, vol. 34, Canberra, pp. 137–160 (1996)

  46. Melrose, R.B., Uhlmann, G.A.: Lagrangian intersection and the Cauchy problem. Commun. Pure Appl. Math. 32(4), 483–519 (1979). https://doi.org/10.1002/cpa.3160320403. MR528633

    Article  MATH  MathSciNet  Google Scholar 

  47. Melrose, R., Wunsch, J.: Propagation of singularities for the wave equation on conic manifolds. Invent. Math. 156(2), 235–299 (2004)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  48. Milnor, J.: Eigenvalues of the Laplace operator on certain manifolds. Proc. Nat. Adad. Sci. USA 51(4), 542 (1964)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  49. Pavloff, N., Schmit, C.: Diffractive orbits in quantum billiards. Phys. Rev. Let. 75(1), 61–64 (1995)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  50. Petkov, V.M., Stoyanov, L.N.: Geometry of reflecting rays and inverse spectral problems. In: Pure and Applied Mathematics MR1172998, Wiley, New York (1992)

  51. Popov, G., Topalov, P.: From K.A.M. tori to isospectral invariants and spectral rigidity of billiard tables. arXiv: 1602.03155 (2016)

  52. Sommerfeld, A.: Mathematische theorie der diffraction. Math. Ann. 47(2–3), 317–374 (1896)

    Article  MATH  MathSciNet  Google Scholar 

  53. Shubin, M.A.: Pseudodifferential operators and spectral theory, 2nd edn. Springer, Berlin (2001) (Translated from the 1978 Russian original by Stig I. Andersson)

  54. Sunada, T.: Riemannian coverings and isospectral manifolds. Ann. Math. 121(1), 169–186 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  55. Taylor, M. E.: Pseudodifferential operators and nonlinear PDE. In: Progress in Mathematics, vol. 100, Birkhäuser Boston, Inc., Boston (1991)

  56. Vignéras, M.-F.: Exemples de sous-groupes discrets non conjugués de \({\rm PSL}(2,{\bf R})\) qui ont même fonction zéta de Selberg. C. R. Acad. Sci. Paris Ser. A-B 287(2), A47–A49 (1978). (French, with English summary)

    Google Scholar 

  57. Watanabe, K.: Plane domains which are spectrally determined. Ann. Global Anal. Geom. 18(5), 447–475 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  58. Witt, E.: Eine Identität zwischen Modulformen zweiten Grades. Abh. Math. Sem. Univ. Hambg. 14, 323–337 (1941)

    Article  MATH  Google Scholar 

  59. Wunsch, J.: A Poisson relation for conic manifolds. Math. Res. Lett. 9(5–6), 813–828 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  60. Zelditch, S.: Survey on the inverse spectral problem. ICCM Not. 2(2), 1–20 (2014). https://doi.org/10.4310/ICCM.2014.v2.n2.a1. MR3314780

    MathSciNet  Google Scholar 

  61. Zelditch, S.: Inverse spectral problem for analytic domains. II. Z2-symmetric domains. Ann. Math 170(1), 205–269 (2009)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Hamid Hezari.

Additional information

Communicated by Stephane Nonnenmacher.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hezari, H., Lu, Z. & Rowlett, J. The Neumann Isospectral Problem for Trapezoids. Ann. Henri Poincaré 18, 3759–3792 (2017). https://doi.org/10.1007/s00023-017-0617-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00023-017-0617-7

Navigation