Billiards and boundary traces of eigenfunctions

Steve Zelditch

Journées équations aux dérivées partielles (2003)

  • page 1-22
  • ISSN: 0752-0360

Abstract

top
This is a report on recent results with A. Hassell on quantum ergodicity of boundary traces of eigenfunctions on domains with ergodic billiards, and of work in progress with Hassell and Sogge on norms of boundary traces. Related work by Burq, Grieser and Smith-Sogge is also discussed.

How to cite

top

Zelditch, Steve. "Billiards and boundary traces of eigenfunctions." Journées équations aux dérivées partielles (2003): 1-22. <http://eudml.org/doc/93442>.

@article{Zelditch2003,
abstract = {This is a report on recent results with A. Hassell on quantum ergodicity of boundary traces of eigenfunctions on domains with ergodic billiards, and of work in progress with Hassell and Sogge on norms of boundary traces. Related work by Burq, Grieser and Smith-Sogge is also discussed.},
author = {Zelditch, Steve},
journal = {Journées équations aux dérivées partielles},
keywords = {Lipschitz domain; boundary value problems; boundary traces of eigenfunctions; quantum ergodicity},
language = {eng},
pages = {1-22},
publisher = {Université de Nantes},
title = {Billiards and boundary traces of eigenfunctions},
url = {http://eudml.org/doc/93442},
year = {2003},
}

TY - JOUR
AU - Zelditch, Steve
TI - Billiards and boundary traces of eigenfunctions
JO - Journées équations aux dérivées partielles
PY - 2003
PB - Université de Nantes
SP - 1
EP - 22
AB - This is a report on recent results with A. Hassell on quantum ergodicity of boundary traces of eigenfunctions on domains with ergodic billiards, and of work in progress with Hassell and Sogge on norms of boundary traces. Related work by Burq, Grieser and Smith-Sogge is also discussed.
LA - eng
KW - Lipschitz domain; boundary value problems; boundary traces of eigenfunctions; quantum ergodicity
UR - http://eudml.org/doc/93442
ER -

References

top
  1. [AG] D. Alonso and P. Gaspard, ˉh expansion for the periodic orbit quantization of chaotic systems. Chaos 3 (1993), no. 4, 601-612. Zbl1055.81541MR1256314
  2. [B] A. Backer, Numerical aspects of eigenvalue and eigenfunction computations for chaotic quantum systems, Mathematical Aspects of Quantum Maps, M. Degli Esposti and S. Graffi (Eds.), Springer Lecture Notes in Physics 618 (2003). Zbl1046.81040
  3. [BS] A. Backer and R. Schubert, Chaotic eigenfunctions in momentum space. J. Phys. A 32 (1999), no. 26, 4795-4815 Zbl0946.81030MR1718807
  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. Zbl0207.40202MR270008
  5. [BB2] R. Balian and C. Bloch, Distribution of eigenfrequencies for the wave equation in a finite domain. III. Eigenfrequency density oscillations. Ann. Physics 69 (1972), 76-160. Zbl0226.35070MR289962
  6. [BFSS] A. Backer, S. Furstberger, R. Schubert, and F. Steiner, Behaviour of boundary functions for quantum billiards. J. Phys. A 35 (2002), no. 48, 10293-10310. Zbl1048.81025MR1947308
  7. [BLR] C. Bardos, G. Lebeau, and J. Rauch, Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary. SIAM J. Control Optim. 30 (1992), no. 5, 1024-1065. Zbl0786.93009MR1178650
  8. [Bi] Bialy, Misha(IL-TLAV) Convex billiards and a theorem by E. Hopf. Math. Z. 214 (1993), no. 1, 147-154 Zbl0790.58023MR1234604
  9. [Bu] N. Burq, Quantum ergodicity of boundary values of eigenfunctions: a control theory approach, arXiv:math.AP/0301349, 2003. 
  10. [DS] M. Dimassi and J. Sjostrand, Spectral asymptotics in the semi-classical limit. London Mathematical Society Lecture Note Series, 268. Cambridge University Press, Cambridge, 1999. Zbl0926.35002MR1735654
  11. [GP] B. Georgeot and R.E. Prange, Exact and quasiclassical Fredholm solutions of quantum billiards. Phys. Rev. Lett. 74 (1995), no. 15, 2851-2854. Zbl1020.81589MR1324958
  12. [GL] P. Gerard and E. Leichtnam, Ergodic properties of eigenfunctions for the Dirichlet problem, Duke Math. J. 71 (1993), 559-607. Zbl0788.35103MR1233448
  13. [G] D. Grieser, Uniform bounds for eigenfunctions of the Laplacian on manifolds with boundary. Comm. Partial Differential Equations 27 (2002), no. 7-8, 1283-1299. Zbl1034.35085MR1924468
  14. [GM] V. Guillemin and R. B. Melrose, The Poisson summation formula for manifolds with boundary. Adv. in Math. 32 (1979), no. 3, 204-232. Zbl0421.35082MR539531
  15. [THS] T. Harayama, A. Shudo, and S. Tasaki, Semiclassical Fredholm determinant for strongly chaotic billiards. Nonlinearity 12 (1999), no. 4, 1113-1149. Zbl0985.81039MR1710464
  16. [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. MR1459088
  17. [HT] A. Hassell and T. Tao, Upper and lower bounds for normal derivatives of Dirichlet eigenfunctions. Math. Res. Lett. 9 (2002), no. 2-3, 289-305. Zbl1014.58015MR1909646
  18. [HSZ] A. Hassell, C. Sogge and S. Zelditch, Billiards and boundary traces of eigenfunctions, (in preparation). 
  19. [HZ] A. Hassell and S. Zelditch, Ergodicity of boundary values of eigenfunctions, preprint (2002). 
  20. [I] V. Ivrii, The second term of the spectral asymptotics for a Laplace-Beltrami operator on manifolds with boundary. (Russian) Funktsional. Anal. i Prilozhen. 14 (1980), no. 2, 25-34 Zbl0453.35068MR575202
  21. [M] R. B. Melrose, The trace of the wave group. Microlocal analysis (Boulder, Colo., 1983), 127-167, Contemp. Math., 27, Amer. Math. Soc., Providence, RI, 1984. Zbl0547.35095MR741046
  22. [O] S. Ozawa, Asymptotic property of eigenfunction of the Laplacian at the boundary. Osaka J. Math. 30 (1993), 303-314. Zbl0808.35090MR1233512
  23. [O2] S. Ozawa, Hadamard's variation of the Green kernels of heat equations and their traces. I. J. Math. Soc. Japan 34 (1982), no. 3, 455-473. Zbl0476.35039MR659615
  24. [SV] Y. Safarov and D. Vassiliev, The asymptotic distribution of eigenvalues of partial differential operators. Translations of Mathematical Monographs, 155. American Mathematical Society, Providence, RI, 1997. Zbl0898.35003MR1414899
  25. [S] C. D. Sogge, Eigenfunction and Bochner Riesz estimates on manifolds with boundary. Math. Res. Lett. 9 (2002), no. 2-3, 205-216. Zbl1017.58016MR1903059
  26. [SS] C. Sogge and H. Smith (in preparation). 
  27. [SZ] C. D. Sogge and S. Zelditch, Riemannian manifolds with maximal eigenfunction growth. Duke Math. J. 114 (2002), no. 3, 387-437. Zbl1018.58010MR1924569
  28. [T] D. Tataru, On the regularity of boundary traces for the wave equation. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 26 (1998), no. 1, 185-206. Zbl0932.35136MR1633000
  29. [TV] J. M Tualle and A. Voros, Normal modes of billiards portrayed in the stellar (or nodal) representation. Chaos Solitons Fractals 5 (1995), no. 7, 1085-1102. Zbl0912.58030MR1357264
  30. [W] Wojtkowski, Maciej P.(1-AZ) Two applications of Jacobi fields to the billiard ball problem. (English. English summary) J. Differential Geom. 40 (1994), no. 1, 155-164 Zbl0812.58067MR1285532
  31. [Z1] S. Zelditch, The inverse spectral problem for analytic plane domains, I: Balian-Bloch trace formula (arXiv: math.SP/0111077). 
  32. [ZZw] S. Zelditch and M. Zworski, Ergodicity of eigenfunctions for ergodic billiards. Comm. Math. Phys. 175 (1996), 673-682. Zbl0840.58048MR1372814

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.