Advertisement

The European Physical Journal Special Topics

, Volume 145, Issue 1, pp 271–286 | Cite as

Nodal lines, ergodicity and complex numbers

  • S. Zelditch
Article

Abstract.

This article reviews two rigorous results about the complex zeros of eigenfunctions of the Laplacian, that is, the zeros of the analytic continuation of the eigenfunctions to the complexification of the underlying space. Such a complexification of the problem is analogous to studying the complex zeros of polynomials with real coefficients. The first result determines the limit distribution of complex zeros of `ergodic eigenfunctions' such as eigenfunctions of classically chaotic systems. The second result determines the expected distribution of complex zeros for complexifications of Gaussian random waves adapted to the Riemannian manifold. The resulting distribution is the same in both cases. It is singular along the set of real points.

Keywords

Riemannian Manifold European Physical Journal Special Topic Real Hypersurface Real Zero Nodal Line 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. A. Bäcker, S. Fürstberger, R. Schubert, Phys. Rev. E 70, 036204 (2004) Google Scholar
  2. A.H. Barnett, Comm. Pure Appl. Math. 59, 1457 (2006) Google Scholar
  3. M.V. Berry, J. Phys. A 10, 2083 (1977) Google Scholar
  4. G. Blum, S. Gnutzmann, U. Smilansky, Phys. Rev. Lett. 88, 114101 (2002) Google Scholar
  5. E. Bogomolny, C. Schmit, Phys. Rev. Lett. 88, 114102-114102-4 (2002) Google Scholar
  6. L. Boutet de Monvel, C. R. Acad. Sci. Paris Sér. A-B 287, A855 (1978) Google Scholar
  7. F. Bruhat, H. Whitney, Comment. Math. Helv. 33, 132 (1959) Google Scholar
  8. K. Burns, V.J. Donnay, Internat. J. Bifur. Chaos Appl. Sci. Eng. 7, 1509 (1997) Google Scholar
  9. H. Donnelly, C. Fefferman, Invent. Math. 93, 161 (1988) Google Scholar
  10. H. Donnelly, C. Fefferman, J. Amer. Math. Soc. 3, 333 (1990) Google Scholar
  11. V. Guillemin, M. Stenzel, J. Diff. Geom. 34, 561 (1991) Google Scholar
  12. V. Guillemin, M. Stenzel, J. Diff. Geom. 35, 627 (1992) Google Scholar
  13. J. Hadamard, Lectures on Cauchy's problem in linear partial differential equations (Dover Publications, New York, 1953) Google Scholar
  14. A. Hassell, S. Zelditch, Comm. Math. Phys. 248, 119 (2004) Google Scholar
  15. H. Hezari, Complex zeros of eigenfunctions of 1D Schrodinger operators (preprint, 2006) Google Scholar
  16. D. Jerison, G. Lebeau, Nodal sets of sums of eigenfunctions, Ch. 14 of Harmonic analysis and partial differential equations (Chicago, IL, 1996), p. 223, Chicago Lectures in Math. (Univ. Chicago Press, Chicago, IL, 1999) Google Scholar
  17. L. Lempert, R. Szöke, Math. Ann. 290, 689 (1991) Google Scholar
  18. L. Lempert, R. Szöke, Canad. Math. Bull. 44, 70 (2001) Google Scholar
  19. D. Mumford, Algebraic geometry. I. Complex projective varieties, Classics in Mathematics (Springer-Verlag, Berlin, 1995) Google Scholar
  20. N. Nadirashvili, J.A. Toth, D. Yakobson, Uspekhi Mat. Nauk 56, 67 (2001); Trans. Russ. Math. Surv. 56, 1085 (2001) Google Scholar
  21. J. Neuheisel, The asymptotic distribution of nodal sets on spheres, Ph.D. thesis (Johns Hopkins University, 2000) Google Scholar
  22. S. Nonnenmacher, A. Voros, J. Statist. Phys. 92, 431 (1998) Google Scholar
  23. G. Patrizio, P.M. Wong, Math. Ann. 289, 355 (1991) Google Scholar
  24. M. Riesz, Acta Math. 81, 1 (1949) Google Scholar
  25. O. Schramm, S. Sheffield, Contour lines of the two-dimensional discrete Gaussian free field, math.PR/0605337 Google Scholar
  26. B. Shiffman, S. Zelditch, Comm. Math. Phys. 200, 661 (1999) Google Scholar
  27. B. Shiffman, S. Zelditch (in preparation) Google Scholar
  28. A.I. Shnirelman, Usp. Mat. Nauk. 29, 181 (1974) Google Scholar
  29. J.A. Toth, S. Zelditch (in preparation) Google Scholar
  30. S. Zelditch, Invent. Math. (to appear) Google Scholar
  31. S. Zelditch, Comm. Math. Phys. 146, 61 (1992); Internat. Math. Res. Not. 115 (1996) Google Scholar
  32. S. Zelditch, Encyclopedia of Mathematical Physics, edited by J.-P. Françoise, G.L. Naber, S.T. Tsou (Oxford: Elsevier, 2006), math-ph/0503026 Google Scholar
  33. Ya.B. Zel'dovich, A.A. Ruzmaikin, D.D. Sokoloff, The Almighty Chance. World Scientific Lecture Notes in Physics, 20 World Scientific Publishing Co. Inc. (River Edge, NJ, 1990) Google Scholar

Copyright information

© EDP Sciences/Società Italiana di Fisica/Springer-Verlag 2007

Authors and Affiliations

  1. 1.Department of MathematicsJohns Hopkins UniversityBaltimoreUSA

Personalised recommendations