Weak localization in simple domains

1. Weak localization in simple domains Binh NGUYEN, Denis GREBENKOV Laboratoire de Physique de la Matière Condensée Ecole Polytechnique, FRANCE

2. Plan of the talk • Historical overview and related problems • Low-frequency localization • High-frequency localization • Summary.

3. Whispering Gallery Modes Saint Paul Cathedral Inside Saint Paul Cathedral Whispering Gallery Modes C. V. Raman et al, Nature, 108, 42, 1921 Goong Chen et al, SIAM Review, 36, 453, 1994 Lord Rayleigh, Scientific paper 5, p. 615, J. Keller, Annals of Physics 9, 24-75 (1960)

4. Anderson localization Random potential may lead to localization of wave functions ! Potential

5. Localized wave observed in ultrasound experiments H. Hu et al, Nature Physics4, 945 (2008).

6. Laplacian eigenfunctions No potential !

7. Laplacian eigenfunctions Since 1990s, many studies of vibrations of irregular or fractal drums by B. Sapoval et al. Even et al, Phys. Rev. Let., 83, 726 (1999)

8. Laplacian eigenfunctions Since 1990s, many studies of vibrations of irregular or fractal drums by B. Sapoval et al. Even et al, Phys. Rev. Let. 83, 726 (1999)

9. Laplacian eigenfunctions Geometrical irregularity may lead to the localizaton of eigenfunctions! S. Felix et al, J. Sound. Vibr. 299, 965 (2007).

10. Laplacian eigenfunctions Since 1990s, many studies of vibrations of irregular or fractal from by B. Sapoval et al. …towards one of many practical applications The Fractal Wall, product of Colas Inc., French patient No. 0203404 Fractal Wall Model in PMC Laboratory, Ecole Polytechnique

11. Plan of the talk • Historical overview and related problems • Low-frequency localization • High-frequency localization • Summary.

12. What is the meaning of localization? Is the geometrical irregularity IMPORTANT or NOT ? Non-localization Localization

13. Bottle-neck localization 1 0.5 1 2 1 1 0.5   = 1 No localization ! Bottle-neck domain

14. Bottle-neck localization 1 0.5 1 1 0.5   = 0.5 More localized ! Bottle-neck domain

15. Bottle-neck localization 1 0.5 1 1 0.5   = 0.3 More and more localized ! Bottle-neck domain

16. Bottle-neck localization 1 0.5 1 1 0.5   = 0.3 Some eigenfunctions are not localized ! Bottle-neck domain

17. Bottle-neck localization 1 0.5 1 1 0.5   = 0.3 Bottle-neck domain

18. Bottle-neck localization 1 0.5 1 1 0.5   = 0.3 Bottle-neck domain

19. Bottle-neck localization 1 0.5 1 1 0.5   = 0.1 Bottle-neck localization only happens when  is small enough !!! Only a fraction of eigenfunctions is localized !!!

20. Domains with branches

21. Domains with branches This is our definition !

22. Domains with branches A. Delytsin, B. T. Nguyen, D. Grebenkov, Exponential Decay of Laplacian eigenfunctions in domains with branches (submitted )

23. Domains with branches

24. Localization in a convex polygon Localization in a triangle Localization in a quadrangle

25. Localization in a convex polygon Localization in a triangle Localization in a quadrangle

26. Localization in a convex polygon Low-frequency localization happens in many convex polygons! B. T. Nguyen, D. Grebenkov, Localization in triangles (in preparation)

27. Localization by a “dust” barrier 1 0.8 a

28. Localization by a “dust” barrier 1 0.8

29. Localization by a “dust” barrier Uniform distribution in “dust” barrier leads to low-frequency localization ! Uniform distribution Non-uniform distribution

30. Plan of the talk • Historical overview and related problems • Low-frequency localization • High-frequency localization • Summary.

31. From Shnirelman theorem… dense subsequence N. Burq, M. Zworski, SIAM Rev., 47, 43 (2005)

32. Localization in a disk… Disks are “localizable” ! Dirichlet boundary condition Neumann boundary condition

33. Can high-frequency localization happen ? Can high-frequency localization happen? V

34. Localization in convex, smooth domains Theorem (*): In a convex, smooth and bounded domain, there always exist some eigenmodes, called whispering gallery modes. These eigenfunctions are mainly distributed near the boundary, and decay exponentially inside. (*) Lazutkin , MathUSSRIzv 7, 439 (1973). (*) J. B. Keller, Annals of Physics 9, 24-75 (1960)

35. Localization in a rectangle? b  No localization in this domain ! 0 a

36. Localization in a rectangle? b  0 a

37. Localization in a rectangle? b  V V 0 a N. Burq, M. Zworski, SIAM Rev., 47, 43 (2005)

38. Localization in an equilateral triangle ?

39. Localization in an equilateral triangle ?

40. Localization in an equilateral triangle ? M. Pinsky, SIAM J.Math.Anal, 11, 819 (1980) M. Pinsky, SIAM J.Math.Anal, 16, 848 (1985)

41. Localization in an equilateral triangle ? All symmetric eigenfunctions are non-localized ! B. T. Nguyen, D. Grebenkov, Weak Localization in Simple Domains (in preparation)

42. Plan of the talk • Historical overview and related problems • Low-frequency localization • High-frequency localization. • Summary.

43. Summary Low frequency Non-convex domains Convex, smooth domains Convex polygons V V - Exist “bottle-neck” eigenfunctions in some domains. - Always exist “whispering gallery modes” in all domains. High frequency - Happens in disks, ellipses. Others ?

44. Questions • Does localization exist in equilateral polygons ? • Is there a relation to the curvature of the boundary ? • Is it related to scarring and chaotic systems? • Does localization happen in Neumann boundary condition or others ? What is localization ?

45. Thank you for your attention !