Weak localization in simple domains Binh NGUYEN Denis

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

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

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)

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

Localized wave observed in ultrasound experiments H. Hu et al, Nature Physics 4, 945 (2008).

Laplacian eigenfunctions No potential !

$Laplacian eigenfunctions Since 1990 s, many studies of vibrations of irregular or fractal drums$

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

$Laplacian eigenfunctions Since 1990 s, many studies of vibrations of irregular or fractal drums$

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

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

$Laplacian eigenfunctions Since 1990 s, many studies of vibrations of irregular or fractal from$

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

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

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

Bottle-neck localization 1 0. 5 1 2 1 0. 5 1 = 1 Bottle-neck domain No localization !

Bottle-neck localization 1 1 0. 5 1 = 0. 5 Bottle-neck domain More localized !

Bottle-neck localization 1 1 0. 5 1 = 0. 3 Bottle-neck domain More and more localized !

Bottle-neck localization 1 1 0. 5 1 = 0. 3 Bottle-neck domain Some eigenfunctions are not localized !

Bottle-neck localization 1 1 0. 5 1 = 0. 3 Bottle-neck domain

Bottle-neck localization 1 1 0. 5 1 = 0. 1 Bottle-neck localization only happens when is small enough !!! Only a fraction of eigenfunctions is localized !!!

Domains with branches

Domains with branches This is our definition !

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

Domains with branches

Localization in a convex polygon Localization in a triangle Localization in a quadrangle

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

Localization by a “dust” barrier 1 0. 8 a

Localization by a “dust” barrier 1 0. 8

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

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

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

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

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

Localization in convex, smooth domains (*) Lazutkin , Math. USSRIzv 7, 439 (1973). (*) J. B. Keller, Annals of Physics 9, 24 -75 (1960) 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.

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

Localization in a rectangle? b 0 a

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

Localization in an equilateral triangle ?

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

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

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

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

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

Thank you for your attention !