Weak localization in simple domainsBinh 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

C. V. Raman et al, Nature, 108, 42, 1921

Lord Rayleigh, Scientific paper 5, p. 615,

Goong Chen et al, SIAM Review, 36, 453, 1994

J. Keller, Annals of Physics 9, 24-75 (1960)

Whispering Gallery Modes

Anderson localization

Potential

Random potential may lead to localization of wave functions !

Localized wave observed in ultrasound experiments

H. Hu et al, Nature Physics 4, 945 (2008).

Laplacian eigenfunctions

No potential !

Laplacian eigenfunctionsSince 1990s, many studies of vibrations of irregular or fractal drums by B. Sapoval et al.

Even et al, Phys. Rev. Let., 83, 726 (1999)

Laplacian eigenfunctionsSince 1990s, 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

S. Felix et al, J. Sound. Vibr. 299, 965 (2007).

Geometrical irregularity maylead to the localizaton of

eigenfunctions!

Laplacian eigenfunctionsSince 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. 0203404Fractal Wall Model in PMC Laboratory, Ecole Polytechnique

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

What is the meaning of localization?

Localization Non-localization

Is the geometrical irregularity IMPORTANT or NOT ?

Bottle-neck localization

1

1 1

0.5

0.5

No localization !Bottle-neck domain

= 1

1

2

Bottle-neck localization

1

1 1

0.5

0.5

More localized !Bottle-neck domain

= 0.5

Bottle-neck localization

1

1 1

0.5

0.5

More and more localized !Bottle-neck domain

= 0.3

Bottle-neck localization

1

1 1

0.5

0.5

Some eigenfunctions are not localized !

Bottle-neck domain

= 0.3

Bottle-neck localization

1

1 1

0.5

0.5

Bottle-neck domain

= 0.3

Bottle-neck localization

1

1 1

0.5

0.5

Bottle-neck domain

= 0.3

Bottle-neck localization

1

1 1

0.5

0.5

Bottle-neck localization only happens when is small enough !!!Only a fraction of eigenfunctions is localized !!!

= 0.1

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

Localization in a triangle Localization in a quadrangle

Localization in a convex polygon

B. T. Nguyen, D. Grebenkov, Localization in triangles (in preparation)

Low-frequency localization happens in many convex polygons!

Localization by a “dust” barrier

a

1

0.8

Localization by a “dust” barrier

0.8

1

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…

N. Burq, M. Zworski, SIAM Rev., 47, 43 (2005)

dense subsequence

Localization in a disk…

Dirichlet boundary condition Neumann boundary condition

Disks are “localizable” !

Can high-frequency localization happen ?

VCan high-frequency localization happen?

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)

Localization in a rectangle?

0 a

b

No localization in this domain !

Localization in a rectangle?

0 a

b

Localization in a rectangle?

0 a

b

V

N. Burq, M. Zworski, SIAM Rev., 47, 43 (2005)

V

Localization in an equilateral triangle ?

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 ?

B. T. Nguyen, D. Grebenkov, Weak Localization in Simple Domains (in preparation)

All symmetric eigenfunctions are non-localized !

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

Summary

Non-convex domains

Convex, smooth domains Convex polygons

- Exist “bottle-neck” eigenfunctions in some domains. - Always exist “whispering

gallery modes” in all domains.

- Happens in disks, ellipses.Others ?

VV

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 ?

Thank you for your attention !