Nguyen by chenmeixiu

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									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
                 wave
localization of Potential functions !
                     Localized wave observed
                    in ultrasound experiments




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




                     No potential !
                 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)
                  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)
                 Laplacian eigenfunctions

           Geometrical irregularity may
            lead to the localizaton of
                 eigenfunctions!

S. Felix et al, J. Sound. Vibr. 299, 965 (2007).
             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
               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
     1              0.5
                                 2




1             0.5        1

              

                                           =1

    Bottle-neck domain               No localization !
          Bottle-neck localization
     1              0.5



1             0.5        1

              

                                   = 0.5

    Bottle-neck domain       More localized !
          Bottle-neck localization
     1              0.5



1             0.5        1

              

                                        = 0.3

    Bottle-neck domain       More and more localized !
          Bottle-neck localization
     1              0.5



1             0.5        1

              

                                       = 0.3

    Bottle-neck domain       Some eigenfunctions are
                                 not localized !
          Bottle-neck localization
     1              0.5



1             0.5        1

              

                                = 0.3

    Bottle-neck domain
          Bottle-neck localization
     1              0.5



1             0.5        1

              

                                = 0.3

    Bottle-neck domain
          Bottle-neck localization
     1               0.5



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




  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
 V


           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?
b


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


     



0         a
                  Localization in a rectangle?
  b


                      

                V             V
  0                                    a




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


    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

                           Non-convex           Convex, smooth domains           Convex polygons
                           domains
frequency frequency




                      - Exist “bottle-neck”
                      eigenfunctions in some
          Low




                      domains.                 - Always exist “whispering
                                               gallery modes” in all
                                               domains.
                                                 - Happens in disks, ellipses.
High




                                                          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 !

								
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