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Weak localization in simple domains Binh NGUYEN, Denis GREBENKOV Laboratoire de Physique de la Matière Condensée Ecole Polytechnique, FRANCE
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Plan of the talk Historical overview and related problems Low-frequency localization High-frequency localization Summary.
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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
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Anderson localization Potential Random potential may lead to localization of wave functions !
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Localized wave observed in ultrasound experiments H. Hu et al, Nature Physics 4, 945 (2008).Nature Physics 4, 945 (2008)
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Laplacian eigenfunctions No potential !
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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)
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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)
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Laplacian eigenfunctions S. Felix et al, J. Sound. Vibr. 299, 965 (2007). Geometrical irregularity may lead to the localizaton of eigenfunctions!
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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
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Plan of the talk Historical overview and related problems Low-frequency localization High-frequency localization Summary.
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What is the meaning of localization? Localization Non-localization Is the geometrical irregularity IMPORTANT or NOT ?
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Bottle-neck localization 1 11 0.5 No localization ! Bottle-neck domain = 1 1 2
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Bottle-neck localization 1 11 0.5 More localized ! Bottle-neck domain = 0.5
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Bottle-neck localization 1 11 0.5 More and more localized ! Bottle-neck domain = 0.3
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Bottle-neck localization 1 11 0.5 Some eigenfunctions are not localized ! Bottle-neck domain = 0.3
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Bottle-neck localization 1 11 0.5 Bottle-neck domain = 0.3
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Bottle-neck localization 1 11 0.5 Bottle-neck domain = 0.3
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Bottle-neck localization 1 11 0.5 Bottle-neck localization only happens when is small enough !!! Only a fraction of eigenfunctions is localized !!! = 0.1
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Domains with branches
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This is our definition !
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Domains with branches A. Delytsin, B. T. Nguyen, D. Grebenkov, Exponential Decay of Laplacian eigenfunctions in domains with branches (submitted )
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Domains with branches
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Localization in a convex polygon Localization in a triangleLocalization in a quadrangle
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Localization in a convex polygon Localization in a triangleLocalization in a quadrangle
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Localization in a convex polygon B. T. Nguyen, D. Grebenkov, Localization in triangles (in preparation) Low-frequency localization happens in many convex polygons!
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Localization by a “dust” barrier a 1 0.8
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Localization by a “dust” barrier 0.8 1
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Localization by a “dust” barrier Uniform distribution in “dust” barrier leads to low-frequency localization ! Uniform distributionNon-uniform distribution
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Plan of the talk Historical overview and related problems Low-frequency localization High-frequency localization Summary.
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From Shnirelman theorem… N. Burq, M. Zworski, SIAM Rev., 47, 43 (2005) dense subsequence
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Localization in a disk… Dirichlet boundary conditionNeumann boundary condition Disks are “localizable” !
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Can high-frequency localization happen ? V
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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)
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Localization in a rectangle? 0 a b No localization in this domain !
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Localization in a rectangle? 0 a b
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0 a b V N. Burq, M. Zworski, SIAM Rev., 47, 43 (2005) V
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Localization in an equilateral triangle ?
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M. Pinsky, SIAM J.Math.Anal, 11, 819 (1980) M. Pinsky, SIAM J.Math.Anal, 16, 848 (1985)
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Localization in an equilateral triangle ? B. T. Nguyen, D. Grebenkov, Weak Localization in Simple Domains (in preparation) All symmetric eigenfunctions are non-localized !
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Plan of the talk Historical overview and related problems Low-frequency localization High-frequency localization. Summary.
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Summary Non-convex domains Convex, smooth domains Convex polygons Low frequency High frequency - Exist “bottle-neck” eigenfunctions in some domains. - Always exist “whispering gallery modes” in all domains. - Happens in disks, ellipses. Others ? V V
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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 ?
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Thank you for your attention !
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