Localization of Laplacian Eigenfunctions in Circular, Spherical, and Elliptical Domains

We consider Laplacian eigenfunctions in circular, spherical, and elliptical domains in order to discuss three kinds of high-frequency localization: whispering gallery modes, bouncing ball modes, and focusing modes. Although the existence of these modes has been known for a class of convex domains, the separation of variables for circular, spherical, and elliptical domains helps us to better understand the “mechanism” of localization, i.e., how an eigenfunction is getting distributed in a small region of the domain and decays rapidly outside this region. Using the properties of Bessel and Mathieu functions, we derive inequalities which imply and clearly illustrate localization. Moreover, we provide an example of a nonconvex domain (an elliptical annulus) for which the high-frequency localized modes are still present. At the same time, we show that there is no localization in most rectangle-like domains. This observation leads us to formulate an open problem of localization in polygonal domains and, more ge...

[1]  Denis S. Grebenkov,et al.  Exponential decay of Laplacian eigenfunctions in domains with branches of variable cross-sectional profiles , 2011, 1109.3408.

[2]  C. V. Raman,et al.  On the whispering-gallery phenomenon , 1922 .

[3]  I. Graham,et al.  Condition number estimates for combined potential integral operators in acoustics and their boundary element discretisation , 2010, 1007.3074.

[4]  A V Hershey,et al.  Computation of Special Functions , 1978 .

[5]  Dr. M. G. Worster.  Methods of Mathematical Physics , 1947, Nature.

[6]  L. Milne‐Thomson.  A Treatise on the Theory of Bessel Functions , 1945, Nature.

[7]  Lev Kaplan,et al.  Linear and Nonlinear Theory of Eigenfunction Scars , 1998, chao-dyn/9809011.

[8]  John William Strutt,et al.  Scientific Papers: The Problem of the Whispering Gallery , 2009 .

[9]  Pawel Kröger,et al.  On the ground state eigenfunction of a convex domain in Euclidean space , 1996 .

[10]  E. Kirkpatrick.  Tables of values of the modified Mathieu functions , 1960 .

[11]  Joseph B. Keller,et al.  Asymptotic solution of eigenvalue problems , 1960 .

[12]  Robert S. Strichartz,et al.  Localized Eigenfunctions: Here You See Them, There You Don't , 2009, 0909.0783.

[13]  F. W. J. Olver,et al.  Some new asymptotic expansions for Bessel functions of large orders , 1952, Mathematical Proceedings of the Cambridge Philosophical Society.

[14]  Denis S. Grebenkov,et al.  Trapped modes in finite quantum waveguides , 2011, 1112.1160.

[15]  V F Lazutkin,et al.  THE EXISTENCE OF CAUSTICS FOR A BILLIARD PROBLEM IN A CONVEX DOMAIN , 1973 .

[16]  K. Siegel.  An inequality involving Bessel functions of argument nearly equal to their order , 1953 .

[17]  Nicolas Burq,et al.  Bouncing Ball Modes and Quantum Chaos , 2003, SIAM Rev..

[18]  Fayez A. Alhargan,et al.  A Complete Method for the Computations of Mathieu Characteristic Numbers of Integer Orders , 1996, SIAM Rev..

[19]  Dmitry Jakobson,et al.  Geometric properties of eigenfunctions , 2001 .

[20]  H. Stöckmann,et al.  Quantum Chaos: An Introduction , 1999 .

[21]  Vladimir F. Lazutkin,et al.  Kam Theory and Semiclassical Approximations to Eigenfunctions , 1993 .

[22]  Bernard Sapoval,et al.  Localizations in Fractal Drums: An Experimental Study , 1999 .

[23]  Jorge V. José,et al.  Chaos in classical and quantum mechanics , 1990 .

[24]  James Ralston,et al.  On the construction of quasimodes associated with stable periodic orbits , 1979 .

[25]  B. Sapoval,et al.  Vibrations of fractal drums. , 1991, Physical review letters.

[26]  V. M. Babich,et al.  Eigenfunctions Concentrated Near a Closed Geodesic , 1968 .

[27]  G. A. Watson.  A treatise on the theory of Bessel functions , 1944 .

[28]  Ll. G. Chambers,et al.  An upper bound for the first zero of Bessel functions , 1982 .

[29]  Jianxin Zhou,et al.  Visualization of Special Eigenmode Shapes of a Vibrating Elliptical Membrane , 1994, SIAM Rev..

[30]  J. V. Ralston,et al.  Approximate eigenfunctions of the Laplacian , 1977 .

[31]  Randall B. Shirts,et al.  The computation of eigenvalues and solutions of Mathieu's differential equation for noninteger order , 1993, TOMS.

[32]  Gertrude Blanch,et al.  On the computation of Mathieu functions , 1946 .

[33]  B. Sapoval,et al.  Localization and increased damping in irregular acoustic cavities , 2007 .

[34]  F. W. J. Olver,et al.  A further method for the evaluation of zeros of Bessel functions and some new asymptotic expansions for zeros of functions of large order , 1951, Mathematical Proceedings of the Cambridge Philosophical Society.

[35]  R. A. Rimmer,et al.  Modes of elliptical waveguides: a correction , 1990 .

[36]  Ronald Smith.  Bouncing Ball Waves , 1974 .

[37]  N. Mclachlan.  Theory and Application of Mathieu Functions , 1965 .

[38]  D. Frenkel,et al.  Algebraic methods to compute Mathieu functions , 2001 .

[39]  Peter Sarnak,et al.  Recent progress on the quantum unique ergodicity conjecture , 2011 .

[40]  F. B..  Introduction to Bessel Functions , 1939, Nature.

[41]  Walter R. Leeb,et al.  Algorithm 537: Characteristic Values of Mathieu's Differential Equation [S22] , 1979, TOMS.

[42]  Árpád Elbert,et al.  Some recent results on the zeros of Bessel functions and orthogonal polynomials , 2001 .