The Sound of Symmetry

Abstract The inverse spectral problem was popularized by M. Kac's 1966 article in THIS MONTHLY “Can one hear the shape of a drum?” Although the answer has been known for over twenty years, many open problems remain. Intended for general audiences, readers are challenged to complete exercises throughout this interactive introduction to inverse spectral theory. The main techniques used in inverse spectral problems are collected and discussed, then used to prove that one can hear the shape of: parallelograms, acute trapezoids, and the regular n-gon. Finally, we show that one can realistically hear the shape of the regular n-gon amongst all convex n-gons because it is uniquely determined by a finite number of eigenvalues; the sound of symmetry can really be heard!

[1]  H. Weinberger,et al.  Some isoperimetric inequalities for membrane frequencies and torsional rigidity , 1961 .

[2]  M. Berg,et al.  Heat equation for a region in R2 with a polygonal boundary , 1988 .

[3]  Carolyn S. Gordon.  You Can’t Hear the Shape of a Manifold , 1992 .

[4]  S. Rosenberg.  The Laplacian on a Riemannian Manifold: The Laplacian on a Riemannian Manifold , 1997 .

[5]  S. J. Chapman,et al.  Drums That Sound the Same , 1995 .

[6]  Lloyd N. Trefethen,et al.  Schwarz-Christoffel Mapping , 2002 .

[7]  P. Buser.  Isospectral Riemann surfaces , 1986 .

[8]  Luc Hillairet,et al.  Contribution of periodic diffractive geodesics , 2005 .

[9]  G. Pólya,et al.  Isoperimetric inequalities in mathematical physics , 1951 .

[10]  Leonid Friedlander,et al.  On the spectrum of the Dirichlet Laplacian in a narrow strip , 2007 .

[11]  J. Steiner,et al.  Einfache Beweise der isoperimetrischen Hauptsätze. , 1838 .

[12]  R. Courant,et al.  Methoden der mathematischen Physik , .

[13]  Ben Andrews,et al.  Proof of the fundamental gap conjecture , 2010, 1006.1686.

[14]  Carolyn S. Gordon.  When you can’t hear the shape of a manifold , 1989 .

[15]  Kevin Barraclough,et al.  I and i , 2001, BMJ : British Medical Journal.

[16]  P. Freitas.  Precise bounds and asymptotics for the first Dirichlet eigenvalue of triangles and rhombi , 2007 .

[17]  On the inverse spectral problem for polygonal domains , 1988 .

[18]  David L. Webb,et al.  One cannot hear the shape of a drum , 1992, math/9207215.

[19]  E. Witt,et al.  Eine Identität zwischen Modulformen zweiten Grades , 1941 .

[20]  Zhiqin Lu,et al.  The Fundamental Gap and One-Dimensional Collapse , 2014 .

[21]  J. Milnor,et al.  EIGENVALUES OF THE LAPLACE OPERATOR ON CERTAIN MANIFOLDS. , 1964, Proceedings of the National Academy of Sciences of the United States of America.

[22]  Viktor Blåsjö,et al.  The Isoperimetric Problem , 2005, Am. Math. Mon..

[23]  Toshikazu Sunada,et al.  Riemannian coverings and isospectral manifolds , 1985 .

[24]  Pedro R. S. Antunes,et al.  On the inverse spectral problem for Euclidean triangles , 2011, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[25]  Å. Pleijel.  A study of certain Green's functions with applications in the theory of vibrating membranes , 1954 .

[26]  D. Grieser,et al.  Hearing the shape of a triangle , 2012, 1208.3163.

[27]  I. Chavel.  Eigenvalues in Riemannian geometry , 1984 .

[28]  David L. Webb,et al.  Isospectral plane domains and surfaces via Riemannian orbifolds , 1992 .

[29]  H. Weyl.  Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen (mit einer Anwendung auf die Theorie der Hohlraumstrahlung) , 1912 .

[30]  H. McKean,et al.  Curvature and the Eigenvalues of the Laplacian , 1967 .

[31]  V. Guillemin,et al.  The spectrum of positive elliptic operators and periodic bicharacteristics , 1975 .

[32]  Claudio Perez Tamargo.  Can one hear the shape of a drum , 2008 .

[33]  Pedro Freitas,et al.  New Bounds for the Principal Dirichlet Eigenvalue of Planar Regions , 2006, Exp. Math..