On Recovering the Shape of a Domain from the Trace of the Heat Kernel

The problem of recovering geometric properties of a domain from the trace of the heat kernel for an initial-boundary value problem arises in NMR microscopy and other applications. It is similar to the problem of "hearing the shape of a drum," for which a Poisson-type summation formula relates geometric properties of the domain to the eigenvalues of the Dirichlet or Neumann problems for the Laplace equation. It is well known that the area, circumference, and the number of holes in a planar domain can be recovered from the short-time asymptotics of the solution of the initial-boundary value problem for the heat equation. It is alsoknown that the length spectrum of closed billiard ball trajectories in the domain is contained in the spectral density of the Laplace operator with the given boundary conditions in the domain, from which the short-time hyperasymptotics of the trace of the heat kernel can be obtained by the Laplace transform. However, the problem of recovering these lengths from measured values of ...

[1]  Joseph B. Keller,et al.  ASYMPTOTIC METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS: THE REDUCED WAVE EQUATION AND MAXWELL'S EQUATION , 1995 .

[2]  J. Keller,et al.  Asymptotic Analysis of Diffusion Equations in Population Genetics , 1978 .

[3]  P. Greiner.  An asymptotic expansion for the heat equation , 1971 .

[4]  V. Guillemin,et al.  The Poisson Summation Formula for Manifolds with Boundary , 1979 .

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

[6]  Jack K. Cohen,et al.  A Ray Method for the Asymptotic Solution of the Diffusion Equation , 1967 .

[7]  H. Pfeifer.  Principles of Nuclear Magnetic Resonance Microscopy , 1992 .

[8]  Joseph B. Keller,et al.  Geometrical Theory of Diffraction in Inhomogeneous Media , 1959 .

[9]  R. Balian,et al.  Distribution of eigenfrequencies for the wave equation in a finite domain: III. Eigenfrequency density oscillations , 1972 .

[10]  J. Keller,et al.  Geometrical theory of diffraction. , 1962, Journal of the Optical Society of America.

[11]  M. Kac.  Can One Hear the Shape of a Drum , 1966 .

[12]  Y. Colin.  Spectre du laplacien et longueurs des géodésiques périodiques. I , 2006 .

[13]  Joseph B. Keller,et al.  Short time asymptotic expansions of solutions of parabolic equations , 1972 .

[14]  Zeev Schuss,et al.  Theory and Applications of Stochastic Differential Equations , 1980 .

[15]  Steve Zelditch,et al.  Spectral determination of analytic bi-axisymmetric plane domains , 2000 .

[16]  M. Berry,et al.  High orders of the Weyl expansion for quantum billiards: resurgence of periodic orbits, and the Stokes phenomenon , 1994, Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences.

[17]  L. Arnold.  Stochastic Differential Equations: Theory and Applications , 1992 .

[18]  K. Stewartson,et al.  On hearing the shape of a drum: further results , 1971, Mathematical Proceedings of the Cambridge Philosophical Society.