Access

You are not currently logged in.

Access your personal account or get JSTOR access through your library or other institution:

login

Log in to your personal account or through your institution.

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

Z. Schuss and A. Spivak
SIAM Journal on Applied Mathematics
Vol. 66, No. 1 (Oct. - Nov., 2005), pp. 339-360
Stable URL: http://www.jstor.org/stable/4096172
Page Count: 22
  • Subscribe ($19.50)
  • Cite this Item
On Recovering the Shape of a Domain from the Trace of the Heat Kernel
Preview not available

Abstract

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 also known 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 the trace of the heat kernel (the "resurgence" problem) is unresolved. In this paper we develop a simple algorithm for extracting the lengths from the short-time hyperasymptotic expansion of the trace. We give an alternative construction of the short-time expansion of the trace by constructing a ray approximation to the heat kernel for a planar domain with Dirichlet or Neumann boundary conditions. We evaluate the trace by introducing the rays as global coordinates.

Page Thumbnails

  • Thumbnail: Page 
339
    339
  • Thumbnail: Page 
340
    340
  • Thumbnail: Page 
341
    341
  • Thumbnail: Page 
342
    342
  • Thumbnail: Page 
343
    343
  • Thumbnail: Page 
344
    344
  • Thumbnail: Page 
345
    345
  • Thumbnail: Page 
346
    346
  • Thumbnail: Page 
347
    347
  • Thumbnail: Page 
348
    348
  • Thumbnail: Page 
349
    349
  • Thumbnail: Page 
350
    350
  • Thumbnail: Page 
351
    351
  • Thumbnail: Page 
352
    352
  • Thumbnail: Page 
353
    353
  • Thumbnail: Page 
354
    354
  • Thumbnail: Page 
355
    355
  • Thumbnail: Page 
356
    356
  • Thumbnail: Page 
357
    357
  • Thumbnail: Page 
358
    358
  • Thumbnail: Page 
359
    359
  • Thumbnail: Page 
360
    360