NMR Microscopy and Hearing the Shape of a Drum 

 

Abstract

NMR microscopy measures the time decay of the intensity of magnetic radiation of polarized excited dipoles in a given region (e.g., in a biological cell). This intensity is proportional to the integral over the region of the probability density of return to the point of departure of a Brownian particle diffusing in the domain with absorption or reflection at the boundary. This equals the trace of Green's function of the initial-boundary-value problem for the heat equation in the domain (i.e., its integral on the diagonal). The mathematical problem of NMR microscopy is to recover geometric properties of the domain from the measured trace. The problem of ``hearing the shape of a drum'' is to recover the shape of a drum membrane from the sound it emits. Mark Kac has shown that the area, circumference, and the number of holes in the membrane can be recovered from the expansion in powers of t of the trace of the heat equation in the membrane. Additional geometric properties have been conjectured from the large s expansion of the Laplace transform of the trace (the Weyl series). These include the lengths of 2-periodic orbits of a billiard ball in the membrane. We use the ray method to construct a short time expansion ``beyond all orders'' of Green's function and use it to find that the exponents in the transcendentally small terms in the short time expansion of the trace are the squares of half the lengths of the extremal periodic orbits of billiard balls in the domain. This confirms and extends a conjecture of Berry and Howles.