Unique Identification of Eigenvalues and Coefficients in a Parabolic Problem

This paper discusses uniqueness questions for identification of coefficients in a second-order, linear, one-dimensional, parabolic partial differential equation. Here, the unknowns are spatially-varying coefficients appearing in the equation. The solution of an initial-boundary value problem is observed at one point over a finite time interval. Conditions are given under which the eigenvalues associated with the problem are uniquely determined by such an observation. The coefficients are not uniquely determined. If, however, the equation is in normal form, the single coefficient which appears is, in certain cases, uniquely determined. This can be established by obtaining the spectral function or by obtaining the eigenvalues for two different boundary value problems and applying existing results (I. M. Gelfand and B. M. Levitan (1959), N. Levison (1949)).