We prove that we can uniquely recover the coefficient of a one dimensional heat equation from a finite set of measurements and provide a constructive procedure for its recovery. The algorithm is based on the well known Gelfand-Levitan-Gasymov inverse spectral theory of Sturm-Liouville operators. By using a hot spot, as a first initial condition, we determine nearly all except maybe a finite number of spectral data. A counting procedure helps detect the number of missing data which is then unraveled by a finite number of measurements.

In this article we recover a coefficient in a multidimensional inverse problem for a heat equation. We show that a sequence of measurements taken at the same point on the boundary but at different times is enough to determine the coefficient uniquely. We provide an identifiability algorithm for both Dirichlet and Neumann lateral boundary conditions and we examine the smoothness of the recovered coefficient.

In this paper we derive a trace formula for the Schrödinger operator on the half-line. As a consequence we obtain a Schmincke type inequality with sharp constant. The main tool in our investigation is the inverse spectral Gelfand-Levitan theory, which allows us to compare two Schrödinger operators whose spectra differ by few eigenvalues.

In the first part of this article, we show that we can recover the coefficient q in the one-dimensional wave equation from a finite number of special lateral measurements. Moreover, if some estimates on the size of q are available, then q can be recovered from a single boundary measurement. In the second part we treat the multidimensional case and show how we can reconstruct the coefficient q from a sequence of boundary measurements taken at one point only.