Solving the dynamical inverse problem for the Schrödinger equation by the boundary control method

Abstract

We consider the inverse problem of determining the potential in the one-dimensional Schrödinger equation from dynamical boundary observations, which are the range values of the Neumann-to-Dirichlet map. Dynamical boundary data have not been used in the inverse problem for the Schrödinger equation, since the traditional Gelfand–Levitan–Marchenko approach reconstructs the potential from spectral or scattering data. Here we show that one can completely recover the spectral data from the dynamical boundary data. The construction of the spectral data uses new results on exact and spectral controllability for the Schrödinger equation, which we obtain by using the properties of exponential Riesz bases (nonharmonic Fourier series). From the spectral data, we solve the inverse problem using the boundary control method, which—unlike other identification methods based on control and optimization—is consistently linear and, in principle, independent of dimensionality.