Reconstructing the potential for the 1D Schrödinger equation from boundary measurements

International audienceWe consider the inverse problem of determining the potential in the dynamical Schrödinger equation on the interval by the measurement on the boundary. We use the Boundary Control method to recover the spectrum of the problem from the observation at either left or right end points. Using the specificity of the one-dimensional situation we recover the spectral function, reducing the problem to the classical one which could be treated by known methods. We apply the algorithm to the situation when only the finite number of eigenvalues are known and prove the convergence of the method

Spin dependent point potentials in one and three dimensions

We consider a system realized with one spinless quantum particle and an array of $N$ spins 1/2 in dimension one and three. We characterize all the Hamiltonians obtained as point perturbations of an assigned free dynamics in terms of some ``generalized boundary conditions''. For every boundary condition we give the explicit formula for the resolvent of the corresponding Hamiltonian. We discuss the problem of locality and give two examples of spin dependent point potentials that could be of interest as multi-component solvable models.Comment: 15 pages, some misprints corrected, one example added, some references modified or adde