Nonlocal Schrödinger Problem with Time Dependent Self-Adjoint Operator

In this paper, in an arbitrary Hilbert space nonlocal boundary value problem for the Schrödinger equation with time dependent self-adjoint operator is studied. Stability estimates for the solution of this problem is established. To find an approximate solution of nonlocal boundary value problem for the Schrödinger equation with time dependent self-adjoint operator first order of accuracy Rothe difference scheme and second order of accuracy Crank-Nicholson difference scheme are constructed. Stability estimates of these difference schemes have been obtained. To obtain stability estimates, the theory of spectral representation of self-adjoint operator is used. In order to support theory, one dimensional in space variable, nonlocal in time variable and with a time dependent self-adjoint operator a numerical example for the Schrödinger problem is given. A modified Gauss elimination method is used to solve the difference schemes.