Determination of a Control Parameter for the Difference Schrödinger Equation

Abstract

The first order of accuracy difference scheme for the numerical solution of the boundary value problem for the differential equation with parameter $p$, $i(du(t)/dt)+Au(t)+iu(t)=f(t)+p$, , $u(0)=\phi$, $u(T)=\psi$, in a Hilbert space $H$ with self-adjoint positive definite operator $A$ is constructed. The well-posedness of this difference scheme is established. The stability inequalities for the solution of difference schemes for three different types of control parameter problems for the Schrödinger equation are obtained.