On the Cauchy problem for the Schrödinger equation degenerating outside a segment: properties of solutions and spectral aspects of the regularization

The evolutionary Schrödinger equation with a second-order generator is considered on the line. For the Schrödinger equation with a degenerate operator, the characteristic form of which vanishes outside a segment $I=[-l,l]\subset\mathbb R$, a well-posed setting of the Cauchy problem is investigated. We find conditions for the initial-value data of the problem that are necessary and sufficient for its unique solvability in a given interval of time. A sequence of regularized Cauchy problems with uniformly elliptic operators is considered as well; we study the convergence of the sequence of solutions for nondegenerate problems to the solution of the degenerate problem as well as the convergence of regularized semigroups of transformations in the strong operator topology. We prove that any arbitrary sequence of solutions of regularized problems with initial data that does not satisfy the existence condition for the solution diverges. However, one cannot exclude that there exists a subsequence of the regularization parameters such that the corresponding sequence of regularized semigroups converges (in the strong operator topology) uniformly on each segment. We provide a description of the set of all possible partial limits for the sequence of regularized semigroups; this description is given in terms of a collection of selfadjoint extensions for the degenerate operator. It is still an open question if all those partial limits are accessible.
The Cauchy problem for the Schrödinger equation, the generator of which is a symmetric linear differential operator in the Hilbert space $H=L_2(\mathbb R)$, is considered as well. We investigate if the behavior of the sequence of regularized semigroups depends on the choice of the regularization for the generator. We define a linear selfadjoint regularization of the Cauchy problem with a degenerate operator as a directed set of well-posed problems approximating the original one. We define a correct regularization as a linear selfadjoint regularization of the degenerate operator such that its index of error determines the well-posedness and the convergence and weak convergence of the sequence of regularized solutions. We find necessary and sufficient conditions for the convergence (in the strong and weak operator topologies) of the sequence of correctly regularized semigroups.