Regularization of the Degenerate Schrödinger Equation as a Random Process

We study random processes with values in the space of quantum states related to the Cauchy problem for the Schrödinger equation with discontinuous degenerate coefficients and its elliptic regularization. We obtain conditions that allow one to determine the process by its expected value at an arbitrary instant.     

Formulation and well-posedness of the Cauchy problem for a diffusion equation with discontinuous degenerating coefficients

The choice of a differential diffusion operator with discontinuous coefficients that corresponds to a finite flow velocity and a finite concentration is substantiated. For the equation with a uniformly elliptic operator and a nonzero diffusion coefficient, conditions are established for the existence and uniqueness of a solution to the corresponding Cauchy problem. For the diffusion equation with degeneration on a half-line, it is proved that the Cauchy problem with an arbitrary initial condition has a unique solution if and only if there is no flux from the degeneration domain to the ellipticity domain of the operator. Under this condition, a sequence of solutions to regularized problems is proved to converge uniformly to the solution of the degenerate problem in L ₁(R) on each interval.     

On Properties of Solutions of the Cauchy Problem for the Schrodinger Equation Degenerate on the Half-Line

We consider the Schrodinger equation on the half-line describing a particle with mass depending on its location. We study the Cauchy problem for the Schrodinger equation with degenerate operator whose characteristic form vanishes on the half-line. A sequence of regularizing Cauchy problems with uniformly elliptic operators is considered, and the convergence of the sequence of solutions of nondegenerate problems to the solution of the degenerate problem is examined.__________Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 10, Suzdal Conference-4, 2003.     

On Self-Adjoint Extensions of Schrödinger Operators Degenerating on a Pair of Half-Lines and the Corresponding Markovian Cocycles

Mathematical Notes - We consider the Schrödinger equation for a quantum particle whose mass depends on the position of the particle on the real line. The well-posedness of the Cauchy problem...     

Degeneration and regularization of the operator in the Cauchy problem for the Schrödinger equation

The Cauchy problem for the Schrödinger equation whose operator degenerates on a half-line is studied. In order to approximate a solution to the problem with degeneracy by solutions to well-posed problems, the notion of regularization for an operator with degeneracy is introduced; an approximative solution to a problem with degeneracy is defined as the limit of a sequence of regularized problems. The dependence of the approximative solution on the choice of the class of admissible regularizations is studied. The weak compactness of sequences of states determined by sequences of solutions to regularized problems in the topologies determined by the space of all bounded linear operators and by subspaces of mutually commuting bounded linear operators is investigated.