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

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 is studied for the Schrödinger equation with characteristic form degenerating outside the finite segment $I=[-l,l]\subset\mathbb R$. We show that this problem generates a unitary Markovian cocycle.