Geodesic
On Self-Adjoint Extensions of Schrödinger Operators Degenerating on a~Pair of Half-Lines and the Corresponding Markovian Cocycles
Matematičeskie zametki, Tome 76 (2004) no. 3, pp. 335-343

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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. 
@article{MZM_2004_76_3_a2,
     author = {G. G. Amosov and V. Zh. Sakbaev},
     title = {On {Self-Adjoint} {Extensions} of {Schr\"odinger} {Operators} {Degenerating} on {a~Pair} of {Half-Lines} and the {Corresponding} {Markovian} {Cocycles}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {335--343},
     publisher = {mathdoc},
     volume = {76},
     number = {3},
     year = {2004},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2004_76_3_a2/}
}
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G. G. Amosov; V. Zh. Sakbaev. On Self-Adjoint Extensions of Schrödinger Operators Degenerating on a~Pair of Half-Lines and the Corresponding Markovian Cocycles. Matematičeskie zametki, Tome 76 (2004) no. 3, pp. 335-343. http://geodesic.mathdoc.fr/item/MZM_2004_76_3_a2/