Geodesic
Semiclassical Asymptotics of the Solution to the Cauchy Problem for the Schr\"odinger Equation with a Delta Potential Localized on a Codimension 1 Surface
Informatics and Automation, Selected issues of mathematics and mechanics, Tome 310 (2020), pp. 322-331

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We describe the semiclassical asymptotics of the solution to the Cauchy problem for the Schrödinger equation with a delta potential localized on a codimension $1$ surface. The initial condition represents a rapidly oscillating wave packet. We show that the asymptotics is expressed in terms of the Maslov canonical operator on a pair of Lagrangian manifolds in the extended phase space; the form of the delta potential defines a mapping between these manifolds that describes the reflection and scattering of the wave packet. 
@article{TRSPY_2020_310_a21,
     author = {A. I. Shafarevich and O. A. Shchegortsova},
     title = {Semiclassical {Asymptotics} of the {Solution} to the {Cauchy} {Problem} for the {Schr\"odinger} {Equation} with a {Delta} {Potential} {Localized} on a {Codimension} 1 {Surface}},
     journal = {Informatics and Automation},
     pages = {322--331},
     publisher = {mathdoc},
     volume = {310},
     year = {2020},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TRSPY_2020_310_a21/}
}
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A. I. Shafarevich; O. A. Shchegortsova. Semiclassical Asymptotics of the Solution to the Cauchy Problem for the Schr\"odinger Equation with a Delta Potential Localized on a Codimension 1 Surface. Informatics and Automation, Selected issues of mathematics and mechanics, Tome 310 (2020), pp. 322-331. http://geodesic.mathdoc.fr/item/TRSPY_2020_310_a21/