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Semiclassical asymptotics of the solution to the Cauchy problem for the Schrödinger equation with a delta potential localized on a codimension 1 surface. (English. Russian original) Zbl 1455.35213

Proc. Steklov Inst. Math. 310, 304-313 (2020); translation from Tr. Mat. Inst. Steklova 310, 322-331 (2020).
The authors consider the Schrödinger equation \(ih\frac{\partial \psi }{ \partial t}=-\frac{h^{2}}{2}\Delta \psi +V(x)\psi +\frac{q(y)}{h}\delta _{M}\psi \), in \(\mathbb{R}^{n}\), where \(\delta _{M}\) is a delta function defined on a smooth and oriented \((n-1)\)-dimensional surface \(M\), \(q(y)\) is a smooth real function defined on the surface \(M\), and \(V(x)\) is a smooth real function. The initial condition \(\psi (x,0)=e^{iS_{0}(x)/h}\phi _{0}(x)\) is imposed, where \(S_{0}(x)\) and \(\phi _{0}(x)\) are smooth real functions, the support of \(\phi _{0}\) being compact and lying at a positive distance from \(M\). Introducing the operator \(\widehat{H}=-\frac{h^{2}}{2}\Delta +V(x)+ \frac{q(y)}{h}\delta _{M}\) defined as a self-adjoint extension of the operator \(H_{0}=-\frac{h^{2}}{2}\Delta +V(x)\) restricted to the functions vanishing on \(M\), the authors observe that its domain consists of functions satisfying the boundary conditions on \(M\): \(\psi (y-0,t)=\psi (y+0,t)\), \(h( \frac{\partial \psi }{\partial m}(y-0,t)-\frac{\partial \psi }{\partial m} (y+0,t))=q(y)\psi (y,t)\). The purpose of the paper is to describe the asymptotics of the solution to the Cauchy problem as \(h\rightarrow 0+\). The authors quote from the book by V.P. Maslov and M.V. Fedoryuk [Semi-classical approximation in quantum mechanics. Dordrecht-Boston-London: D. Reidel Publishing Company (1981; Zbl 0458.58001)], the WKB asymptotics of the solution to the Cauchy problem \(ih\frac{\partial \psi }{\partial t}=-\frac{ h^{2}}{2}\Delta \psi +V(x)\psi \), \(\psi (x,0)=e^{iS_{0}(x)/h}\varphi ^{0}(x)\) , where \(V(x)\) is a smooth function, in the absence and then in the presence of focal points. Considering the Cauchy problem with delta function, the authors first suppose the absence of focal points. They prove a representation result of the solution, introducing the solutions of Hamilton-Jacobi and transport equations. In the presence of focal points, they construct two Lagrangian manifolds respectively corresponding to the transmitted and reflected waves and they again prove a representation of the solution to the Cauchy problem.

MSC:

35Q41 Time-dependent Schrödinger equations and Dirac equations
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
81Q20 Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory
53D12 Lagrangian submanifolds; Maslov index
35B40 Asymptotic behavior of solutions to PDEs

Citations:

Zbl 0458.58001
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References:

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