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Initial value problem for the time-dependent linear Schrödinger equation with a point singular potential by the unified transform method. (English) Zbl 1406.35316

Summary: We study an initial value problem for the one-dimensional non-stationary linear Schrödinger equation with a point singular potential. In our approach, the problem is considered as a system of coupled initial-boundary value (IBV) problems on two half-lines, to which we apply the unified approach to IBV problems for linear and integrable nonlinear equations, also known as the Fokas unified transform method. Following the ideas of this method, we obtain the integral representation of the solution of the initial value problem.

MSC:

35Q41 Time-dependent Schrödinger equations and Dirac equations
35E15 Initial value problems for PDEs and systems of PDEs with constant coefficients
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
65R10 Numerical methods for integral transforms
35A22 Transform methods (e.g., integral transforms) applied to PDEs
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References:

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