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The time-dependent Schrödinger equation with piecewise constant potentials

Part of: Representations of solutions Equations of mathematical physics and other areas of application Partial differential equations

Published online by Cambridge University Press:  18 September 2018

NATALIE E. SHEILS  and
BERNARD DECONINCK
NATALIE E. SHEILS
Affiliation:
School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA email: nesheils@umn.edu
BERNARD DECONINCK
Affiliation:
Department of Applied Mathematics, University of Washington, Seattle, WA 98195-3925, USA email: deconinc@uw.edu
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Abstract

The linear Schrödinger equation with piecewise constant potential in one spatial dimension is a well-studied textbook problem. It is one of only a few solvable models in quantum mechanics and shares many qualitative features with physically important models. In examples such as ‘particle in a box’ and tunnelling, attention is restricted to the time-independent Schrödinger equation. This paper combines the unified transform method and recent insights for interface problems to present fully explicit solutions for the time-dependent problem.

Keywords

Unified Transform MethodFokas MethodSchrödinger equation

MSC classification

Primary: 35B30: Dependence of solutions on initial and boundary data, parameters
Secondary: 35B40: Asymptotic behavior of solutions 35C05: Solutions in closed form 35C15: Integral representations of solutions 35Q40: PDEs in connection with quantum mechanics 35Q41: Time-dependent Schrödinger equations, Dirac equations
Type
Papers
Information
European Journal of Applied Mathematics , Volume 31 , Issue 1 , February 2020 , pp. 57 - 83
Copyright
© Cambridge University Press 2018 

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