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On the stationary Schrödinger equation in the semi‐classical limit: Asymptotic blow‐up at a turning point
Author(s) -
Döpfner Kirian,
Arnold Anton
Publication year - 2019
Publication title -
pamm
Language(s) - English
Resource type - Journals
ISSN - 1617-7061
DOI - 10.1002/pamm.201900004
Subject(s) - bounded function , constant (computer programming) , turning point , stationary point , mathematical analysis , limit (mathematics) , function (biology) , schrödinger equation , planck constant , physics , mathematics , energy (signal processing) , interval (graph theory) , mathematical physics , quantum mechanics , combinatorics , evolutionary biology , computer science , acoustics , period (music) , quantum , biology , programming language
We consider a model for the wave function of an electron, injected at a fixed energy E into an electronic device with stationary potential V ( x ). This wave function is the solution of the stationary 1D Schrödinger equation. The scattering problem is modeled on an interval where the potential varies. Moreover, V ( x ) is assumed constant in the exterior, i.e. in the leads of the device. Here we are interested in including turning points – points x ¯ where the potential and the energy of the particle coincide, i.e. E = V ( x ¯ ) . We show that including a turning point lets the wave function blow‐up asymptotically as the scaled Planck constant ε → 0. This is an essential difference to the uniformly bounded wave function if turning points are excluded.