K. Döpfner, A. Arnold:

"On the stationary Schrödinger equation in the semi-classical limit: Asymptotic blow-up at a turning point";

in: "ASC Report 32/2020", issued by: Institute for Analysis and Scientific Computing; Vienna University of Technology, Wien, 2020, ISBN: 978-3-902627-13-1, 1 - 2.

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, and it 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.

http://www.asc.tuwien.ac.at/preprint/2020/asc32x2020.pdf

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