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

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.


Model
We consider a scattering problem for the stationary Schrödinger equation in 1D, which is a relevant quantum dynamical model for the electron transport in a diode. The diode covers the interval [x 0 , 1], having leads to both sides. Electrons are injected from the right lead in the form of a plane wave (of unit amplitude, e.g.). The scattering problem formulation for the wave function ψ(x), with a(x) := E − V (x), and the scaled Planck constant ε := √ 2m reads: The two transparent boundary conditions correspond to constant potentials in the exterior problems, i.e. for x ≤ x 0 and x ≥ 1. The model includes a first order turning point (i.e. a zero of a(x)) at x = 0.
The goal of this note is to describe the asymptotic behavior of ψ ε in the semi-classical limit -in particular close to the turning point. This is an important input information for the numerical treatment of (1). In [2] both of these questions were discussed for a very similar scattering problem, but having a linear potential V (x) := E − x in the whole left lead. Here, we extend this to the more realistic case of constant potentials in both leads.
Away from the turning point, an efficient numerical treatment of the highly oscillatory problem (1) can be based on first eliminating analytically the dominant oscillations (using asymptotic WKB-approximations of the solution). Then, the resulting smoother problem can be solved numerically on a coarse grid [1,2], with an error that is uniform in ε. Since this approach is not valid near a turning point, we assumed here and in [2], as a simplification, that the potential is linear in the vicinity of the turning point. ψ ε can then be obtained as the numerical solution of (1) on [x 1 , 1], coupled to the explicit (analytic) solution on [x 0 , x 1 ]: 2 of 2 Section 18: Numerical methods of differential equations with some normalization constant α ε , and y(x) := − x ε 2/3 , where A ε := Ai (y(x 0 ) + y(x 0 ) Ai(y(x 0 ) , B ε := Bi (y(x 0 ) + y(x 0 ) Bi y(x 0 ) .
Here, Ai and Bi are the fundamental solutions to the Airy equation, i.e. Airy functions.
2 Asymptotic blow-up at a turning point Example 2.1 Consider (1) with x 0 = −0.3 and a(x) = x for x ∈ [x 0 , 1] and 0 < ε < 1. Then the explicit solution is given by (2) on all of [x 0 , 1], and the normalization is chosen such that ψ ε satisfies the right boundary condition: .
For the detailed proof see [3]; it is an adaption of the proof for Proposition 4.2 in [2].