A Quantum Transmitting Schrödinger–poisson System

Abstract

We consider a stationary Schrödinger-Poisson system on a bounded interval of the real axis. The Schrödinger operator is defined on the bounded domain with transparent boundary conditions. This allows to model a non-zero current flow trough the boundary of the interval. We prove that the system always admits a solution and give explicit a priori estimates for the solutions.

A 1D coupled Schrödinger drift-diffusion model including collisions

“…[5,4,1,11,10]. For p > 0, ψ p denote the right-going scattering states, whereas ψ p , p < 0, denote the left-going scattering states.…”

“…Using the asymptotics (2.2) of the scattering states ψ p , one can derive boundary conditions for ψ p at the boundaries x 1 and x 2 , see [6,7,10,1]. Furthermore, note that the transmission and reflection amplitudes are given in terms of ψ p (x i ) and ψ p (x i ), i = 1, 2, see [6,1] for details.…”

“…Furthermore, note that the transmission and reflection amplitudes are given in terms of ψ p (x i ) and ψ p (x i ), i = 1, 2, see [6,1] for details. This allows a numerical treatment of the Schrödinger equation.…”

R-matrix Formalism for Electron Scattering in Two Dimensions with Applications to Nanostructures with Quantum Dots

“…Writing explicitly the position of the interfaces ±d at the exponent has advantages for the analytical treatment of the scattering problem [22,39]. As it is discussed in Refs.…”

Functional Model of Dissipative Fourth Order Differential Operators

“…Nagy-Foias dilation theory [19] and Lax-Phillips scattering theory [18]. Pavlov's approach ( [21][22][23]) to the model construction of dissipative extensions of symmetric operators was followed by Allahverdiev in his works [1][2][3][4][5] and some others, and by the group of authors [6][7][8], where the theory of the dissipative Schrödinger operator on a finite interval was applied to the problems arising in the semiconductor physics. In [9][10][11][12], Pavlov's functional model was extended to (general) dissipative operators which are finite dimensional extensions of a symmetric operator, and the corresponding dissipative and Lax-Phillips scattering problems were investigated in some detail.…”