Numerical simulation of quantum waveguides

This chapter is a review of the research of the authors from the last decade and focuses on the mathematical analysis of the Schrödinger model for nano-scale semiconductor devices. We discuss transparent boundary conditions (TBCs) for the time-dependent Schrödinger equation on a two dimensional domain. First we derive the two dimensional discrete TBCs in conjunction with a conservative Crank-Nicolson-type finite difference scheme and a compact nine-point scheme. For this difference equations we derive discrete transparent boundary conditions (DTBCs) in order to get highly accurate solutions for open boundary problems. The presented discrete boundary-valued problem is unconditionally stable and completely reflection-free at the boundary. Then, since the DTBCs for the Schrödinger equation include a convolution w.r.t. time with a weakly decaying kernel, we construct approximate DTBCs with a kernel having the form of a finite sum of exponentials, which can be efficiently evaluated by recursion. In several numerical tests we illustrate the perfect absorption of outgoing waves independent of their impact angle at the boundary, the stability, and efficiency of the proposed method. Finally, we apply inhomogeneous DTBCs to the transient simulation of quantum waveguides with a prescribed electron inflow

On spectrum of a dissipatively perturbed Schrödinger operator on the real semi-axis

In this work, we consider self-adjoint Schrödinger operators in one dimension, with potentials which are either compactly supported or periodi

Analysis of Water Waves in the Presence of Geometry and Damping

The evolution of waves on the surface of a body of water (or another approximately inviscid liquid) is governed by the free-surface Euler equations; that is, the incompressible Euler equations coupled with a kinematic and a dynamic boundary condition on the free surface. We assume that the flow has zero vorticity in the bulk of the fluid domain and so consider the irrotational free-surface Euler equations (the water waves system). Two major themes are present in our study of the water waves system. The first is the consideration of flows in the presence of substantial geometric features. The second theme is the consideration of the effects of damping, which is an essential tool in the numerical study of water waves. In both contexts, our objective is to consider the local-in-time well-posedness of the water waves system and to study the lifespan of solutions (i.e., the timescales on which solutions to the water waves system persist).Doctor of Philosoph

Symmetries and currents of the ideal and unitary Fermi gases

The maximal algebra of symmetries of the free single-particle Schroedinger equation is determined and its relevance for the holographic duality in non-relativistic Fermi systems is investigated. This algebra of symmetries is an infinite dimensional extension of the Schroedinger algebra, it is isomorphic to the Weyl algebra of quantum observables, and it may be interpreted as a non-relativistic higher-spin algebra. The associated infinite collection of Noether currents bilinear in the fermions are derived from their relativistic counterparts via a light-like dimensional reduction. The minimal coupling of these currents to background sources is rewritten in a compact way by making use of Weyl quantisation. Pushing forward the similarities with the holographic correspondence between the minimal higher-spin gravity and the critical O(N) model, a putative bulk dual of the unitary and the ideal Fermi gases is discussed.Comment: 67 pages, 2 figures; references added, minor improvements in the presentation, version accepted for publication in JHE