B. Toomire

Time‐dependent dissipation in nonlinear Schrödinger systems

A coupled nonlinear Schrodinger–Poisson equation is considered which contains a time‐dependent dissipation function as a specific model of dissipation effects in nonlinear quantum transport theory and other areas. The Wigner–Poisson equation associated with this system is derived. Using conservation and quasiconservation laws and certain growth assumptions for the nonlinearities and the dissipation function, global existence of solutions to the Cauchy problem of the time‐dependent Schrodinger–Poisson system is shown both for small (attractive case) or arbitrary data (repulsive case).

An overview of Schrödinger-Poisson problems☆

Abstract Recent advances to the “Wigner-Poisson” problem are presented. Existence, uniqueness, and regularity of solutions is shown in various settings; these include periodic, stationary, and ‘mixed’ boundary value problems. Results for Wigner equations modified to describe dissipation and nonlinear field effects are also presented; finally, conditions for the existence of a classical limit to the Wigner function are discussed.

Inflow Boundary Conditions in Quantum Transport Theory

A linear (given potential) steady-state Wigner equation is considered in conjunction with inflow boundary conditions and relaxation-time terms. A brief review of the use of inflow conditions in the classical case is also discussed. An analytic expansion of solutions is shown and a recursion relation derived for the given problem under certain regularity assumptions on the given inflow data. The uniqueness of the physical current of the solutions is shown and a brief discussion of the lack of charge conservation associated with the relaxation-time is given.

Quantum BGK modes for the Wigner-poisson system

Abstract It is shown that an infinite family of solutions to the stationary Wigner equation can be constructed as arbitrary functions of a complete set of commuting observables. These are generalizations of the famous BGK modes of classical Vlasov theory; their existence was first proposed by M. Buchanan. For one specific function corresponding to the canonical distribution, a pseudo-differential equation-the Bloch equation-can be written down for the Wigner function. This equation is known to have a unique solution. For other functions, an equation which the Wigner function obeys remains to be discovered.

On quasi-linear Schrödinger-Poisson systems

We study a high-field version of the periodic Schrodinger–Poisson system, for which the Poisson equation includes nonlinear terms corresponding to a field-dependent dielectric constant. Using a Galerkin scheme, we prove global existence and uniqueness, and present the matrix equations for the numerical evaluation of the potential.

A note on conservation laws

Abstract In proving existence of solutions to the Wigner-Poisson (WP) system1,2,3 and, indeed, to many systems of partial differential equations, conservation laws are often used to prove that a local solution does not blow up in a finite time interval, and hence is a global-in-time solution. For the (WP) system, one proves1,2 a relevant conservation principle referred to as “energy conservation” which takes the form1,2,3

Boundary conditions in quantum transport theory

Abstract The problem of finding proper boundary conditions for the Wigner equation on the interval is considered. It is shown that self-adjoint boundary conditions in the Schrodinger equation lead in general to zero current It is proposed to use instead Bohzmann boundary conditions of inflow type.

QUANTUM TRANSPORT THEORY

Abstract Procedures are discussed for calculating the steady-state current for linear quantum transport problems subject to inflow boundary conditions. A review of classical kinetic models is given, and techniques (based on quantum BGK modes) for solving the stationary Wigner equation with inflow BCs are discussed. Lastly, uniqueness of the steady-state current is shown.

A Survey oh Wigner-Poisson Problems

In 1932, E.Wigner [1] introduced a phase-space method of computing physical observables in quantum (statistical) mechanics; some surveys have been published recently [2], [3]. See also a review article by Carruthers [4]. Here we give a brief synopsis of the method to make this paper self-contained, referring the reader to the references for details, including proofs.