P. F. Zweifel

WIGNER METHOD IN QUANTUM STATISTICAL MECHANICS.

The Wigner method of transforming quantum‐mechanical operators into their phase‐space analogs is reviewed with applications to scattering theory, as well as to descriptions of the equilibrium and dynamical states of many‐particle systems. Inclusion of exchange effects is discussed.

Radiative Transfer. II

This paper presents an exact solution to the equations of radiative transfer for a generalization of the Uniform‐Picket‐Fence model discussed in a previous work. Here the absorption coefficient is allowed to take N different values over the frequency spectrum. Cases method is used to construct the normal mode solutions to the set of N coupled integral equations. Then half‐range completeness and orthogonality theorems are proved that enable one to solve typical half‐space problems. Explicitly, the asymptotic solution to the Milne problem is developed, including the extrapolated end point, while implicitly the complete solution is available.

Vlasov theory of plasma oscillations: Linear approximation

A functional analytic approach to the linearized collisionless Vlasov equation is presented utilizing a resolvent integration technique on the resolvent of the transport operator evaluated at a particular point. Formulae for the eigenfunction expansion are found for cases in which the plasma disperion function Λ has first and second order zeroes. Special care is taken in the study of real zeroes of Λ culminating in new results for this case. For a simple zero of Λ with nonvanishing imaginary part the van Kampen–Case discrete modes are reproduced. The results are used to obtain the solution to the initial value problem.

Extension of the Case formulas to Lp. Application to half and full space problems

The singular eigenfunction expansions originally applied by Case to solutions of the transport equation are extended from the space of Holder‐continuous functions to the function spaces Xp = {f‖μf (μ) ‐ Lp}, where the expansions are now to be interpreted in the Xp norm. The spectral family for the transport operator is then obtained explicitly, and is used to solve transport problems with Xp sources and incident distributions.

Multigroup neutron transport. I. Full range

A functional analytic approach to the N‐group, isotropic scattering, particle transport problem is presented. A full‐range eigenfunction expansion is found in a particularly compact way, and the stage is set for the determination of the half‐range expansion, which is discussed in a companion paper. The method is an extension of the work of Larsen and Habetler for the one‐group case.

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).

Wigner Approach to the Two-Band Kane Model for a Tunneling Diode

Abstract In this article we study a simple model for describing the electron behavior in a heterogeneous material characterized by two energy bands. The model, the so-called Kane system, is composed of two Schrödinger-like equations coupled by a k · P term which describes the interband tunneling. We give a physical justification of the Kane system and reformulate it in terms of Wigner functions. The well-posedness of the corresponding Wigner system is investigated in a suitable Hilbert space.

A discrete-velocity, stationary Wigner equation

This paper is concerned with the one-dimensional stationary linear Wigner equation, a kinetic formulation of quantum mechanics. Specifically, we analyze the well-posedness of the boundary value problem on a slab of the phase space with given inflow data for a discrete-velocity model. We find that the problem is uniquely solvable if zero is not a discrete velocity. Otherwise one obtains a differential-algebraic equation of index 2 and, hence, the inflow data make the system overdetermined.

The Wigner transform and the Wigner-poisson system

Abstract The Wigner method in (quantum) statistical mechanics was introduced in 19321 as an alternative to the standard formulation2 which involves computation of eigenvalues and eigenvectors (and traces) of linear operators in Hilbert spaces. The Wigner approach uses only integrations over phase space –R6N in the case of an N-particle system–of ordinary functions f:R6n → C.

A Fokker-Planck equation for growing cell populations

Closed form solutions are obtained for a Fokker-Planck model for cell growth as a function of maturation velocity and degree of maturation. For reproduction rules where daughter cells inherit their parents maturation velocity the complete solution is derived in terms of Airy functions. For more complicated reproduction rules partial results are obtained. Emphasis is given to the relationship of these problems to time dependent linear transport theory.