Toru Miyazawa

Algebraic structure of diffusion and scattering in one dimension

The one‐variable Fokker–Planck equation is studied from an algebraic point of view. Diffusion in an external potential can be interpreted as a multiple scattering process. A description of the one‐dimensional scattering in terms of boson operators is introduced, and it is generalized to a form that does not depend on representation. An expression for the Green’s function is derived in terms of generators of the OSp(1/2) superalgebra. It is shown that many seemingly different descriptions of the diffusion problem are just different representations of this general expression.

Boson representations of one-dimensional scattering

A new description of one-dimensional scattering processes in terms of boson operators is presented, and the Schrodinger equation in a general form is analysed in this description on the basis of the factorization scheme. As an example of application, a low-energy expansion formula of the Green function is derived within the framework of this formalism.

Algebraic approach to scattering theory of the Fokker–Planck equation

The one-variable Fokker–Planck equation is studied in a scattering formalism. Scattering processes are represented by paths in one-dimensional space, and the paths are treated as algebraic objects that constitute infinite-dimensional representations of SL(2, C). Various expressions for the scattering coefficients are derived in a systematic way by means of algebraic methods with considerations on symmetries.

Low-energy expansion formula for one-dimensional Fokker―Planck and Schrödinger equations with periodic potentials

We study the low-energy behavior of the Green function for one-dimensional Fokker–Planck and Schrodinger equations with periodic potentials. We derive a formula for the power series expansion of reflection coefficients in terms of the wave number, and apply it to the low-energy expansion of the Green function.

Low-energy asymptotic expansion of the Green function for one-dimensional Fokker–Planck and Schrödinger equations

We consider Schrodinger equations and Fokker–Planck equations in one dimension, and study the low-energy asymptotic behavior of the Green function using a new method. In this method, the coefficient of the expansion in powers of the wave number can be systematically calculated to arbitrary order, and the behavior of the remainder term can be analyzed on the basis of an expression in terms of transmission and reflection coefficients. This method is applicable to a wide variety of potentials which may not necessarily be finite as x → ±∞.

Generalized low-energy expansion formula for Green’s function of the Fokker–Planck equation

The frequency expansion formula for the one-dimensional Fokker–Planck equation derived in a previous paper is generalized to an expansion formula for arbitrary powers of the Green’s function. Based on this expansion, the small-frequency behavior of the Green’s function is studied for the cases where the potential V(x) tends to infinity at both x→+∞ and x→−∞.

Green’s function of the Fokker–Planck equation: General formula of frequency expansion

The one-variable Fokker–Planck equation is studied in its general form by means of an algebraic method. An expression of the Green’s function is derived as an expansion in powers of the square root of frequency. The expansion coefficient of arbitrary order is expressed as a functional of the potential in terms of integrals.

Low-energy expansion formula for one-dimensional Fokker–Planck and Schrödinger equations with asymptotically periodic potentials

We consider one-dimensional Fokker–Planck and Schrodinger equations with a potential that approaches a periodic function at spatial infinity. We extend the low-energy expansion method, which was introduced in previous papers, to be applicable to such asymptotically periodic cases. Using this method, we study the low-energy behavior of the Green function.

Expressions of the Green function in terms of reflection coefficients

We study the one-dimensional Schrodinger equation and derive exact expressions for the Green function in terms of reflection coefficients which are defined for semi-infinite intervals. We also discuss the relation between our results and the WKB approximation.

Analysis of reflection coefficients for the Fokker–Planck equation

The mathematical structure of the reflection coefficients for the one-dimensional Fokker–Planck equation is studied. A new formalism using differential operators is introduced and applied to the analysis in the high- and low-energy regions. Formulae for high-energy and low-energy expansions are derived, and expressions for the coefficients of the expansion, as well as the remainder terms, are obtained for general forms of the potential. Conditions for the validity of these expansions are discussed on the basis of the analysis of the remainder terms.