Toru Miyazawa

We study low-energy expansion and high-energy expansion of reflection coefficients for one-dimensional Schrödinger equation, from which expansions of the Green function can be obtained. Making use of the equivalent Fokker-Planck equation, we develop a generalized formulation of a method for deriving these expansions in a unified manner. In this formalism, the underlying algebraic structure of the problem can be clearly understood, and the basic formulas necessary for the expansions can be derived in a natural way. We also examine the validity of the expansions for various asymptotic behaviors of the potential at spatial infinity.

We study low-energy expansion and high-energy expansion of reflection coefficients for one-dimensional Schrodinger equation, from which expansions of the Green function can be obtained. Making use of the equivalent Fokker-Planck equation, we develop a generalized formulation of a method for deriving these expansions in a unified manner. In this formalism, the underlying algebraic structure of the problem can be clearly understood, and the basic formulas necessary for the expansions can be derived in a natural way. We also examine the validity of the expansions for various asymptotic behaviors of the potential at spatial infinity. (C) 2015 AIP Publishing LLC.

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.

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 -> +/-infinity.

A new formalism is presented for high- energy analysis of the Green function for Fokker-Planck and Schrodinger equations in one dimension. Formulae for the asymptotic expansion in powers of the inverse wave number are derived, and conditions for the validity of the expansion are studied through the analysis of the remainder term. This method is applicable to a large class of potentials, including the cases where the potential V (x) is infinite as x ->+/-infinity. The short-time expansion of the Green function is also discussed.

We study the one-dimensional Schrödinger 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. © 2006 IOP Publishing Ltd.

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.

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. © 2006 IOP Publishing Ltd.

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.

A new description of one-dimensional scattering processes in terms of boson operators is presented, and the Schrödinger 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.

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. © 1999 American Institute of Physics.

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. © 1995 American Institute of Physics.

Structure of the reflection coefficient of the Fokker-Planck equation is investigated. Asymptotic expressions of the reflection coefficient for small wavenumber (low frequency) and large wavenumber (high frequency) are presented. As an application of this analysis, a method for calculating the eigenvalues of the Fokker-Planck equation is derived.

Behavior of solutions to the one-dimensional Fokker-Planck equation with potentials U(x) which diverge logarithmically as x→ ± ∞ is investigated through the analysis of the frequency component of the Green's function. It is shown that solutions of the equation decrease for large time t either logarithmically or as t-a, where a may take arbitrary positive value depending on a parameter. © 1992 American Institute of Physics.

A general theory of the one-variable Fokker-Planck equation is presented. The emphasis is placed on the calculation of the Green function, from which the nonstationary solution with arbitrary initial condition can be obtained. The frequency component of the Green function is expressed in a simple form. The behavior of the time-dependent Green function is studied on the basis of this expression. The short-time behavior is expressed as a power series in terms of time. The long-time behavior is also explored. The eigenvalue problem and other properties of the Fokker-Planck equation are discussed as well. © 1989 The American Physical Society.

Formal solution of the diffusion equation (/t)V=(/x)[D(x)(/x)V] is obtained as a functional of D(x). The solution is expressed by means of certain operators, which are somewhat similar to the time-ordering operator of quantum mechanics. This formalism provides us with a good perspective of the problem and serves as a basis for approximation methods. In particular, the equation is exactly solvable when D(x) is a discrete (stepwise) function. Coarse graining of the diffusion coefficient is also discussed. It is shown that the reciprocal of the coarse-grained diffusion coefficient is equal to the arithmetical average of 1/D(x). Finally, a stochastic equation obtained by adding a decay term and noise terms to the diffusion equation is considered. Spatial correlation function is calculated exactly for the case where D(x) is a step function. © 1987 The American Physical Society.