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Toru Miyazawa

Publications and source records attributed to Toru Miyazawa.

9 recordsLinked to original sources
arXiv2018

SL(3,C) structure of one-dimensional Schrödinger equation

We present a new formalism for describing solutions of the one-dimensional stationary Schrödinger equation in terms of the Lie group SL(3,C) and its Lie algebra. In this formalism, we obtain a universal expression for the Green function which can be used in any representation of SL(3,C) and also expressions for various quantities involving products of Green functions. Specifically, we introduce an infinite-dimensional representation of SL(3,C) that provides a natural description of multiple scattering of waves. Using this particular representation, we can derive formulas which are useful for the analysis of the Green function.

math-ph↗
arXiv2015

Formulation of a unified method for low- and high-energy expansions in the analysis of reflection coefficients for one-dimensional Schrödinger equation

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.

math-ph↗
arXiv2014

To react or not to react? Intrinsic stochasticity of human control in virtual stick balancing

Understanding how humans control unstable systems is central to many research problems, with applications ranging from quiet standing to aircraft landing. Increasingly much evidence appears in favor of event-driven control hypothesis: human operators only start actively controlling the system when the discrepancy between the current and desired system states becomes large enough. The event-driven models based on the concept of threshold can explain many features of the experimentally observed dynamics. However, much still remains unclear about the dynamics of human-controlled systems, which likely indicates that humans employ more intricate control mechanisms. The present paper argues that control activation in humans may be not threshold-driven, but instead intrinsically stochastic, noise-driven. Specifically, we suggest that control activation stems from stochastic interplay between the operator's need to keep the controlled system near the goal state on one hand and the tendency to postpone interrupting the system dynamics on the other hand. We propose a model capturing this interplay and show that it matches the experimental data on human balancing of virtual overdamped stick. Our results illuminate that the noise-driven activation mechanism plays a crucial role at least in the considered task, and, hypothetically, in a broad range of human-controlled processes.

physics.bio-ph↗
arXiv2011

Analysis of reflection coefficients for the Fokker-Planck equation

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 high- and low-energy regions. Formulas 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.

math-ph↗
arXiv2011

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

A new formalism is presented for high-energy analysis of the Green function for Fokker-Planck and Schrödinger equations in one dimension. Formulas 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. The short-time expansion of the Green function is also discussed.

math-ph↗
arXiv2011

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

We consider Schrödinger 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.

math-ph↗
arXiv2011

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

We consider one-dimensional Fokker-Planck and Schrödinger equations with a potential which 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.

math-ph↗