Yoshio Uwano

The four‐dimensional conformal Kepler problem reduces to the three‐dimensional Kepler problem with a centrifugal potential and Dirac’s monopole field. Classical theory

The four‐dimensional conformal Kepler problem is reduced by an S1 action, when the associated momentum mapping takes nonzero fixed values. The reduced Hamiltonian system proves to be the three‐dimensional Kepler problem along with a centrifugal potential and Dirac’s monopole field. The negative‐energy surface turns out to be diffeomorphic to S3×S2, on which the symmetry group SO(4) acts. Constants of motion of the reduced system are also obtained, which include the total angular momentum vector and a Runge–Lenz‐like vector. The Kepler problem is thus generalized so as to admit the same symmetry group.

The quantised MIC-Kepler problem and its symmetry group for negative energies

The author discusses the quantised MIC-Kepler problem and its symmetry group for negative energies at the Lie algebraic level, on the basis of complex line bundles.

Quantization of the multifold Kepler system

For over a century, the Kepler problem and the harmonic oscillator have been known as the only central force dynamical systems, all of whose bounded motions are periodic. Two of the authors (T. I. and N. K.) have found an infinite number of dynamical systems possessing such a periodicity property, which have been called multi‐fold Kepler systems or ν‐fold Kepler systems, with ν a positive rational number. If ν is allowed to take the real positive numbers, say ν=α, then for the α‐fold Kepler system, all the bounded motions become periodic or not, according to whether the parameter α is a rational number or not. A purpose of this paper is to quantize the α‐fold Kepler system and thereby to figure out a quantum analog of the closed orbit property of the α‐fold Kepler system. It will turn out that the quantized α‐fold Kepler system admits accidental degeneracy in energy levels or not, according to whether α is a rational number or not.

A REDUCE program for the normalization of polynomial Hamiltonians

For a given polynomial Hamiltonian near an equilibrium point the program GITA calculates analytically the normal Birkhoff-Gustavson form in Cartesian as well as angle-action coordinates and the formal integral of the motion. These quantities are presented in the form of a truncated power series, the highest degree of which determines the degree of the approximation. The program performs the calculation of the modified normal Birkhoff-Gustavson form for the Hamiltonian restricted by the resonance condition. The program package is written in the computer algebra system REDUCE.

Certain Integrable System on a Space Associated with a Quantum Search Algorithm

On thinking up a Grover-type quantum search algorithm for an ordered tuple of multiqubit states, a gradient system associated with the negative von Neumann entropy is studied on the space of regular relative configurations of multiqubit states (SR2CMQ). The SR2CMQ emerges, through a geometric procedure, from the space of ordered tuples of multiqubit states for the quantum search. The aim of this paper is to give a brief report on the integrability of the gradient dynamical system together with quantum information geometry of the underlying space, SR2CMQ, of that system.

On Normalization of a Class of Polynomial Hamiltonians: From Ordinary and Inverse Points of View

In this paper, the normalization of a class of polynomial Hamiltonians based on the symbolic computing is disscussed from both the ordinary and the inverse direction-points of view. The truncated three-particle Toda linear chain(3- TLC) and the regularized system of a planar hydrogen atom with the linear Stark effect (HLSE) are taken as examples to demonstrate the symbolic-computational approach to the ordinary and the inverse normalization problems.

Symbolic-numerical algorithm for solving the time-dependent schrödinger equation by split-operator method

A new computational approach is proposed for the solution of the time-dependent Schrodinger equation (TDSE), in which a symbolic algorithm named GATEO and a numerical scheme based on the finite-element method (FEM) are effectively composed. The GATEO generates the multi-layer operator-difference scheme for TDSE and evaluates the effective Hamiltonian from the original time-dependent Hamiltonian by means of the Magnus expansion and the Pade-approximation. In order to solve the TDSE with the effective Hamiltonian thus obtained, the FEM is applied to a discretization of spatial domain which brings the difference scheme in operator form to the one in algebraic form. The efficiency and accuracy of GATEO and the numerical scheme associated with FEM is confirmed in the second-, fourth-, and sixth-order time-step computations for certain integrable atomic models with external fields.

A Comparison of Algorithms for the Normalization and Quantization of Polynomial Hamiltonians

Algorithms and programs for the normalization of polynomial Hamiltonians of classical mechanics by the Birkhoff–Gustavson and Deprit–Hori, as well as quasi-classical quantization procedures for normal forms, are compared. The algorithms and programs are represented in a universal pseudocode and implemented in the computer algebra systems REDUCE, MAPLE, and MATHEMATICA. Examples that illustrate the operation of these algorithms and programs for polynomial Hamiltonians of atomic systems in external electromagnetic fields are considered.

On a bifurcation in the quantum system for an approximation to the Hénon-Heiles system

Abstract It is known, in classical mechanics, that the Henon-Heiles system exhibits bifurcation phenomena in its periodic solutions depending on a parameter. These phenomena are usually analyzed by applying the Birkhoff-Gustavsons normal form method; the Henon-Heiles Hamiltonian is expanded into a series in normal form, and then truncated at a finite order. It is also known that the bifurcation taking place in the truncated system approximates the one taking place in the original Henon-Heiles system in case of sufficiently small energy. In this regard, the aim of this article is to find what will happen when the truncated Henon-Heiles system is quantized. Quantum analogue to the classical bifurcation phenomena are observed as follows: Degeneracy of energy eigenvalues proves to occur at a certain parameter value. At that value the quantum system happens to admit the SO(2)×Dα symmetry, Dα being the dihedral group. However, at a generic parameter value the symmetry reduces to SO(2)×D2. Changes in eigenstates depending on the parameter are observed; a symmetry breaking of zero angular momentum state takes place when the parameter passes that particular value.

Symbolic algorithm for factorization of the evolution operator of the time-dependent Schrödinger equation

A symbolic algorithm for the decomposition of the unitary evolution operator is developed. This algorithm allows one to generate multilayer implicit schemes for solution of the time-dependent Schrödinger equation. Some additional gauge transformations are also implemented in the algorithm. This allows one to distinguish symmetric operators, which are required for constructing efficient evolutionary schemes. The efficiency of the generated schemes is demonstrated by integrable models.