Toshihiro Iwai

A geometric setting for internal motions of the quantum three-body system

Quantum mechanics for internal motions of the three‐body system is set up on the basis of the complex vector bundle theory. The three‐body system is called a triatomic molecule in the Born–Oppenheimer approximation. The internal states of the molecule are described as cross sections in the complex vector bundle assigned by an eigenvalue of the square of the total angular momentum operator. This bundle is equipped with a linear connection, which is a natural consequence of a geometric interpretation of the so‐called Eckart condition. The coupling of the internal motion with the rotation is understood naturally in terms of this connection. The internal Hamiltonian operator is obtained which includes the internal motion–rotation coupling and a centrifugal potential. The complex vector bundle for the triatomic molecule proves to be a trivial bundle, though the geometric setting for the internal motion is independent of whether the bundle is trivial or not.

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.

On extended taub-NUT metrics

Abstract Much attention has been paid to the (Euclidean) Taub-NUT metric because the geodesic on this space describes approximately the motion of two well-separated interacting monopoles. It is also well known that the Taub-NUT metric admits a Kepler-type symmetry. In this paper, the Taub-NUT metric is extended so that it still admits a Kepler-type symmetry. The geodesics of this metric will be investigated. In particular, regularization of singular geodesics is studied by use of a method from dynamical systems. Further, some geometrical properties of the extended Taub-NUT metric are cleared up. In order that the extended Taub-NUT metric either has a self-dual Riemann curvature tensor or is an Einstein metric, it is necessary and sufficient that it coincides with the original Taub-NUT metric up to a constant factor. Furthermore, a class of extended Taub-NUT metrics have a self-dual Weyl curvature tensor is found. This class of metrics, of course, includes the Taub-NUT metric.

A gauge theory for the quantum planar three-body problem

A several‐particle system is called a molecule in the Born–Oppenheimer approximation. The nonrigidity of molecules involves difficulty in molecular dynamics. Guichardet [A. Guichardet, Ann. Inst. H. Poincare 40, 329 (1984)] showed recently that the vibration motion cannot in general be separated from the rotation motion, by using the connection theory in differential geometry. The point of his theory is the observation that a center‐of‐mass system is made into a principal fiber bundle with rotation group as the structure group, and is equipped with a connection by the Eckart condition of rotationless constraint. The base manifold of this bundle is called the internal space. The fact that the connection has nonvanishing curvature gives rise to the nonseparability of vibration from rotation. This is a mathematical meaning of nonrigidity of molecules. As an application of the connection theory due to Guichardet, this paper establishes a gauge theory for nonrigid molecules on the basis of the observation that...

The geometry of the SU(2) Kepler problem

Abstract The SU(2) Kepler problem is defined and analyzed, which is a Hamiltonian system reduced from the conformal Kepler problem on T∗( R 8 − {0}) by the use of the symplectic SU(2) action lifted from the SU(2) left action on the SU(2) bundle R 8 − {0} → R 5 − {0}. This reduced system has a parameter μ ϵ su(2) coming from the value of the moment map associated with the symplectic SU(2) action. If μ ≠ 0, the phase space of this system have a bundle structure with base space T∗( R 5 − {0}) and fibre S2. The fibre, a (co)adjoint orbit through μ for SU(2), represents the internal degrees of freedom, called the isospin, of the particle of this system. The SU(2) Kepler problem with μ ≠ 0 is then interpreted as describing the motion of a classical particle with isospin in the Newtonian potential plus a specific repulsive potential together with a Yang-Mills field. This Yang-Mills field is to be referred to as BPST Yangs monopole field in R 5 − {0};, since it becomes the Belavin-Polyakov-Schwartz-Tyupkin instanton, restricted on S4. If μ = 0, the SU(2) Kepler problem reduces to the ordinary Kepler problem. Like the ordinary Kepler problem, the Hamiltonian flows of the SU(2) Kepler problem of negative energy are all closed. It is shown in an explicit manner that the energy manifolds and isoenergetic orbit spaces for the SU(2) Kepler problem of negative energy are both homogeneous manifolds on which SU(4) acts transitively to the right; those homogeneous manifold are classified into two, according as the parameter μ is zero or not. For a certain value of μ, however, they contracts to the manifold which represents the set of all the equilibrium states. The isoenergetic orbit spaces are finally shown to be symplectomorphic to certain Kirillov-Konstant-Souriau coadjoint orbits for U(4), if μ is not the exceptional value mentioned above.

A new, globally convergent Riemannian conjugate gradient method

This article deals with the conjugate gradient method on a Riemannian manifold with interest in global convergence analysis. The existing conjugate gradient algorithms on a manifold endowed with a vector transport need the assumption that the vector transport does not increase the norm of tangent vectors, in order to confirm that generated sequences have a global convergence property. In this article, the notion of a scaled vector transport is introduced to improve the algorithm so that the generated sequences may have a global convergence property under a relaxed assumption. In the proposed algorithm, the transported vector is rescaled in case its norm has increased during the transport. The global convergence is theoretically proved and numerically observed with examples. In fact, numerical experiments show that there exist minimization problems for which the existing algorithm generates divergent sequences, but the proposed algorithm generates convergent sequences.

Reduction of quantum systems on Riemannian manifolds with symmetry and application to molecular mechanics

This paper deals with a general method for the reduction of quantum systems with symmetry. For a Riemannian manifold M admitting a compact Lie group G as an isometry group, the quotient space Q=M/G is not a smooth manifold in general but stratified into a collection of smooth manifolds of various dimensions. If the action of the compact group G is free, M is made into a principal fiber bundle with structure group G. In this case, reduced quantum systems are set up as quantum systems on the associated vector bundles over Q=M/G. This idea of reduction fails, if the action of G on M is not free. However, the Peter–Weyl theorem works well for reducing quantum systems on M. When applied to the space of wave functions on M, the Peter–Weyl theorem provides the decomposition of the space of wave functions into spaces of equivariant functions on M, which are interpreted as Hilbert spaces for reduced quantum systems on Q. The concept of connection on a principal fiber bundle is generalized to be defined well on the...

Multifold Kepler systems--Dynamical systems all of whose bounded trajectories are closed

According to the Bertrand theorem, the Kepler problem and the harmonic oscillator are the only central force dynamical systems that have closed orbits for all bounded motions. In this article, an infinite number of dynamical systems having such a closed orbit property are found on T*(R3−{0}) by applying a slightly modified Bertrand’s method to a spherical symmetric Hamiltonian with two undetermined functions of the radius. Actually, for any positive rational number ν, there exists a Hamiltonian system with the closed orbit property just mentioned, which system will be called the ν‐fold Kepler system. Each of the systems is completely integrable and further allows the explicit expression of trajectories. The bounded trajectories in the configuration space R3−{0} may have self‐intersection points. Moreover, the ν‐fold Kepler system is reducible to a two‐degrees‐of‐freedom system, which is completely integrable and gives rise to flows on the two‐torus for bounded motions. If ν is allowed to take irrational n...

Two classes of dynamical systems all of whose bounded trajectories are closed

According to the Bertrand theorem, the Kepler problem and the harmonic oscillator are the only central force dynamical systems that have closed orbits for all bounded motions. In this article, other dynamical systems having such a closed orbit property are found on T*(R3−{0}). Consider a natural dynamical system on T*(R4−{0}) whose Hamiltonian function is composed of kinetic and potential energies, and invariant under a SO(2) action. Then one can reduce the system to a Hamiltonian system on T*(R3−{0}) by the use of the Kustaanheimo–Stiefel transformation. If the original potential on R4−{0} is a central one, Bertrand’s method is applicable to the reduced system for determining the potential so that any bounded motions may be periodic. As a result, two types of potential functions will be found; one is linear in the radial variable and the other proportional to the inverse square root of that. The dynamical systems obtained are capable of physical interpretation. In particular, the dynamical system with th...

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.