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  • 1
    Electronic Resource
    Electronic Resource
    A geometric setting for the quantum planar n-body system, and a U(n−1) basis for the internal states (1988)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 29 (1988), S. 1325-1337 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: In a previous paper [J. Math. Phys. 28, 964 (1987)], the author showed that the internal motion of a molecule, a many-body system in the Born–Oppenheimer approximation, can be well described in terms of the gauge theory or of the connection theory in differential geometry. However, the scope of that paper centers on the planar triatomic molecule in order to put forward the gauge theory in an explicit manner. This paper is a continuation of the previous one and gives the generalization to the planar multiatomic molecule. The internal space of the n-atomic molecule proves to be diffeomorphic to R+×CPn−2, the product of the positive real numbers and the complex projective space. The internal states of the molecule are described as cross sections in complex line bundles over the internal space. Introduction of the complex line bundles is a geometric consequence of the angular momentum conservation law, because cross sections in each complex line bundle are in one-to-one correspondence with eigenstates that have a fixed total angular momentum eigenvalue. The internal Hamiltonian operator is obtained, which acts on the cross sections in the complex line bundle. Further, boson calculus is performed to obtain a complete basis of internal states of the molecule, using the harmonic oscillator annihilation and creation operators. As a result, carrier spaces of unitary irreducible representations of the unitary group U(n−1), which are characterized by two integers, are realized as finite-dimensional subspaces of the space of the square integrable cross sections in the complex line bundle.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.527925
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  • 2
    Electronic Resource
    Electronic Resource
    A gauge theory for the quantum planar three-body problem (1987)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 28 (1987), S. 964-974 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: 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. Poincaré 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 vector bundle associated with the principal fiber bundle (the center-of-mass system) provides a setting for quantum mechanics of the "internal'' molecular motion. The interest, however, centers on planar triatomic molecules in order to put forward the gauge theory in an explicit manner. The conclusion is this: The internal space of a planar triatomic molecule is diffeomorphic with R3−{0}, and endowed with Dirac's monopole field which may be interpreted as a Coriolis field induced by the rotation. The angular momentum eigenvalues, which are twice the quantized monopole strengths, assign the complex line bundles over the internal space. The internal states of the molecule are described as the cross sections of the complex line bundle, on which the internal Hamiltonian operator acts in minimally coupling with the monopole field.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.527588
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  • 3
    Electronic Resource
    Electronic Resource
    A geometric setting for internal motions of the quantum three-body system (1987)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 28 (1987), S. 1315-1326 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: 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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.527534
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  • 4
    Electronic Resource
    Electronic Resource
    Quantization of the multifold Kepler system (1996)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 37 (1996), S. 608-624 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: 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. © 1996 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.531431
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  • 5
    Electronic Resource
    Electronic Resource
    Two classes of dynamical systems all of whose bounded trajectories are closed (1994)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 35 (1994), S. 2914-2933 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: 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 the inverse square root potential may be called the twofold Kepler system, whose bounded trajectories have a self-intersection point.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.530494
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  • 6
    Electronic Resource
    Electronic Resource
    Classical and quantum mechanics of jointed rigid bodies with vanishing total angular momentum (1999)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 40 (1999), S. 2381-2399 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: A gauge theoretical treatment proves to have been successful in the study of systems of point particles; the center-of-mass system is made into a principal fiber bundle, on which is defined a natural connection. The gauge theoretical approach may be generalized to be applicable to a system of rigid bodies. The present article deals with a system of two identical axially symmetric cylinders jointed together by a special type of joint. This system is the model made by Kane and Scher and reformulated later by Montgomery, in order to study the falling cats who can land on their legs when released upside down. With the no-twist condition, the system turns out to have the configuration space diffeomorphic with SO(3), which is made into a principal O(2) bundle over RP2, the real projective space of dimension two, and endowed with a natural connection. An optimal control problem for this system with the vanishing total angular momentum is satisfactorily treated in this bundle picture. Along with a certain performance index, the Maximum Principle gives rise to a Hamiltonian system on the cotangent bundle T*(SO(3)) of SO(3). This Hamiltonian system is shown to admit a symmetry group O(2), which is not the structure group, but comes from the material symmetry of the respective cylinders. Moreover, quantization of this "classical" system is carried out, giving rise to a quantum system with the constraints of the vanishing total angular momentum. Through the symmetry by the structure group O(2), the reduction procedure is performed for both the classical and the quantum systems. It then turns out that the respective reduced systems, classical and quantum, admit the material symmetry group O(2), in general. © 1999 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.532871
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  • 7
    Electronic Resource
    Electronic Resource
    Multifold Kepler systems—Dynamical systems all of whose bounded trajectories are closed (1995)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 36 (1995), S. 1790-1811 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: 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 numbers, any flow is shown to be dense in the torus. In conclusion, on the analogy of the Kepler problem, the Runge–Lenz-like vector for the ν-fold Kepler system is touched upon. © 1995 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.531086
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  • 8
    Electronic Resource
    Electronic Resource
    Classical and quantum symmetry groups of a free-fall particle (1985)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 26 (1985), S. 55-61 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: Symmetry of a free-fall particle is studied in quantum as well as classical mechanics. The quantum symmetry group is shown to be a central extension of the classical one. In the case of two degrees of freedom, the action of the quantum symmetry group is expressed in the form of integral transform as a unitary operator on the space of wave functions.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.526797
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  • 9
    Electronic Resource
    Electronic Resource
    The four-dimensional conformal Kepler problem reduces to the three-dimensional Kepler problem with a centrifugal potential and Dirac's monopole field. Classical theory (1986)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 27 (1986), S. 1523-1529 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: 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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.527112
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  • 10
    Electronic Resource
    Electronic Resource
    On reduction of two degrees of freedom Hamiltonian systems by an S1 action, and SO0(1,2) as a dynamical group (1985)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 26 (1985), S. 885-893 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: Reduction by an S1 action is a method of finding periodic solutions in Hamiltonian systems, which is known rather as the method of averaging. Such periodic solutions can be reconstructed as S1 orbits by pulling back the critical points of an associated "reduced Hamiltonian'' on a "reduced phase space'' along the reduction. For Hamiltonian systems of two degrees of freedom, a geometric setting of the reduction is already accomplished in the case where the reduced phase space is a two-sphere in the Euclidean space R3, and the reduced Hamilton's equations of motion are Euler's equations. This article deals with the case where the reduced phase space will be a two-hyperboloid in the three-Minkowski space, and the reduced Hamilton's equations of motion will be Euler's equations with respect to the Lorentz metric. This reduction is associated with SU(1,1) symplectic action on the phase space R4. As a consequence of this association the reduced Hamiltonian system proves to admit a dynamical group SO0(1,2). A well-known reduction by an S1 action occurs in the case of rotational-invariant Hamiltonian systems, which will be associated with SL(2,R) symplectic action on R4. It is shown that the reduction associated with SU(1,1) and with SL(2,R) are symplectically equivalent.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.526544
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