Volume 36, Issue 9, September 1995
Index of content:

A nodal theorem for coupled systems of Schrödinger equations and the number of bound states
View Description Hide DescriptionIt is shown that the number of bound states of coupled systems of radial Schrödinger equations can be found by computing the number of zeros of a suitable determinant. This is related to the concept of conjugate points from Morse theory and is the basis for a numerically efficient method.

The absence of the absolutely continuous spectrum for δ ′ Wannier–Stark ladders
View Description Hide DescriptionA modification of the Kronig–Penney model consisting of equidistantly spaced δ ′ interactions is considered. We prove that an absolutely continuous spectrum of such a system disappears under the influence of an external electric field. The result extends to periodic lattices of nonidentical δ ′ interactions and potentials which are lower unbounded and, up to a bounded term, asymptotically decreasing with bounded first two derivatives.

On the complete basis of Pauli‐allowed states of three‐cluster systems in the Fock–Bargmann space
View Description Hide DescriptionIn the Fock–Bargmann space, the complete harmonic‐oscillator basis for the three‐cluster system is constructed. The indices of the reduction U(6)⊇U(2)×U(3) are chosen as the quantum numbers. The Pauli‐forbidden states are eliminated by the orthogonal transformation of basis functions. The basis states are obtained in terms of hypergeometric functions and the spherical Wigner functions. Their simple form allows one to solve the problem of calculating the matrix elements of the microscopic Hamiltonian needed for the study of three‐cluster systems within the algebraic version of RGM.

Exactness in the Wentzel–Kramers–Brillouin approximation for some homogeneous spaces
View Description Hide DescriptionAnalysis of the Wentzel–Kramers–Brillouin (WKB) exactness in some homogeneous spaces is attempted. CP ^{ N } as well as its noncompact counterpart D _{ N,1} is studied. U(N+1) or U(N,1) based on the Schwinger bosons leads us to CP ^{ N } or D _{ N,1} path integral expression for the quantity tr e ^{−iHT }, with the aid of coherent states. The WKB approximation terminates in the leading order and yields the exact result provided that the Hamiltonian is given by a bilinear form of the creation and the annihilation operators. An argument on the WKB exactness to more general cases is also made.

Group quantization of parametrized systems. I. Time levels
View Description Hide DescriptionA method of quantizing parametrized systems is developed that is based on a kind of ‘‘gauge invariant’’ quantities—the so‐called perennials (a perennial must also be an ‘‘integral of motion’’). The problem of time in its particular form (frozen time formalism, global problem of time, multiple choice problem) is met, as well as a related difficulty characteristic for this type of theory: the paucity of perennials. The present paper is an attempt to find some remedy in the ideas on ‘‘forms of relativistic dynamics’’ by Dirac. Some aspects of Dirac’s theory are generalized to all finite‐dimensional first‐class parametrized systems. The generalization is based on replacing the Poincaré group and the algebra of its generators as used by Dirac by a canonical group of symmetries and by an algebra of elementary perennials. A number of insights are gained; the following are the main results. First, conditions are revealed under which the time evolution of the ordinary quantum mechanics, or a generalization of it, can be constructed. The construction uses a kind of gauge and time choice and it is described in detail. Second, the theory is structured so that the quantum mechanics resulting from different choices of gauge and time are compatible. Third, a practical way is presented of how a broad class of problems can be solved without the knowledge of explicit form of perennials.

Group quantization of parametrized systems. II. Pasting Hilbert spaces
View Description Hide DescriptionThe method of group quantization described in the preceding paper [J. Math. Phys. 36, 4612 (1995)] is extended so that it becomes applicable to some parametrized systems that do not admit a global transversal surface. A simple completely solvable toy model is studied that admits a pair of maximal transversal surfaces intersecting all orbits. The corresponding two quantum mechanics are constructed. The similarity of the canonical group actions in the classical phase spaces on the one hand and in the quantum Hilbert spaces on the other hand suggests how the two Hilbert spaces are to be pasted together. The resulting quantum theory is checked to be equivalent to that constructed directly by means of Dirac’s operator constraint method. The complete system of partial Hamiltonians for any of the two transversal surfaces is chosen and the quantum Schrödinger or Heisenberg pictures of time evolution are constructed.

SU(2) coherent‐state path integral
View Description Hide DescriptionThe SU(2) coherent‐state path integral is used to represent the matrix element of a propagator in the SU(2) coherent‐state basis. It is argued that the continuum representation of this integral is correct provided the necessary boundary term is taken into account. In the case of the SU(2) dynamical symmetry the path integral is explicitly computed by means of a change of variables, the SU(2) motion of the underlying phase space. The correct stationary‐phase expansion for the propagator in terms of the total action including boundary term and classical trajectories is obtained.

Bargmann‐ and Calogero‐type bounds for the Dirac equation
View Description Hide DescriptionThe Dirac equation for a particle of mass m submitted to a spherically symmetric vector potential is considered. Similar to the Schrödinger case, a nodal characterization of solutions for energies in the spectral gap (−m,m) is shown. As a consequence some bounds on the number of bound states which reduce in the nonrelativistic limit to the classical bounds of Bargmann and Calogero are proven.

Markov diffusions in comoving coordinates and stochastic quantization of the free relativistic spinless particle
View Description Hide DescriptionWe revisit the classical approach of comoving coordinates in relativistic hydrodynamics and we give a constructive proof for their global existence under suitable conditions, which is proper for stochastic quantization. We show that it is possible to assign stochastic kinematics for the free relativistic spinless particle as a Markov diffusion globally defined on M^{4}. Then introducing dynamics by means of a stochastic variational principle with Einstein’s action, we are lead to positive‐energy solutions of the Klein–Gordon equation. The procedure exhibits relativistic covariance properties.

Representation of the five‐dimensional harmonic oscillator with scalar‐valued U(5) ⊇ SO(5) ⊇ SO(3)–coupled VCS wave functions
View Description Hide DescriptionVector coherent state methods, which reduce the U(5) ⊇ SO(5) ⊇ SO(3) subgroup chain, are used to construct basis states for the five‐dimensional harmonic oscillator. Algorithms are given to calculate matrix elements in this basis. The essential step is the construction of SO(5) ⊇ SO(3) irreps of type [v,0]. The methodology is similar to that used in two recent papers except that one‐dimensional, as opposed to multidimensional, vector‐valued wave functions are used to give conceptually simpler results. Another significant advance is a canonical resolution of the SO(5) ⊇ SO(3) multiplicity problem.

Completeness of decoherence functionals
View Description Hide DescriptionThe basic ingredients of the ‘‘consistent histories’’ approach to a generalized quantum theory are ‘‘histories’’ and decoherence functionals. The main aim of this program is to find and to study the behavior of consistent sets associated with a particular decoherence functionald. In its recent formulation by Isham [‘‘Quantum logic and the histories approach to quantum theory,’’ J. Math. Phys. 35, 2157–2185 (1994)] it is natural to identify the space UP of propositions about histories with an orthoalgebra or lattice. When UP is given by the lattice of projectors P(V) in some Hilbert spaceV, consistent sets correspond to certain partitions of the unit operator in V into mutually orthogonal projectors {α_{1},α_{2},...}, such that the functiond(α,α) is a probability distribution on the boolean algebragenerated by {α_{1},α_{2},...}. Using the classification theorem for decoherence functionals proven in Isham et al. [‘‘The classification of decoherence functionals: An analog of Gleason’s Theorem,’’ J. Math. Phys. 35, 6360–6370 (1994)] we show that in the case where V is some separable Hilbert space there exists for each partition of the unit operator into a set of mutually orthogonal projectors, and for any probability distributionp(α) on the corresponding boolean algebra, decoherence functionalsd with respect to which this set is consistent and which are such that for the probability functionsd(α,α)=p(α) holds.

A q deformation of the Gauss distribution
View Description Hide DescriptionThe q deformed commutation relation aa^{*} −qa^{*}a=1 for the harmonic oscillator is considered with q∈[−1,1]. An explicit representation generalizing the Bargmann representation of analytic functions on the complex plane is constructed. In this representation the distribution of a+a ^{*} in the vacuum state is explicitly calculated. This distribution is to be regarded as the natural q deformation of the Gaussian.

SU(2) coherent states on an m‐sheeted covering of the sphere
View Description Hide DescriptionSU(2) coherent states on an m‐sheeted covering of the sphere are introduced, and properties like overcompleteness and resolution of the identity are studied. Operators J _{+} ^{(m)}, J _{−} ^{(m)}, J _{ z } ^{(m)} that obey the SU(2) algebra but shift the ‖jn〉 states by m steps are considered, and it is shown that the properties of our coherent states with respect to them are analogous to the properties of the standard SU(2) coherent states with respect to the usual angular momentum operators J _{+}, J _{−}, J _{ z }. An extended SU(2) Bargmann representation on an m‐sheeted sphere is introduced in which the transformations of an m‐sheeted covering of the SU(2) group are implemented as extended Mobius conformal mappings. The formalism is applied in the study of Hamiltonians that contain the operators J _{+} ^{(m)}, J _{−} ^{(m)}, J _{ z } ^{(m)} and describe processes where the state of a particle is shifted by m steps. The method is also used in the context of both the Holstein–Primakoff and Schwinger representation of SU(2).

Large deviations in a semi‐infinite layered ferromagnet with mean‐field interactions: Classical and quantum models
View Description Hide DescriptionWe consider the probability distribution in a classical semi‐infinite layered ferromagnet with mean‐field interactions and prove that this satisfies a large deviation principle, giving the rate function explicitly. For an analogous quantum system we show that the limiting states are product states, which are completely characterized by the classical mean‐field solutions.

Properties of q‐entropies
View Description Hide DescriptionBasic properties of the q‐entropy S _{ q }[ρ]=(q−1)^{−1}(1−tr(ρ^{ q })) (0<q≠1) for states of a quantum system are established: concavity, quasi‐convexity, continuity, and failure of ‘‘additivity’’ and ‘‘subadditivity’’ for composite systems.

The spectrum of adiabatic stellar oscillations
View Description Hide DescriptionThe stabilityanalysis for spherically symmetric stellar equilibrium models with respect to ‘‘small’’ adiabatic Lagrangian perturbations leads to the consideration of a class of densely defined, linear symmetric operators in Hilbert space, which are induced by certain singular vector–integro–partial differential operator. The extension properties of these operators as well as the spectral properties of the linear self‐adjoint extensions which are chosen by physical boundary conditions are investigated. For this, the equilibrium models are assumed to be polytropic, with a constant adiabatic index only near the center and near the boundary of the star. Among others it is shown that the operators of the class having a polytropic index near the boundary which is ≥1 are in particular essentially self‐adjoint and have a closure with a pure point spectrum.

The spectrum of radial adiabatic stellar oscillations
View Description Hide DescriptionThe stabilityanalysis with respect to ‘‘small’’ radial adiabatic perturbations of spherically symmetric stellar equilibrium models which are polytropic with a constant adiabatic index only near the center and the boundary of the star leads to the consideration of a class of singular minimal Sturm–Liouville operators. It is shown that the physical boundary conditions choose in a unique way the corresponding Friedrichs extensions. Moreover, all linear self‐adjoint extensions of the members of the class are determined and are shown to have a purely discrete spectrum.

Bi‐Hamiltonian formulation for the Korteweg–de Vries hierarchy with sources
View Description Hide DescriptionThe first Hamiltonian structure is constructed for the Korteweg–de Vries (KdV) hierarchy with sources and for the respective modified systems. Applying the appropriate Miura map the second Hamiltonian structure for the KdV hierarchy with sources is derived. Stationary projection of considered system gives so‐called restricted flows of the KdV hierarchy as well as its bi‐Hamiltonian structure.

A solvable Hamiltonian system
View Description Hide DescriptionThe initial‐value problem for the dynamical system characterized by the Hamiltonian H=λn∑^{ n } _{ j=1} p _{ j }+μ∑^{ n } _{ j,k=1} (p _{ jp } _{ k })^{1/2} cos[ν(q _{ j }−q _{ k })] is solved in completely explicit form, for arbitrary n. A set of matrices is introduced, whose remarkable properties are related to this problem, and also present an interest of their own.

Solution of three waves interaction type models with nontrivial asymptotic and boundary conditions
View Description Hide DescriptionIn the following we introduce a new way to solve the Cauchy problem for non‐ vanishing potentials at infinity for the spectral problem of the Zakharov and Shabat, Ablowitz, Kaup, Newell, and Segur (ZS‐AKNS) hierarchy. Then we apply the results to solve a class of coupled systems, with this nonvanishing condition on the potential and with quite general boundary conditions on the other fields. Finally we present an application to a physical system governed by a three waves interaction type model.