Volume 37, Issue 12, December 1996
Index of content:

Factorization of scattering matrices due to partitioning of potentials in one‐dimensional Schrödinger‐type equations
View Description Hide DescriptionThe one‐dimensional Schrödinger equation and two of its generalizations are considered, as they arise in quantum mechanics, wave propagation in a nonhomogeneous medium, and wave propagation in a nonconservative medium where energy may be absorbed or generated. Generically, the zero‐energy transmission coefficient vanishes when the potential is nontrivial, but in the exceptional case this coefficient is nonzero, resulting in tunneling through the potential. It is shown that any nontrivial exceptional potential can always be fragmented into two generic pieces. Furthermore, any nontrivial potential, generic or exceptional, can be fragmented into generic pieces in infinitely many ways. The results remain valid when Dirac delta functions are included in the potential and other coefficients are added to the Schrödinger equation. For such Schrödinger equations, factorization formulas are obtained that relate the scattering matrices of the fragments to the scattering matrix of the full problem.

l^{ P } interpolation and optimized bounds on pairwise interacting fermion systems
View Description Hide DescriptionWe derive a set of inequalities which relate the translation invariant problem of N identical particles with pairwise interactions to an independent particle problem. These inequalities apply to attractive power law potentials V(r)=r ^{ q }, 1≤q≤∞, and superpositions of such potentials; they become identities in the harmonic oscillator case q=2. We use the inequalities to derive new upper and lower bounds for the ground state energies of fermion systems, which interact through these potentials. These bounds improve all previous results in the range 1≤q≤∞; they reduce to the exact answer in the harmonic oscillator case.

Interchannel resonances at a threshold
View Description Hide DescriptionFermi’s Golden Rule, the perturbation theoretic formula for calculating the half‐width of a resonance, is not applicable to the case of a transition from a bound state into an open channel, when the energy of the bound state is exactly at the threshold of the continuum. We study solvable models of this phenomenon. The exact results coincide in leading order with the formulas found by modifying Fermi’s Golden Rule.

Transmission of conduction electrons through a symmetric pair of delta‐barriers or delta‐wells embedded in a semiconductor or a metal
View Description Hide DescriptionThe transmission coefficientT(k _{0}) is calculated for conduction electrons incident with a wave vector k _{0} upon a double barrier (double well) formed of two equal delta‐barriers (of two equal delta‐wells) embedded in a one‐dimensional (1‐D) semiconductor or in a 1‐D metal. The stationary Schrödinger–Wannier equationE(−i∂/∂x)ψ+V(x)ψ=Eψ is solved for V(x)=γ[δ(x+a/2)+δ(x−a/2)] (with real and time‐independent parameters γ, a) and E=E(k _{0})>0. (The interband transitions are neglected.) The operator E(−i∂/∂x) corresponds to a given (possibly nonquadratic) dispersion functionE(k) of the conduction electrons [E(0)=0]. It is shown that T(k _{0}) is an oscillating function reaching the maximum value [T(k _{0})→1] on an infinite set {K ^{(j)}} of values of k _{0}. The shape of T(k _{0}) depends on the shape of the dispersion functionE(k) in a simple way: T(k _{0})=T _{par} ( mv(k _{0})/ℏ)) where T _{par}(k _{0}) means the transmission coefficient in the special case when the dispersion function is quadratic, E _{par}(k)=ℏ^{2} k ^{2}/2m, and v(k)=(1/ℏ)∂E(k)/∂k is the group velocity due to E(k). [Here E(k) is taken as an increasing function.]

Localization of the photon on phase space
View Description Hide DescriptionWe obtain phase space representations of the Poincaré group for zero mass particles of all helicities, including photons. A natural quantization scheme for massless particles arises, and a covariant phase space localization operator is found.

Multi‐periodic coherent states and the WKB exactness
View Description Hide DescriptionWe construct the path integral formula in terms of a ‘‘multi‐periodic’’ coherent state as an extension of the Nielsen–Rohrlich formula for spin. We make an exact calculation of the formula and show that, when a parameter corresponding to the magnitude of spin becomes large, the leading order term of the expansion coincides with the exact result. We also give an explicit correspondence between the trace formula in the multi‐periodic coherent state and the one in the ‘‘generalized’’ coherent state.

Hamiltonian structure of Dubrovin’s equation of associativity in 2‐d topological field theory
View Description Hide DescriptionA third order Monge‐Ampère type equation of associativity that Dubrovin has obtained in 2‐d topological field theory is formulated in terms of a variational principle subject to second class constraints. Using Dirac’s theory of constraints this degenerate Lagrangian system is cast into Hamiltonian form and the Hamiltonian operator is obtained from the Dirac bracket. There is a new type of Kac‐Moody algebra that corresponds to this Hamiltonian operator. In particular, it is not a W‐algebra.

Classical and quantum implications of the causality structure of two‐dimensional space–times with degenerate metrics
View Description Hide DescriptionThe causalitystructure of two‐dimensional manifolds with degenerate metrics is analyzed in terms of global solutions of the massless wave equation. Certain novel features emerge. Despite the absence of a traditional Lorentzian Cauchy surface on manifolds with a Euclidean domain, it is possible to uniquely determine a global solution (if it exists), satisfying well‐defined matching conditions at the degeneracy curve, from Cauchy data on certain spacelike curves in the Lorentzian region. In general, however, no global solution satisfying such matching conditions will be consistent with this data. Attention is drawn to a number of obstructions that arise prohibiting the construction of a bounded operator connecting asymptotic single particle states. The implications of these results for the existence of a unitary quantum field theory are discussed.

Scattering in one dimension: The coupled Schrödinger equation, threshold behaviour and Levinson’s theorem
View Description Hide DescriptionWe formulate scattering in one dimension due to the coupled Schrödinger equation in terms of the S matrix, the unitarity of which leads to constraints on the scattering amplitudes. Levinson’s theorem is seen to have the form η(0)=π(n _{ b }+1/2n−1/2N), where η(0) is the phase of the S matrix at zero energy, n _{ b } the number of bound states with nonzero binding energy, n the number of half‐bound states, and N the number of coupled equations. In view of the effects due to the half‐bound states, the threshold behaviour of the scattering amplitudes is investigated in general, and is also illustrated by means of particular potential models.

Special‐relativistic harmonic oscillator modeled by Klein–Gordon theory in anti‐de Sitter space
View Description Hide DescriptionIt is shown that the one‐particle sector of the Klein–Gordon theory in the universal covering space of the anti‐de Sitter space (CAdS) can be interpreted, in a natural way, as a special‐relativistic oscillator in Minkowski space. The quantum wave functions have a significantly different behavior with respect to the nonrelativistic ones. The energy spectrum coincides, up to the ground state energy, with that of the nonrelativistic oscillator. The requirement of having the adequate nonrelativistic limit for the special‐relativistic oscillator theory turns out to be equivalent to the imposition of the Dirichlet‐type boundary condition at spatial infinity on CAdS Klein–Gordon functions.

Prepotential of N=2 SU(2) Yang–Mills gauge theory coupled with a massive matter multiplet
View Description Hide DescriptionWe discuss the N=2 SU(2) Yang–Mills theory coupled with a massive matter in the weak coupling. In particular, we obtain the instanton expansion of its prepotential. Instanton contributions in the mass‐less limit are completely reproduced. We study also the double scaling limit of this massive theory and find that the prepotential with instanton corrections in the double scaling limit coincides with that of N=2 SU(2) Yang–Mills theory without matter.

Symmetry and history quantum theory: An analog of Wigner’s theorem
View Description Hide DescriptionThe basic ingredients of the ‘‘consistent histories’’ approach to quantum theory are a space UP of ‘‘history propositions’’ and a space D of ‘‘decoherence functionals.’’ In this article we consider such history quantum theories in the case where UP is given by the set of projectors P(V) on some Hilbert spaceV. We define the notion of a ‘‘physical symmetry of a history quantum theory’’ (PSHQT) and specify such objects exhaustively with the aid of an analog of Wigner’s theorem. In order to prove this theorem we investigate the structure of D, define the notion of an ‘‘elementary decoherence functional,’’ and show that each decoherence functional can be expanded as a certain combination of these functionals. We call two history quantum theories that are related by a PSHQT ‘‘physically equivalent’’ and show explicitly, in the case of history quantum mechanics, how this notion is compatible with one that has appeared previously.

Lie algebra cohomology and group structure of gauge theories
View Description Hide DescriptionWe explicitly construct the adjoint operator of coboundary operator and obtain the Hodge decomposition theorem and the Poincaré duality for the Lie algebra cohomology of the infinite‐dimensional gauge transformation group. We show that the adjoint of the coboundary operator can be identified with the BRST adjoint generator Q ^{°} for the Lie algebra cohomology induced by BRST generator Q. We also point out an interesting duality relation—Poincaré duality—with respect to gauge anomalies and Wess–Zumino–Witten topological terms. We consider the consistent embedding of the BRST adjoint generator Q ^{°} into the relativistic phase space and identify the noncovariant symmetry recently discovered in QED with the BRST adjoint Nöther charge Q ^{°}.

On q‐deformed supersymmetric classical mechanical models
View Description Hide DescriptionBased on the idea of quantum groups and para‐Grassmannian variables, we present a generalization of supersymmetric classical mechanics with a deformation parameter q=exp(2πi/k) dealing with the k=3 case. The coordinates of the q‐superspace are a commuting parameter t and a para‐Grassmannian variable θ, where θ^{3}=0. The generator and covariant derivative are obtained, as well as the action for some possible superfields.

Inverse problem in nonstationary multidimensional medium
View Description Hide DescriptionThe problem of a scalar wave propagation from the point impulsive source in the layer of a nonstationary multidimensional medium is considered. The boundary problem for the wave equation is reformulated in the problem with the initial condition using the invariant imbedding method. The integral‐differential inverse procedures of the various orders were obtained from the imbedding equations using the singularities method. The order of inverse procedure is defined by the degree of a polynomial in the analytical representation of the medium characteristic near the layer boundary. It was shown that the coefficients of the polynomial are calculated with the help of the differential characteristics of the point impulsive source in the inhomogeneous medium. The cause and character of the multidimensional inverse problem overdefiniteness are considered. The application of the proposed procedure for a statistical problem is discussed.

A temperature and mass dependence of the linear Boltzmann collision operator from group theory point of view
View Description Hide DescriptionThe Lie group of the transformations affecting the parameters of the linear Boltzmann collision operator such as temperature of background gas and ratio of masses of colliding particles and molecules is discovered. The group also describes the conservation laws for collisions and main symmetries of the collision operator. New algebraic properties of the collision operator are derived. Transformations acting on the variables and parameters and leaving the linear Boltzmann kinetic equation invariant are found. For the constant collision frequency the integral representation of solutions for nonuniform case in terms of the distribution function of particles drifting in a gas with zero temperature is deduced. The new exact relaxation solutions are obtained too.

Rigorous estimates of small scales in turbulent flows
View Description Hide DescriptionWe derive rigorous bounds on the length scale of determining local averages (volume elements) for the 3‐D Navier‐Stokes Equations. These length scale estimates are related to Kolmogorov’s notion of a dissipation length scale in turbulent flows.

Explicit solutions of supersymmetric KP hierarchies: Supersolitons and solitinos
View Description Hide DescriptionWide classes of explicit solutions of the Manin‐Radul and Jacobian supersymmetric KP hierarchies are constructed by using line bundles over complex supercurves based on the Riemann sphere. Their construction extends several ideas of the standard KP theory, such as wave functions, ∂̄‐equations and τ‐functions. Thus, supersymmetric generalizations of N‐soliton solutions, including a new purely odd ‘‘solitino’’ solution, as well as rational solutions, are found and characterized.

Lax–Nijenhuis operators for integrable systems
View Description Hide DescriptionThe relationship between Lax and bi‐Hamiltonian formulations of dynamical systems on finite‐ or infinite‐dimensional phase spaces is investigated. The Lax–Nijenhuis equation is introduced and it is shown that every operator that satisfies that equation satisfies the Lenard recursion relations, while the converse holds for an operator with a simple spectrum. Explicit higher‐order Hamiltonian structures for the Toda system, a second Hamiltonian structure of the Euler equation for a rigid body in n‐dimensional space, and the quadratic Adler–Gelfand–Dickey structure for the KdV hierarchy are derived using the Lax–Nijenhuis equation.

A geometrical method towards first integrals for dynamical systems
View Description Hide DescriptionWe develop a method, based on Darboux’s and Liouville’s works, to find first integrals and/or invariant manifolds for a physically relevant class of dynamical systems, without making any assumption on these elements’ forms. We apply it to three dynamical systems: Lotka–Volterra, Lorenz and Rikitake.