Volume 40, Issue 4, April 1999
Index of content:
 QUANTUM PHYSICS; PARTICLES AND FIELDS


New approach to semiclassical analysis in mechanics
View Description Hide DescriptionA new method is proposed for constructing approximate solutions to the Schrödinger equation. In place of the wave function, its Gaussianwindowed Fourier transform is used as the fundamental entity. This allows an intuitively attractive connection to be made with a family of classical trajectories and, at all times, the wave function is inferred from the present state of these trajectories. The fact that the connection between the wave function and the classical trajectories is consistently constructed in phase space allows this method to be free of the limitations of other methods.

Bounded Bose fields in 1+1 dimensions commuting for space and timelike distances
View Description Hide DescriptionWe consider scalar Bose fields Φ in 1+1 dimensions which are bounded [i.e., is a bounded operator], commute for space and timelike distances, and are dilation covariant with scaling dimension We show that their truncated npointfunctions are related to the truncated functions of via with is a free chiral real Fermi field of dimension depending on the light cone coordinates This comes close to the conjecture that under the above assumptions Φ is nothing but a weighted sproduct of

The Floquet analysis and noninteger higher harmonics generation
View Description Hide DescriptionWe consider here the exact solution of a nonrelativistic quantum system composed of a twolevel atom interacting with a laser with arbitrary large frequency and intensity. We use the analogy of this system and a ℏ/2spin particle interacting with a timedependent magnetic field. A systematic use of the dynamical symmetry underlying the physical system is made. Actually the Hamiltonian is a Hermitian element of the SU(2) Lie Algebra. The exact Temporal Evolution Operator in terms of a Generalized Displacement Operator of the group is constructed. It is possible to develop a nonperturbative method that allows us to solve exactly the model for any value of the relevant frequencies (Rabi, Laser, and Atom Frequencies) and in so doing the usual Rotating Wave and Small Interaction approximations are unnecessary. The properties of periodicity of this model and the phenomenon of harmonic generation are considered by using Floquet Analysis. We find that in addition to the so far wellknown spectrum composed by odd harmonics, this model generates another type of noninteger harmonics whose frequencies and amplitudes are determined for any value of the relevant parameters of the system.

Generalization of the Birman–Schwinger method for the number of bound states
View Description Hide DescriptionWe generalize the Birman–Schwinger method, and derive a general upper bound on the number of bound states in the S wave for a spherically symmetric potential. This general bound includes, of course, the Bargmann bound, but also leads, for increasing (negative) potentials, to a Calogero–Cohntype bound. Finally, we show that for a large class among these potentials, one can obtain further improvements.

Energy levels of a quantal particle enclosed in N identical, mirror symmetric wells of a periodic potential. Numerical test of phaseintegral formulas
View Description Hide DescriptionAn interesting structure prevails for the energy levels of a quantal particle in a periodic potential with N (⩾2) mirror symmetric wells separated by mirror symmetric barriers, when the logarithmic derivative of the wave function is given at corresponding (periodically and mirror symmetrically situated) points in the barrier to the left of the first well and in the barrier to the right of the well. It is shown that the quantization conditions that one obtains for these energy levels by means of a careful and rigorous phaseintegral treatment are capable of giving extremely accurate results. The accuracy obtainable is demonstrated for by comparison with numerically exact results, which were obtained by means of the extended version of the phaseamplitude method presented in an Appendix. In the concluding section we summarize the results and point out unexpected features of the energy spectrum and the wave functions. Two different boundary conditions, commonly used in the theory of crystals, and closely related to the present investigation, are also discussed there.

Instability of a pseudorelativistic model of matter with selfgenerated magnetic field
View Description Hide DescriptionFor a pseudorelativistic model of matter, based on the nopair Hamiltonian, we prove that the inclusion of the interaction with the selfgenerated magnetic field leads to instability for all positive values of the fine structure constant. This is true no matter whether this interaction is accounted for by the Breit potential, by an external magnetic field which is chosen to minimize the energy, or by the quantized radiation field.

Gauge symmetries of the master action in the Batalin–Vilkovisky formalism
View Description Hide DescriptionWe study the geometry of the Lagrangian Batalin–Vilkovisky (BV) theory on an antisymplectic manifold. We show that gauge symmetries of the BV theory are essentially the symmetries of an even symplectic structure on the stationary surface of the master action.

On path integral localization and the Laplacian
View Description Hide DescriptionWe introduce a new localization principle, which is a generalized canonical transformation. It unifies BRST localization, the nonAbelian localization principle and a special case of the conformal Duistermaat–Heckman integration formula of Paniak, Semenoff, and Szabo. The heat kernel on compact Lie groups is localized in two ways. First, using a nonAbelian generalization of the derivative expansion localization of Palo and Niemi and second, using the BRST localization principle and a configuration space path integral. In addition, we present some new formulas on homogeneous spaces, which might be useful in a possible localization of Selberg’s trace formula on locally homogeneous spaces.

Twoparticle scattering theory for anyons
View Description Hide DescriptionWe consider potential scattering theory of a nonrelativistic quantum mechanical 2particle system in with anyon statistics. Sufficient conditions are given which guarantee the existence of Møller operators and the unitarity of the Smatrix. As examples the rotationally invariant potential well and the δfunction potential are discussed in detail. In case of a general rotationally invariant potential the angular momentum decomposition leads to a theory of Jost functions. The anyon statistics parameter gives rise to an interpolation for angular momenta analogous to the Regge trajectories for complex angular momenta. Levinson’s theorem is adapted to the present context. In particular we find that in case of a zero energy resonance the statistics parameter can be determined from the scattering phase.

Poisson brackets of normalordered Wilson loops
View Description Hide DescriptionWe formulate Yang–Mills theory in terms of the large limit, viewed as a classical limit, of gaugeinvariant dynamical variables, which are closely related to Wilson loops, via deformation quantization. We obtain a Poisson algebra of these dynamical variables corresponding to normalordered quantum (at a finite value of ℏ) operators. Comparing with a Poisson algebra one of us introduced in the past for Weylordered quantum operators, we find, using ideas closely related to topological graph theory, that these two Poisson algebras are, roughly speaking, the same. More precisely speaking, there exists an invertible Poisson morphism between them.

Differential equations for scaling relation in supersymmetric SU(2) Yang–Mills theory coupled with massive hypermultiplet
View Description Hide DescriptionDifferential equations for the scaling relation of prepotential in supersymmetric SU(2) Yang–Mills theory coupled with massive matter hypermultiplet are proposed and are explicitly demonstrated in one flavor theory. By applying Whitham dynamics, the firstorder derivative of the prepotential over the variable corresponding to the mass of the hypermultiplet, which has a line integral representation, is found to satisfy a differential equation. As a result, the closed form of this derivative can be obtained by solving this equation. In this way, the scaling relation of massive prepotential is established. Furthermore, as an application of another differential equation for the massive scaling relation, the massive prepotential in a strong coupling region is derived.

The fluxacrosssurfaces theorem for short range potentials and wave functions without energy cutoffs
View Description Hide DescriptionThe quantum probability flux of a particle integrated over time and a distant surface gives the probability for the particle crossing that surface at some time. The relation between these crossing probabilities and the usual formula for the scattering cross section is provided by the fluxacrosssurfaces theorem, which was conjectured by Combes, Newton, and Shtokhamer [Phys. Rev. D 11, 366–372 (1975)]. We prove the fluxacrosssurfaces theorem for short range potentials and wave functions without energy cutoffs. The proof is based on the free fluxacrosssurfaces theorem (Daumer et al.) [Lett. Math. Phys. 38, 103–116 (1996)], and on smoothness properties of generalized eigenfunctions: It is shown that if the potential decays like at infinity with then the generalized eigenfunctions of the corresponding Hamiltonian are times continuously differentiable with respect to the momentum variable.

 CLASSICAL MECHANICS AND CLASSICAL FIELDS


On conserved quantities at spatial infinity
View Description Hide DescriptionThere is a wellknown short list of asymptotic conserved quantities for a physical system at spatial infinity. We search for new ones. This is carried out within the asymptotic framework of Ashtekar and Romano, in which spatial infinity is represented as a smooth boundary of space–time. We first introduce, for physical fields on space–time, a characterization of their asymptotic behavior as certain fields on this boundary. Conserved quantities at spatial infinity, in turn, are constructed from these fields. We find, in Minkowski space–time, that each of a Klein–Gordon field, a Maxwell field, and a linearized gravitational field yields an entire hierarchy of conserved quantities. Only certain quantities in this hierarchy survive into curved space–time.

 DYNAMICAL SYSTEMS AND FLUID DYNAMICS


Group invariant solutions for the super Korteweg–de Vries equation
View Description Hide DescriptionThe method of symmetry reduction is used to solve Grassmannvalued differential equations. The supersymmetric Korteweg–de Vries equation is considered. It admits a Lie superalgebra of symmetries of dimension 5. A twodimensional subsuperalgebra is chosen to reduce the number of independent variables in this equation. We are then able to give different types of exact solutions, in particular soliton solutions.

On the integrability of nonlinear partial differential equations
View Description Hide DescriptionWe investigate the integrability of Nonlinear Partial Differential Equations (NPDEs). The concepts are developed by first discussing the integrability of the KdV equation. We proceed by generalizing the ideas introduced for the KdV equation to other NPDEs. The method is based upon a linearization principle that can be applied on nonlinearities that have a polynomial form. The method is further illustrated by finding solutions of the nonlinear Schrödinger equation and the vector nonlinear Schrödinger equation, which play an important role in optical fiber communication. Finally, it is shown that the method can also be generalized to higher dimensions.

Isospectral problem for Schrödinger operator: Evolutional viewpoint
View Description Hide DescriptionAn isospectral transform of the Schrödinger operator is considered as an evolutional problem. For a transform defined by the McKean–Trubowitz flows associated evolutional equations are derived. It is shown that for onelevel and twolevel flows these equations can be split into integrable Liouville equations. A relationship between the Liouville equations and the Darboux transforms is discussed; this analysis suggests that the evolutional equations can be split into the Liouville equations in the general case. A Hamiltonian formulation of the isospectral transform defined by the McKean–Trubowitz flows is presented. It is shown that this transform is performed by a canonical change of variables, which is related to the Darboux transform.

Soliton solutions, Liouville integrability and gauge equivalence of Sasa Satsuma equation
View Description Hide DescriptionExact integrability of the Sasa Satsuma equation (SSE) in the Liouville sense is established by showing the existence of an infinite set of conservation laws. The explicit form of the conserved quantities in terms of the fields are obtained by solving the Riccati equation for the associated Lax operator. The soliton solutions, in particular, one and two soliton solutions, are constructed by the Hirota’s bilinear method. The one soliton solution is also compared with that found through the inverse scattering method. The gauge equivalence of the SSE with a generalized Landau Lifshitz equation is established with the explicit construction of the new equivalent Lax pair.

Hirota bilinear approach to a new integrable differentialdifference system
View Description Hide DescriptionA new integrable differentialdifference system is proposed. By the dependent variable transformation, the system is transformed into multilinear form. By introducing an auxiliary variable, we further transform it into the bilinear form. A corresponding Bäcklund transformation for it is obtained. Furthermore a nonlinear superposition formula is presented. As an application of the obtained results, soliton solutions to the system are derived.

Highfrequency solitonlike waves in a relaxing medium
View Description Hide DescriptionA nonlinear evolution equation is suggested to describe the propagation of waves in a relaxing medium. It is shown that for lowfrequency approach this equation is reduced to the KdVB equation. The highfrequency perturbations are described by a new nonlinear equation. This equation has ambiguous looplike solutions. It is established that a dissipative term, with a dissipation parameter less than some limit value, does not destroy these looplike solutions.

 RELATIVITY AND GRAVITATION


Asymptotically Schwarzschild space–times
View Description Hide DescriptionIt is shown that if an asymptotically flat space–time is asymptotically stationary, in the sense that vanishes at the rate for asymptotically timelike vector field and the energy–momentum tensor vanishes at the rate then the space–time is an asymptotically Schwarzschild space–time. This gives a new aspect of the uniqueness theorem of a black hole.
