Volume 48, Issue 12, December 2007
Index of content:
 QUANTUM MECHANICS (GENERAL AND NONRELATIVISTIC)


Generalized definition of time delay in scattering theory
View Description Hide DescriptionWe advocate for the systematic use of a symmetrized definition of time delay in scattering theory. In twobody scattering processes, we show that the symmetrized time delay exists for arbitrary dilated spatial regions symmetric with respect to the origin. It is equal to the usual time delay plus a new contribution, which vanishes in the case of spherical spatial regions. We also prove that the symmetrized time delay is invariant under an appropriate mapping of time reversal. These results are also discussed in the context of classical scattering theory.

Multiplicativity of the maximal output 2norm for depolarized WernerHolevo channels
View Description Hide DescriptionWe study the multiplicativity of the output 2norm for depolarized WernerHolevo channels and show that multiplicativity holds for a product of two identical channels in this class. Moreover, it is shown that the depolarized WernerHolevo channels do not satisfy the entrywise positivity condition introduced by King and Ruskai (eprint quantph∕0909181v2;eprint quantph∕0981026v1), which suggests that the main result is nontrivial.

Nonrelativistic Lee model in three dimensional Riemannian manifolds
View Description Hide DescriptionIn this work, we construct the nonrelativistic Lee model on some class of three dimensional Riemannian manifolds by following a novel approach introduced by S. G. Rajeev (eprint hepth∕9902025). This approach together with the help of heat kernel allows us to perform the renormalization nonperturbatively and explicitly. For completeness, we show that the ground state energy is bounded from below for different classes of manifolds, using the upper bound estimates on the heat kernel. Finally, we apply a kind of mean field approximation to the model for compact and noncompact manifolds separately and discover that the ground state energy grows linearly with the number of bosons .

Quasiexactly solvable difference equations
View Description Hide DescriptionSeveral explicit examples of quasiexactly solvable “discrete” quantum mechanical Hamiltonians are derived by deforming the wellknown exactly solvable Hamiltonians of one degree of freedom. These are difference analogs of the wellknown quasiexactly solvable systems, the harmonic oscillator (with∕without the centrifugal potential) deformed by a sextic potential, and the potential deformed by a potential. They have a finite number of exactly calculable eigenvalues and eigenfunctions.

Multiparticle quasiexactly solvable difference equations
View Description Hide DescriptionSeveral explicit examples of multiparticle quasiexactly solvable “discrete” quantum mechanical Hamiltonians are derived by deforming the wellknown exactly solvable multiparticle Hamiltonians, the RuijsenaarsSchneidervan Diejen systems. These are difference analogs of the quasiexactly solvable multiparticle systems, the quantum Inozemtsev systems obtained by deforming the wellknown exactly solvable CalogeroSutherland systems. They have a finite number of exactly calculable eigenvalues and eigenfunctions. This paper is a multiparticle extension of the recent paper by one of the authors [R. Sasaki, J. Math. Phys.48, 122104 (2007)] on deriving quasiexactly solvable difference equations of single degree of freedom.

Stochastic representations of Feynman integration
View Description Hide DescriptionFor polynomially bounded potentials such that is essentially selfadjoint on , this essay offers two reconstructions of Feynman’s sum over histories as the unitary image of a genuine integral with respect to Wiener measure of a functional defined on the space of Brownian paths into momentum space . The first representation, based on Feynman’s original argument, “lifts” from a “convolutional Trotter product formula” for the Fouriertransformed image of the timeevolved wave function in . The second—which varies and extends a construction introduced in a slightly different context by Albeverio and HøeghKrohn [Mathematical Theory of Feynman Integrals, Springer Lecture Notes in Mathematics Vol. 523 (Springer, New York, 1976)]—lifts the functional from a “convolutional Dyson expansion” of the timeevolved momentumspace function .

Parametrictime coherent states for SmorodinskyWinternitz potentials
View Description Hide DescriptionIn this study, we construct the coherent states for a particle in the SmorodinskyWinternitz potentials, which are the generalizations of the twodimensional Kepler problem. In the third case, the system is transformed into four ocillators and the parametrictime coherent states are constructed in two coordinate frames. In the fourth case, the system is transformed into two ocillators with the reflection symmetry and the parametrictime coherent states are constructed in two coordinate frames.

 RELATIVISTIC QUANTUM MECHANICS, FIELD THEORY, BRANE THEORY (INCLUDING STRINGS)


The orbifolds of permutation type as physical string systems at multiples of . Cyclic permutation orbifolds
View Description Hide DescriptionI consider the prime freebosonic permutation orbifolds as interacting physical string systems at . As a first step, I introduce twisted tree diagrams which confirm at the interacting level that the physical spectrum of each twisted sector is equivalent to that of an ordinary closed string. The untwisted sectors are surprisingly more difficult to understand, and there are subtleties in the sewing of the loops, but I am able to propose provisional forms for the full modularinvariant cosmological constants and oneloop diagrams with insertions.

Lie fields revisited
View Description Hide DescriptionA class of interacting classical random fields is constructed using deformed ⋆algebras of creation and annihilation operators. The fields constructed are classical random field versions of “Lie fields.” A vacuum vector is used to construct linear forms over the algebras, which are conjectured to be states over the algebras. Assuming this conjecture is true, the fields constructed are “quantum random fields” in the sense that they have Poincaré invariant vacua with a fluctuation scale determined by . A nonlocal particle interpretation of the formalism is shown to be the same as a particle interpretation of a quantum field theory.

 GENERAL RELATIVITY AND GRAVITATION


On the conformal forms of the RobertsonWalker metric
View Description Hide DescriptionAll possible transformations from the RobertsonWalker metric to those conformal to the LorentzMinkowski form are derived. It is demonstrated that the commonly known family of transformations and associated conformal factors are not exhaustive and that there exists another relatively less well known family of transformations with a different conformal factor in the particular case that . Simplified conformal factors are derived for the special case of maximally symmetric spacetimes. The full set of all possible cosmologically compatible conformal forms is presented as a comprehensive table. A product of the analysis is the determination of the settheoretical relationships between the maximally symmetric spacetimes, the RobertsonWalker spacetimes, and functionally more general spacetimes. The analysis is preceded by a short historical review of the application of conformal metrics to cosmology.

 DYNAMICAL SYSTEMS


Connection between quantum mechanical and classical time evolution of certain dissipative systems via a dynamical invariant
View Description Hide DescriptionIn a former publication [D. Schuch and M. Moshinsky, Phys. Rev. A73, 062111 (2006)] we have shown that the time evolution of a quantum system with at most quadratic Hamiltonian that can be described by different methods, namely, the timedependent Schrödinger equation, the time propagator or Feynman kernel, and the Wigner function is connected via a dynamical invariant, the socalled Ermakov invariant. Since exact invariants of this type also exist for certain effective descriptions of dissipative quantum systems, we will show the relations between these descriptions and the corresponding invariants and will discuss how the results obtained for the conservative systems must be modified in the presence of a linear velocity dependent frictional force.

 CLASSICAL MECHANICS AND CLASSICAL FIELDS


The support of the limit distribution of optimal Riesz energy points on sets of revolution in
View Description Hide DescriptionLet be a compact point set in the right half of the plane and the set in obtained by rotating about the axis. We investigate the support of the limit distribution of minimal energy point charges on that interact according to the Riesz potential , , where is the Euclidean distance between points. Potential theory yields that this limit distribution coincides with the equilibrium measure on which is supported on the outer boundary of . We show that there are sets of revolution such that the support of the equilibrium measure on is not the complete outer boundary, in contrast to the Coulomb case . However, the support of the limit distribution on the set of revolution as goes to infinity is the full outer boundary for certain sets, in contrast to the logarithmic case .

On convex surfaces with minimal moment of inertia
View Description Hide DescriptionWe investigate the problem of minimizing the moment of inertia among convex surfaces in having a specified surface area. First, we prove that a minimizing surface exists, and derive a necessary condition holding at points of positive curvature. Then we show that an equilateral triangular prism is the optimal triangular prism, that the cube is the optimal rectangular prism, and that the sphere is (locally) optimal among ellipsoids. Many examples of convex surfaces are examined, among which the lowest moment of inertia is achieved by a truncated tetrahedron. The problem of finding the global minimizing surface remains open. The analogous problem in two dimensions has been solved by Sachs and later by Hall, who showed that the equilateral triangle minimizes the moment of inertia, among all convex curves with given length.

 FLUIDS


An exact fluid model for relativistic electron beams: The many moment case
View Description Hide DescriptionAn interesting and satisfactory fluid model has been proposed in literature for the description of relativistic electron beams. It was obtained with 14 independent variables by imposing the entropy principle and the relativity principle. Here the case is considered with an arbitrary number of independent variables, still satisfying the above mentioned two principles; these lead to conditions whose general solution is here found. We think that the results satisfy also a certain ordering with respect to a smallness parameter measuring the dispersion of the velocity about the mean; this ordering generalizes that appearing in literature for the 14 moments case.

Shock reflection and oblique shock waves
View Description Hide DescriptionThe linear stability of steady attached oblique shock wave and pseudosteady regular shock reflection is studied for the nonviscous full Euler system of equations in aerodynamics. A sufficient and necessary condition is obtained for their linear stability under threedimensional perturbation. The result confirms the sonic point condition in the study of transition point from regular reflection to Mach reflection, in contrast to the von Neumann condition and detachment condition predicted from mathematical constraint.

 STATISTICAL PHYSICS


Large deviations and Chernoff bound for certain correlated states on a spin chain
View Description Hide DescriptionIn this paper we extend the results of Lenci and ReyBellet [J. Stat. Phys.119, 715 (2005)] on the large deviation upper bound of the distribution measures of local Hamiltonians with respect to a Gibbs state in the setting of translationinvariant finiterange interactions. We show that a certain factorization property of the reference state is sufficient for a large deviation upper bound to hold and that this factorization property is satisfied by Gibbs states of the above kind as well as finitely correlated states. As an application of the methods, the Chernoff bound for correlated states with factorization property is studied. In the specific case of the distributions of the ergodic averages of a onesite observable with respect to an ergodic finitely correlated state, the spectral theory of positive maps is applied to prove the full large deviation principle.

Interacting quantum gases in confined space: Two and threedimensional equations of state
View Description Hide DescriptionIn this paper, we calculate the equations of state and the thermodynamic quantities for two and threedimensional hardsphere Bose and Fermi gases in finitesize containers. The approach we used to deal with interacting gases is to convert the effect of interparticle hardsphere interaction to a kind of boundary effect, and then the problem of a confined hardsphere quantum gas is converted to the problem of a confined ideal quantum gas with a complex boundary. For this purpose, we first develop an approach for calculating the boundary effect on dimensional ideal quantum gases and then calculate the equation of state for confined quantum hardsphere gases. The thermodynamic quantities and their lowtemperature and highdensity expansions are also given. In higherorder contributions, there are cross terms involving both the influences of the boundary and of the interparticle interaction. We compare the effect of the boundary and the effect of the interparticle interaction. Our result shows that, at low temperatures and high densities, the ratios of the effect of the boundary to the effect of the interparticle interaction in two dimensions are essentially different to those in three dimensions: in two dimensions, the ratios for Bose systems and for Fermi systems are the same and are independent of temperatures, while in three dimensions, the ratio for Bose systems depends on temperatures, but the ratio for Fermi systems is independent of temperatures. Moreover, for threedimensional Fermi cases, compared with the contributions from the boundary, the contributions from the interparticle interaction to entropies and specific heats are negligible.

 METHODS OF MATHEMATICAL PHYSICS


Determination of small amplitude perturbations for the electric permittivity from partial dynamic boundary measurements
View Description Hide DescriptionThe paper deals with reconstruction of small perturbation of the uniform isotropic complex electric permittivity from dynamic measurements of the tangential component of the magnetic field on the boundary (or a part of the boundary) of a domain. The method is based on derived asymptotic inverse Fourier transform of the perturbation in the complex permittivity. Through construction of appropriate test functions by a geometrical control method, we provide a rigorous derivation of the inverse Fourier transform of the perturbations in the electric permittivity as the leading order of an appropriate averaging of the partial dynamic boundary perturbations.

Coupling constant behavior of eigenvalues of ZakharovShabat systems
View Description Hide DescriptionWe consider the eigenvalues of the nonselfadjoint ZakharovShabat systems as the coupling constant of the potential is varied. In particular, we are interested in eigenvalue collisions and eigenvalue trajectories in the complex plane. We identify shape features in the potential that are responsible for the occurrence of collisions and we prove asymptotic formulas for large coupling constants that tell us where eigenvalues collide or where they emerge from the continuous spectrum. Some examples are provided which show that the asymptotic methods yield results that compare well with exact numerical computations.

Random matrices, nonbacktracking walks, and orthogonal polynomials
View Description Hide DescriptionSeveral wellknown results from the random matrix theory, such as Wigner’s law and the MarchenkoPastur law, can be interpreted (and proved) in terms of nonbacktracking walks on a certain graph. Orthogonal polynomials with respect to the limiting spectral measure play a role in this approach.
