Volume 45, Issue 3, March 2004
Index of content:
 QUANTUM MECHANICS (GENERAL AND NONRELATIVISTIC)


“Squashed entanglement”: An additive entanglement measure
View Description Hide DescriptionIn this paper, we present a new entanglement monotone for bipartite quantum states. Its definition is inspired by the socalled intrinsic information of classical cryptography and is given by the halved minimum quantum conditional mutual information over all tripartite state extensions. We derive certain properties of the new measure which we call “squashed entanglement”: it is a lower bound on entanglement of formation and an upper bound on distillable entanglement. Furthermore, it is convex, additive on tensor products, and superadditive in general. Continuity in the state is the only property of our entanglement measure which we cannot provide a proof for. We present some evidence, however, that our quantity has this property, the strongest indication being a conjectured Fannestype inequality for the conditional von Neumann entropy. This inequality is proved in the classical case.

Quantum mechanics of damped systems. II. Damping and parabolic potential barrier
View Description Hide DescriptionWe investigate the resonant states for the parabolic potential barrier known also as inverted or reversed oscillator. They correspond to the poles of meromorphic continuation of the resolvent operator to the complex energy plane. As a byproduct we establish an interesting relation between parabolic cylinder functions (representing energy eigenfunctions of our system) and a class of Gel’fand distributions used in our recent paper.

Galerkin analysis for Schrödinger equation by wavelets
View Description Hide DescriptionWe consider the perturbed Schrödinger equation, which is an elliptic operator with unbounded coefficients. We use wavelets adapted to the Schrödinger operator to deal with problems on the unbounded domain. The wavelets are constructed from Hermite functions, which characterizes the space generated by the Schrödinger operator. We show that the Galerkin matrix can be preconditioned by a diagonal matrix so that its condition number is uniformly bounded. Moreover, we introduce a periodic pseudodifferential operator and show that its discrete Galerkin matrix under periodic wavelet system is equal to the Galerkin matrix for the equation with unbounded coefficients under the Hermite system. The convergence is proved in the topology.

A random matrix approach to the crossover of energylevel statistics from Wigner to Poisson
View Description Hide DescriptionWe analyze a class of parametrized random matrix models, introduced by Rosenzweig and Porter, which is expected to describe the energy level statistics of quantum systems whose classical dynamics varies from regular to chaotic as a function of a parameter. We compute the generating function for the correlations of energy levels, in the limit of infinite matrix size. The crossover between Poisson and Wigner statistics is measured by a renormalized coupling constant. The model is exactly solved in the sense that, in the limit of infinite matrix size, the energylevel correlation functions and their generating function are given in terms of a finite set of integrals.

Time dependent transformations in deformation quantization
View Description Hide DescriptionWe study the action of time dependent canonical and coordinate transformations in phase space quantum mechanics. We extend the covariant formulation of the theory by providing a formalism that is fully invariant under both standard and time dependent coordinate transformations. This result considerably enlarges the set of possible phase space representations of quantum mechanics and makes it possible to construct a causal representation for the distributional sector of Wigner quantum mechanics.

Contextual approach to quantum mechanics and the theory of the fundamental prespace
View Description Hide DescriptionWe constructed a Hilbert space representation of a contextual Kolmogorov model. This representation is based on two fundamental observables—in the standard quantum model these are the position and momentum observables. This representation has all distinguishing features of the quantum model. Our representation is not standard model with hidden variables. In particular, this is not a reduction of the quantum model to the classical one.

Bound states in two spatial dimensions in the noncentral case
View Description Hide DescriptionWe derive a bound on the total number of negative energy bound states in a potential in two spatial dimensions by using an adaptation of the Schwinger method to derive the Birman–Schwinger bound in three dimensions. Specifically, counting the number of bound states in a potential for is replaced by counting the number of for which zero energy bound states exist, and then the kernel of the integral equation for the zeroenergy wave functon is symmetrized. One of the keys of the solution is the replacement of an inhomogeneous integral equation by a homogeneous integral equation.

Pseudounitary operators and pseudounitary quantum dynamics
View Description Hide DescriptionWe consider pseudounitary quantum systems and discuss various properties of pseudounitary operators. In particular we prove a characterization theorem for blockdiagonalizable pseudounitary operators with finitedimensional diagonal blocks. Furthermore, we show that every pseudounitary matrix is the exponential of times a pseudoHermitian matrix, and determine the structure of the Lie groups consisting of pseudounitary matrices. In particular, we present a thorough treatment of 2×2 pseudounitary matrices and discuss an example of a quantum system with a 2×2 pseudounitary dynamical group. As other applications of our general results we give a proof of the spectral theorem for symplectic transformations of classical mechanics, demonstrate the coincidence of the symplectic group with the real subgroup of a matrix group that is isomorphic to the pseudounitary group and elaborate on an approach to second quantization that makes use of the underlying pseudounitary dynamical groups.

Equal rank embedding and its related construction to superconformal field theories
View Description Hide DescriptionThe lowest lines of Euler multiplets corresponding to massless and massive supersymmetric representations are classified. At the level of representation theory, the Euler multiplets constructed by the GKRS method of equal rank embedding of semisimple complex Lie algebras are found to be the intrinsic ground states of superconformal field theories according to the Kazama–Suzuki construction.

On linearity of separating multiparticle differential Schrödinger operators for identical particles
View Description Hide DescriptionWe show that hierarchies of differential Schrödinger operators for identical particles which are separating for the usual (anti)symmetric tensor product, are necessarily linear, and offer some speculations on the source of quantum linearity.

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


Casimir energy inside a cavity with triangular cross section
View Description Hide DescriptionFor a certain class of triangles (with angles proportional to we formulate an image method by making use of the group generated by reflections with respect to the three lines which form the triangle under consideration. We formulate the regularization procedure by classification of subgroups of and corresponding fixed points in the triangle. We then also calculate Casimir energy for a cavity of infinite height with triangular cross section for scalar massless fields. More detailed calculation is given for odd

Soliton solutions on noncommutative orbifold
View Description Hide DescriptionIn this paper, we explicitly construct a series of projectors on integrable noncommutative orbifold by extended construction. They include integration of two arbitary functions with symmetry. Our expression possesses manifest symmetry. It is proven that the expression includes all projectors with minimal trace and in their standard expansions, the eigenvalue functions of coefficient operators are continuous with respect to the arguments and Based on the integral expression, we alternately show the derivative expression in terms of the similar kernal to the integral one. Since projectors correspond to soliton solutions of the field theory on the noncommutative orbifold, we thus present a series of corresponding solitons.

 GENERAL RELATIVITY AND GRAVITATION


The BMS group and generalized gravitational instantons
View Description Hide DescriptionThe ordinary Bondi–Metzner–Sachs (BMS) group B is the best candidate for the fundamental symmetry group of General Relativity. It has been shown that B admits generalizations to real space–times of any signature, and also to complex space–times. It has been suggested that certain continuous unitary irreducible representations (IRs) of B and of its generalizations correspond to gravitational instantons. Here I make this correspondence more precise and I take this suggestion one step further by arguing that a subclass of IRs of B and of its generalizations correspond to generalized gravitational instantons. Some of these generalized gravitational instantons involve in their definition certain subgroups of the Cartesian product group where is the cyclic group of order r. With this motivation, I give the subgroups of explicitly.

 DYNAMICAL SYSTEMS


Hamiltonians separable in Cartesian coordinates and thirdorder integrals of motion
View Description Hide DescriptionWe present in this article all Hamiltonian systems in that are separable in Cartesian coordinates and that admit a thirdorder integral, both in quantum and in classical mechanics. Many of these superintegrable systems are new, and it is seen that there exists a relation between quantum superintegrable potentials, invariant solutions of the Korteweg–de Vries equation and the Painlevé transcendents.

Equations of arbitrary order invariant under the Kadomtsev–Petviashvili symmetry group
View Description Hide DescriptionBy means of a simple new approach, a general Kadomtsev–Petviashvili (KP) family with an arbitrary function of group invariants of arbitrary order is proposed. It is proved that the general KP family possesses a common infinite dimensional Kac–Moody–Virasoro Lie point symmetry algebra. The known fourth order one can be reobtained as a special example. The finite transformation group is presented in a clearer form. The Kac–Moody–Virasoro group invariant solutions and the Kac–Moody group invariant solutions of the KP family are determined by the Boussinesq and KdV families, respectively.

Hyperelliptic Nambu flow associated with integrable maps
View Description Hide DescriptionWe study hyperelliptic Nambu flows associated with some dimensional maps and show that discrete integrable systems can be reproduced as flows of this class.

 CLASSICAL MECHANICS AND CLASSICAL FIELDS


Geometric integrators and nonholonomic mechanics
View Description Hide DescriptionA geometric derivation of nonholonomic integrators is developed. It is based in the classical technique of generating functions adapted to the special features of nonholonomic systems. The theoretical methodology and the integrators obtained are different from those obtained in Cortés and Martı́nez [“Nonholonomic integrators,” Nonlinearity 14, 1365–1392 (2001)]. In the case of mechanical systems with linear constraints a family of geometric integrators preserving the nonholonomic constraints is given.

Nonintegrability of the Suslov problem
View Description Hide DescriptionIn this paper we investigate the Suslov problem in the case when the vector of nonholonomic constraint coincides with the third principal axis of the body, and the fixed point of the body lies in the principal plane defined by the third and the first principal axes but is out of these axes. We called this version of the Suslov problem the generalized Kozlov case, and we prove that in this case a third real meromorphic first integral functionally independent together with the energy and geometrical integrals does not exist.

An extension of the classical theory of algebraic invariants to pseudoRiemannian geometry and Hamiltonian mechanics
View Description Hide DescriptionWe develop a new approach to the study of Killing tensors defined in pseudoRiemannian spaces of constant curvature that is ideologically close to the classical theory of invariants. The main idea, which provides the foundation of the new approach, is to treat a Killing tensor as an algebraic object determined by a set of parameters of the corresponding vector space of Killing tensors under the action of the isometry group. The spaces of group invariants and conformal group invariants of valence two Killing tensors defined in the Minkowski plane are described. The group invariants, which are the generators of the space of invariants, are applied to the problem of classification of orthogonally separable Hamiltonian systems defined in the Minkowski plane. Transformation formulas to separable coordinates expressed in terms of the parameters of the corresponding space of Killing tensors are presented. The results are applied to the problem of orthogonal separability of the Drach superintegrable potentials.

 STATISTICAL PHYSICS


The Potts model on with countable set of spin values
View Description Hide DescriptionThe Potts model with countable set Φ of spin values on is considered. It is proved that with respect to Poisson distribution on Φ the set of limiting Gibbs measures is not empty.
