Volume 45, Issue 4, April 2004
Index of content:
 QUANTUM MECHANICS (GENERAL AND NONRELATIVISTIC)


On solvable potentials related to SO(2,2). II. Natanzon potentials
View Description Hide DescriptionGeneral Natanzon potentials related to the SO(2,2) group are studied. The matrices for systems under consideration are related to intertwining operators between Weyl equivalent most degenerate principal series representations of SO(2,2).

Spectral properties of a shortrange impurity in a quantum dot
View Description Hide DescriptionThe spectral properties of the quantum mechanical system consisting of a quantum dot with a shortrange attractive impurity inside the dot are studied in the zerorange limit. The Green function of the system is obtained in an explicit form. In the case of a spherically symmetric quantum dot, the dependence of the spectrum on the impurity position and strength of the impurity potential is analyzed in detail. The recovering of the confinement potential of the dot from the spectroscopy data is proven; the consequences of the hidden symmetry breaking by the impurity are considered. The effect of the positional disorder is analyzed.

Decoherence in a twoparticle model
View Description Hide DescriptionWe consider a simple onedimensional quantum system consisting of a heavy and a light particle interacting via a point interaction. The initial state is chosen to be a product state, with the heavy particle described by a coherent superposition of two spatially separated wave packets with opposite momentum and the light particle localized in the region between the two wave packets. We characterize the asymptotic dynamics of the system in the limit of small mass ratio, with an explicit control of the error. We derive the corresponding reduced density matrix for the heavy particle and explicitly compute the (partial) decoherence effect for the heavy particle induced by the presence of the light one for a particular set up of the parameters.

Continuum singularities of a meanfield theory of collisions
View Description Hide DescriptionConsider a complex energy for an particle Hamiltonian and let χ be any wave packet accounting for any channel flux. The timeindependent meanfield (TIMF) approximation of the inhomogeneous, linear equation consists of replacing Ψ by a product or Slater determinant φ of singleparticle states This results, under the Schwinger variational principle, in selfconsistent TIMF equations in singleparticle space. The method is a generalization of the Hartree–Fock (HF) replacement of the body homogeneous linear equation by singleparticle HF diagonalizations We show how, despite strong nonlinearities in this meanfield method, threshold singularities of the inhomogeneous TIMF equations are linked to solutions of the homogeneous HF equations.

Quantum indistinguishability from general representations of
View Description Hide DescriptionA treatment of the spinstatistics relation in nonrelativistic quantum mechanics due to Berry and Robbins [Proc. R. Soc. London Ser. A 453, 1771–1790 (1997)] is generalized within a grouptheoretical framework. The construction of Berry and Robbins is reformulated in terms of certain locally flat vector bundles over particle configuration space. It is shown how families of such bundles can be constructed from irreducible representations of the group The construction of Berry and Robbins, which leads to a definite connection between spin and statistics (the physically correct connection), is shown to correspond to the completely symmetric representations. The spinstatistics connection is typically broken for general representations, which may admit, for a given value of spin, both Bose and Fermi statistics, as well as parastatistics. The determination of the allowed values of the spin and statistics reduces to the decomposition of certain zeroweight representations of a (generalized) Weyl group of A formula for this decomposition is obtained using the Littlewood–Richardson theorem for the decomposition of representations of into representations of

Bound states in coupled guides. I. Two dimensions
View Description Hide DescriptionBound states that can occur in coupled quantum wires are investigated. We consider a twodimensional configuration in which two parallel waveguides (of different widths) are coupled laterally through a finite length window and construct modes which exist local to the window connecting the two guides. We study both modes above and below the first cutoff for energy propagation down the coupled guide. The main tool used in the analysis is the socalled residue calculus technique, in which complex variable theory is used to solve a system of equations which is derived from a modematching approach. For bound states below the first cutoff a single existence condition is derived, but for modes above this cutoff (but below the second cutoff), two conditions must be satisfied simultaneously. A number of results have been presented which show how the boundstate energies vary with the other parameters in the problem.

Bound states in coupled guides. II. Three dimensions
View Description Hide DescriptionWe compute boundstate energies in two threedimensional coupled waveguides, each obtained from the twodimensional configuration considered in paper I [J. Math. Phys. 45, 1359–1379 (2004)] by rotating the geometry about a different axis. The first geometry consists of two concentric circular cylindrical waveguides coupled by a finite length gap along the axis of the inner cylinder, and the second is a pair of planar layers coupled laterally by a circular hole. We have also extended the theory for this latter case to include the possibility of multiple circular windows. Both problems are formulated using a modematching technique, and in the cylindrical guide case the same residue calculus theory as used in paper I is employed to find the boundstate energies. For the coupled planar layers we proceed differently, computing the zeros of a matrix derived from the matching analysis directly.

The magnetic Weyl calculus
View Description Hide DescriptionIn the presence of a variable magnetic field, the Weyl pseudodifferential calculus must be modified. The usual modification, based on “the minimal coupling principle” at the level of the classical symbols, does not lead to gauge invariant formulas if the magnetic field is not constant. We present a gauge covariant quantization, relying on the magnetic canonical commutation relations. The underlying symbolic calculus is a deformation, defined in terms of the magnetic flux through triangles, of the classical Moyal product.

On the absolutely continuous and negative discrete spectra of Schrödinger operators on the line with locally integrable globally square summable potentials
View Description Hide DescriptionFor onedimensional Schrödinger operators with potentials subject to we prove that the absolutely continuous spectrum is [0,∞), extending the 1999 result due to Dieft–Killip. As a byproduct we show that under the same condition the sequence of the negative eigenvalues is 3/2summable improving the relevant result by Lieb–Thirring.

Quantummechanical scattering in exterior domains with impenetrable periodic boundaries and shortrange potentials
View Description Hide DescriptionWe study the scattering of a nonrelativistic particle in an exterior domain (=open connected subset) containing a halfspace and contained in another halfspace, and having an impenetrable periodic boundary ∂Ω. “Impenetrable” means that (generalized) homogeneous Dirichlet conditions are imposed on ∂Ω. We prove the existence and completeness of the wave operators corresponding to the scattering of a nonrelativistic particle in Ω by the combined effect of the boundary and a shortrange potential present in Ω. Here is the negative distributional Laplacian in the Hilbert space being the Dirichlet Laplacian in the Hilbert space an operator of multiplication in H by a bounded measurable function on Ω having the periodicity of the boundary, and an identification operator. The operators model the quantummechanical scattering of lowenergy atoms by crystal surfaces, with modeling the interaction between the incident particles and the surface atoms. This interaction is idealized by assuming that depends solely on when a being a sufficiently large positive constant, and the component of directed perpendicularly to the surfaces of the above two halfspaces. Under this and other hypotheses on Ω and stated precisely in the paper, we prove that exist as partially isometric operators whose initial sets have a transparent physical meaning. Moreover, we prove the following: (a) and (b) are asymptotically complete, in the sense that Here and are suitably defined subspaces of scattering and surface states of H. These results are proved by using directintegral techniques, asymptotic methods from the theory of ODEs, and methods analogous to those of Lyford. The present paper generalizes an earlier one by the author for the case

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


Supersymmetric fieldtheoretic models on a supermanifold
View Description Hide DescriptionWe propose the extension of some structural aspects that have successfully been applied in the development of the theory of quantum fields propagating on a general space–time manifold so as to include superfield models on a supermanifold. We only deal with the limited class of supermanifolds which admit the existence of a smooth body manifold structure. Our considerations are based on the Catenacci–Reina–Teofillatto–Bryant approach to supermanifolds. In particular, we show that the class of supermanifolds constructed by Bonora–Pasti–Tonin satisfies the criteria which guarantee that a supermanifold admits a Hausdorff body manifold. This construction is the closest to the physicist’s intuitive view of superspace as a manifold with some anticommuting coordinates, where the odd sector is topologically trivial. The paper also contains a new construction of superdistributions and useful results on the wavefront set of such objects. Moreover, a generalization of the spectral condition is formulated using the notion of the wavefront set of superdistributions, which is equivalent to the requirement that all of the component fields satisfy, on the body manifold, a microlocal spectral condition proposed by Brunetti–Fredenhagen–Köhler.

Variational derivation of relativistic fermion–antifermion wave equations in QED
View Description Hide DescriptionWe present a variational method for deriving relativistic twofermion wave equations in a Hamiltonian formulation of QED. A reformulation of QED is performed, in which covariant Green functions are used to solve for the electromagnetic field in terms of the fermion fields. The resulting modified Hamiltonian contains the photon propagator directly. The reformulation permits one to use a simple Fockspace variational trial state to derive relativistic fermion–antifermion wave equations from the corresponding quantum field theory. We verify that the energy eigenvalues obtained from the wave equation agree with known results for positronium.

 GENERAL RELATIVITY AND GRAVITATION


Ideally embedded space–times
View Description Hide DescriptionDue to the growing interest in embeddings of space–time in higherdimensional spaces we consider a specific type of embedding. After proving an inequality between intrinsically defined curvature invariants and the squared mean curvature, we extend the notion of ideal embeddings from Riemannian geometry to the indefinite case. Ideal embeddings are such that the embedded manifold receives the least amount of tension from the surrounding space. Then it is shown that the de Sitter spaces, a Robertson–Walker space–time and some anisotropic perfect fluid metrics can be ideally embedded in a fivedimensional pseudoEuclidean space.

Homotopy structure of 5d vacua
View Description Hide DescriptionIt is shown that flat zeroenergy solutions (vacua) of the 5d Kaluza–Klein theory admit a nontrivial homotopy structure generated by certain Kaluza–Klein excitations. These vacua consist of an infinite set of homotopically different space–times denoted by among which and are especially identified as and the vacuum states of the 5d Kaluza–Klein theory and the 5d general relativity, respectively (where represents the kdimensional Minkowski space).

Classification of static plane symmetric space–times according to their matter collineations
View Description Hide DescriptionIn this paper we classify static plane symmetric space–times according to their matter collineations. These have been studied for both cases when the energy–momentum tensor is nondegenerate and also when it is degenerate. It turns out that the nondegenerate case yields either four, five, six, seven, or ten independent matter collineations in which four are isometries and the rest are proper. There exists three interesting cases where the energy–momentum tensor is degenerate but the group of matter collineations is finitedimensional. The matter collineations in these cases are either four, six, or ten.

Symmetries of the energymomentum tensor of cylindrically symmetric static space–times
View Description Hide DescriptionWe investigate matter symmetries of cylindrically symmetric static space–times. These are classified for both cases when the energymomentum tensor is nondegenerate and also when it is degenerate. It is found that the nondegenerate energymomentum tensor gives either three, four, five, six, seven or ten independent matter collineations in which three are isometries and the rest are proper. The worth mentioning cases are those where we obtain the group of matter collineations finite dimensional, even the energymomentum tensor is degenerate. These are either three, four, five or ten. Some examples are constructed satisfying the constraints on the energymomentum tensor.

 DYNAMICAL SYSTEMS


Geodesic flow on (super) Bott–Virasoro group and Harry Dym family
View Description Hide DescriptionWe show various Harry Dym type equations and Super Harry Dym equations, introduced by Brunelli, Das, and Popowicz [2003 Supersymmetric extensions of the Harry Dym hierarchy, J. Math. Phys. 44, 4756–4767 (2003)], follow from the geodesic flows on the Bott–Virasoro group and its supersymmetric generalization, superconformal group. In fact, their biHamiltonian structures can be derived from the Lie Poisson structures on the (super) Bott–Virasoro orbit. We also show that that or Calogero–Degasperis equation is also connected to the Bott–Virasoro group.

 STATISTICAL PHYSICS


Superfluidity of a Fermi liquid from the viewpoint of a hierarchy of equations for reduced density matrices
View Description Hide DescriptionThe hierarchy of equations for reduced density matrices relevant to thermodynamic equilibrium with account taken of the spin obtained earlier is modified in order to describe the state of a Fermi system with a condensate. Although the procedure is to some extent analogous with the one carried out by the author earlier for a Bose liquid peculiarities relevant to Fermi statistics complicate considerably the treatment. As in the case of the Bose liquid the condensate phase can be superfluid as well as nonsuperfluid, the physical causes of superfluidity being identical. A new mechanism of fermion pairing that acts even in the case of a purely repulsive Hamiltonian is pointed out. Special attention is given to the thermodynamics of a superfluid Fermi system. The example of a hardsphere system is used to find out the form of phase diagrams, the character of the phase transition to a condensate phase and the properties of the last. Noticeable dissimilarities from a Bose system with the same Hamiltonian are revealed. Application of the present approach to superconductivity is discussed as well.

Existence of Gibbs state for continuous gas with manybody interaction
View Description Hide DescriptionA continuous infinite system of point particles interacting via finiterange manybody potentials of superstable type is considered in the framework of classical statistical mechanics. We prove that for any temperature and chemical activity there exists at least one Gibbs state.

Equilibrium states for the Bose gas
View Description Hide DescriptionThe generating functional of the cyclic representation of the canonical commutation relations (CCR) representation for the thermodynamic limit of the grand canonical ensemble of the free Bose gas with attractive boundary conditions is rigorously computed. We use it to study the condensate localization as a function of the homothety point for the thermodynamic limit using a sequence of growing convex containers. The Kac function is explicitly obtained proving nonequivalence of ensembles in the condensate region in spite of the condensate density being zero locally.
