Volume 44, Issue 1, January 2003
 QUANTUM MECHANICS (GENERAL AND NONRELATIVISTIC)


On the spectral and wave propagation properties of the surface Maryland model
View Description Hide DescriptionWe study the discrete Schrödinger operator in with the surface potential of the form where for we write ω∈[0,1). We first consider the case where the components of the vector α are rationally independent, i.e., the case of the quasiperiodic potential. We prove that the spectrum of on the interval (coinciding with the spectrum of the discrete Laplacian) is absolutely continuous. Then we show that generalized eigenfunctions, corresponding to this interval, have the form of volume (bulk) waves, which are oscillating and nondecreasing (or slow decreasing) in all variables. They are the sum of the incident plane wave and of an infinite number of reflected or transmitted plane waves, scattered by the subspace These eigenfunctions are orthogonal, complete and verify a natural analog of the Lippmann–Schwinger equation. We discuss also the case where and is a rational number, i.e., a periodic surface potential. In this case we show that the spectrum is absolutely continuous and besides the volume (Bloch) waves there are also the surface waves, whose amplitude decays exponentially as The part of the spectrum corresponding to the surface waves consists of a finite number of bands. For large the bands outside of are exponentially small in and converge in a natural sense to the pure point spectrum that was found [B. Khoruzhenko and L. Pastur, Phys. Rep. 288, 109–125 (1997)] in the case of the Diophantine α’s.

Asymptotics of information entropies of some Todalike potentials
View Description Hide DescriptionThe spreading of the quantum probability density for the highlyexcited states of a singleparticle system with an exponentialtype potential on the positive semiaxis is quantitatively determined in both position and momentum spaces by means of the Boltzmann–Shannon informationentropy. This problem boils down to the calculation of the asymptotics of the entropylike integrals of the modified Bessel function of the second kind (also called the Mcdonald function or Basset function). The dependence of the two physical entropies on the large quantum number is given in detail. It is shown that the semiclassical (WKB) position–space entropy grows slower than the corresponding quantity of not only the harmonic oscillator but also the singleparticle systems with any powertype potential of the form and The momentum–space entropy, calculated with a method based on the properties of the Mcdonald function, is rigorously found to have a behavior of the form in strong contrast with the corresponding quantity of other onedimensional systems known up to now (powertype potentials, infinite well).

Spin coherentstate path integrals and the instanton calculus
View Description Hide DescriptionWe use an instanton approximation to the continuoustime spin coherentstate path integral to obtain the tunnel splitting of classically degenerate ground states. We show that provided the fluctuation determinant is carefully evaluated, the path integral expression is accurate to order We apply the method to the LMG model and to the molecular magnet in a transverse field.

Central limit theorems for large graphs: Method of quantum decomposition
View Description Hide DescriptionA new method is proposed for investigating spectral distribution of the combinatorial Laplacian (adjacency matrix) of a large regular graph on the basis of quantum decomposition and quantum central limit theorem. General results are proved for Cayley graphs of discrete groups and for distanceregular graphs. The Coxeter groups and the Johnson graphs are discussed in detail by way of illustration. In particular, the limit distributions obtained from the Johnson graphs are characterized by the Meixner polynomials which form a oneparameter deformation of the Laguerre polynomials

The essential spectrum of Schrödinger operators with asymptotically constant magnetic fields on the Poincaré upperhalf plane
View Description Hide DescriptionWe study the essential spectrum of the magnetic Schrödinger operators on the Poincaré upperhalf plane and establish a hyperbolic analog of Iwatsuka’s result [J. Math. Kyoto Univ. 23(3), 475–480 (1983)] on the stability of the essential spectrum under perturbations from constant magnetic fields.
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 RELATIVISTIC QUANTUM MECHANICS, FIELD THEORY, BRANE THEORY (INCLUDING STRINGS)


Noncommutative geometry of angular momentum space su(2))
View Description Hide DescriptionWe study the standard angular momentumalgebra as a noncommutative manifold We show that there is a natural 4D differential calculus and obtain its cohomology and Hodge ^{*} operator. We solve the spin 0 wave equation and some aspects of the Maxwell or electromagnetictheory including solutions for a uniform electric current density, and we find a natural Dirac operator We embed inside a 4D noncommutative space–time which is the limit of Minkowski space and show that has a natural quantum isometry group given by the quantum double which is a singular limit of the Lorentz group. We view as a collection of all fuzzy spheres taken together. We also analyze the semiclassical limit via minimum uncertainty states approximating classical positions in polar coordinates.

Chern–Simons field theories with nonsemisimple gauge group of symmetry
View Description Hide DescriptionThe subject of this work is a class of Chern–Simons field theories with nonsemisimple gauge group, which may well be considered as the most straightforward generalization of an Abelian Chern–Simons field theory. As a matter of fact, these theories, which are characterized by a nonsemisimple group of gauge symmetry, have cubic interactions like those of nonAbelian Chern–Simons field theories, but are free from radiative corrections. Moreover, at the tree level in the perturbative expansion, there are only two connected tree diagrams, corresponding to the propagator and to the three vertex originating from the cubic interaction terms. For such theories it is derived here a set of BRST invariant observables, which lead to metric independent amplitudes. The vacuum expectation values of these observables can be computed exactly. From their expressions it is possible to isolate the Gauss linking number and an invariant of the Milnor type, which describes the topological relations among three or more closed curves.

Path integrals evaluation in twodimensional de Sitter space
View Description Hide DescriptionThe propagator in de Sitter space is calculated based on the path integrals. The method of evaluation of path integrals for particles with spin is proposed. The calculations are compared with the quantum mechanical ones.

Towards vacuum superstring field theory: The supersliver
View Description Hide DescriptionWe extend some aspects of vacuum string field theory to superstring field theory in Berkovits’ formulation, and we study the star algebra in the fermionic matter sector. After clarifying the structure of the interaction vertex in the operator formalism of Gross and Jevicki, we provide an algebraic construction of the supersliver state in terms of infinite–dimensional matrices. This state is an idempotent string field and solves the matter part of the equation of motion of superstring field theory with a pure ghost BRST operator. We determine the spectrum of eigenvalues and eigenvectors of the infinite–dimensional matrices of Neumann coefficients in the fermionic matter sector. We then analyze coherent states based on the supersliver and use them in order to construct higher–rank projector solutions, as well as to construct closed subalgebras of the star algebra in the fermionic matter sector. Finally, we show that the geometric supersliver is a solution to the superstring field theoryequations of motion, including the (super)ghost sector, with the canonical choice of vacuum BRST operator recently proposed by Gaiotto, Rastelli, Sen and Zwiebach.

An exact fluid model for relativistic electron beams
View Description Hide DescriptionAn interesting and satisfactory fluid model has been proposed in the literature for the description of relativistic electron beams. It is obtained by imposing the entropy principle up to a certain order with respect to a smallness parameter ε measuring the dispersion of the velocity about the mean. Here the general exact solution is found, satisfying the entropy principle and the relativity principle up to whatever order.

On two dimensional coupled bosons and fermions
View Description Hide DescriptionWe study complex bosons and fermions coupled through a generalized Yukawa type coupling in the large limit following ideas of Rajeev [Int. J. Mod. Phys. A 9, 5583 (1994)]. We study a linear approximation to this model. We show that in this approximation we do not have boson–antiboson and fermion–antifermion bound states occuring together. There is a possibility of having only fermion–antifermion bound states. We support this claim by finding distributional solutions with energies lower than the two mass treshold in the fermion sector. This has implications from the point of view of scattering theory to this model. We discuss some aspects of the scattering above the two mass treshold of boson pairs and fermion pairs. We also briefly present a gauged version of the same model and write down the linearized equations of motion.
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 DYNAMICAL SYSTEMS


Magnetoencephalography in ellipsoidal geometry
View Description Hide DescriptionAn exact analytic solution for the forward problem in the theory of biomagnetics of the human brain is known only for the (1D) case of a sphere and the (2D) case of a spheroid, where the excitation field is due to an electric dipole within the corresponding homogeneous conductor. In the present work the corresponding problem for the more realistic ellipsoidal brain model is solved and the leading quadrupole approximation for the exterior magnetic field is obtained in a form that exhibits the anisotropic character of the ellipsoidal geometry. The results are obtained in a straightforward manner through the evaluation of the interior electric potential and a subsequent calculation of the surface integral over the ellipsoid, using Lamé functions and ellipsoidal harmonics. The basic formulas are expressed in terms of the standard elliptic integrals that enter the expressions for the exterior Lamé functions. The laborious task of reducing the results to the spherical geometry is also included.

The investigation into new integrable systems of equations in dimensions
View Description Hide DescriptionA new integrable class of systems of nonlinear partial differential equations (NPDEs) in dimensions is derived from the matrix Nizhnik–Novikov–Veselov (NVV) equation by means of an asymptotically exact nonlinear reduction method based on Fourier expansion and spatiotemporal rescaling. The integrability by the inverse scattering method is explicitly demonstrated, by applying the reduction technique also to the Lax pair of the starting matrix equation and thereby obtaining the Lax pair for the new class of systems of equations. A reduction to a system of two interacting complex fields is briefly described.

Deformations of the Monge/Riemann hierarchy and approximately integrable systems
View Description Hide DescriptionDispersive deformations of the Monge equation are studied using ideas originating from topological quantum field theory and the deformation quantization program. It is shown that, to a high order, the symmetries of the Monge equation may also be appropriately deformed, and that, if they exist at all orders, they are uniquely determined by the original deformation. This leads to either a new class of integrable systems or to a rigorous notion of an approximate integrable system. QuasiMiura transformations are also constructed for such deformed equations.
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 CLASSICAL MECHANICS AND CLASSICAL FIELDS


Constructing solutions of Hamilton–Jacobi equations for 2D fields with one component by means of Bäcklund transformations
View Description Hide DescriptionThe Hamilton–Jacobi formalism generalized to twodimensional field theories according to Lepage’s canonical framework is applied to several relativistic real scalar fields, e.g., massless and massive Klein–Gordon, Sinh and Sine–Gordon, Liouville and theories. The relations between the Euler–Lagrange and the Hamilton–Jacobi equations are discussed in DeDonder and Weyl’s and the corresponding wave fronts are calculated in Carathéodory’s formulation. Unlike mechanics one has to impose certain integrability conditions on the velocity fields to guarantee the transversality relations and especially the dynamical equivalence between Hamilton–Jacobi wave fronts and fields of extremals embedded therein. Bäcklund transformations play a crucial role in solving the resulting system of coupled nonlinear PDEs.
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 METHODS OF MATHEMATICAL PHYSICS


matrix presentation for superYangians
View Description Hide DescriptionWe give a RTT presentation of superYangians for thereby unifying the formalism with the cases of and

On powers of Bessel functions
View Description Hide DescriptionA formula for the Taylor series expansion of the power of the modified Bessel function is derived for arbitrary The result is expressed in terms of a recursive formula for a class of polynomials, which facilitates the systematic construction of the expansion of

A Poincaré–Birkhoff–Witt commutator lemma for
View Description Hide DescriptionWe present and prove in detail a Poincaré–Birkhoff–Witt commutator lemma for the quantum superalgebra

On certain geometric aspects of harmonic maps
View Description Hide DescriptionA Weierstrasstype system of equations corresponding to harmonic maps is presented. It constitutes a generalization of the previously constructed systems for and fields. From the linear spectral problem for the model a set of conserved quantities is derived and used for a construction of a generalized Weierstrass representation for conformally parametrized surfaces immersed in multidimensional Euclidean spaces. Based on this representation a possible geometrical interpretation of harmonic maps is discussed.

The extended Lotka–Volterra lattice and affine Jacobi varieties of spectral curves
View Description Hide DescriptionBased on the work by Smirnov and Zeitlin, we study a simple realization of the matrix construction of the affine Jacobi varieties. We find that the realization is given by a classical integrable model, the extended Lotka–Volterra lattice. We investigate the integrable structure of the representative for the gauge equivalence class of matrices, which is isomorphic to the affine Jacobi variety, and make use of it to discuss the solvability of the model.
