Volume 44, Issue 10, October 2003
 QUANTUM MECHANICS (GENERAL AND NONRELATIVISTIC)


Pauli approximations to the selfadjoint extensions of the Aharonov–Bohm Hamiltonian
View Description Hide DescriptionIt is well known that the formal Aharonov–Bohm Hamiltonian operator, describing the interaction of a charged particle with a magnetic vortex, has a fourparameter family of selfadjoint extensions, which reduces to a twoparameter family if one requires that the Hamiltonian commutes with the angular momentum operator. The question we study here is which of these selfadjoint extensions can considered as limits of regularized Aharonov–Bohm Hamiltonians, that is Pauli Hamiltonians in which the magnetic field corresponds to a flux tube of nonzero diameter. We show that not all the selfadjoint extensions in this twoparameter family can be obtained by these approximations, but only two oneparameter subfamilies. In these two cases we can choose the gyromagnetic ratio in the approximating Pauli Hamiltonian in such a way that we get convergence in the norm resolvent sense to the corresponding selfadjoint extension.

Stratified reduction of manybody kinetic energy operators
View Description Hide DescriptionThe centerofmass system of many bodies admits a natural action of the rotation group SO(3). According to the orbit types for the SO(3) action, the centerofmass system is stratified into three types of strata. The principal stratum consists of nonsingular configurations for which the isotropy subgroup is trivial, and the other two types of strata consist of singular configurations for which the isotropy subgroup is isomorphic with either SO(2) or SO(3). Depending on whether the isotropy subgroup is isomorphic with SO(2) or SO(3), the stratum in question consists of collinear configurations or of a single configuration of the multiple collision. It is shown that the kinetic energy operator is expressed as the sum of rotational and vibrational energy operators on each stratum except for the stratum of multiple collision. The energy operator for nonsingular configurations has singularity at singular configurations. However, the singularity is not essential in the sense that both of the rotational and vibrational energy integrals have a finite value. This can be proved by using the boundary conditions of wave functions at singular configurations for threebody systems, for simplicity. It is shown, in addition, that the energy operator for collinear configurations has also singularity at the multiple collision, but the singularity is not essential either in the sense that the kinetic energy integral is not divergent at the multiple collision. Reduction procedure is applied to the respective energy operators for the nonsingular and the collinear configurations to obtain respective reduced operators, both of which are expressed in terms of internal coordinates.

Dense Dirac combs in Euclidean space with pure point diffraction
View Description Hide DescriptionRegular model sets, describing the point positions of ideal quasicrystallographic tilings, are mathematical models of quasicrystals. An important result in mathematical diffraction theory of regular model sets, which are defined on locally compact Abelian groups, is the pure pointedness of the diffraction spectrum. We derive an extension of this result, valid for dense point sets in Euclidean space, which is motivated by the study of quasicrystallographic random tilings.

On the pseudoHermitian nondiagonalizable Hamiltonians
View Description Hide DescriptionWe consider a class of (possibly nondiagonalizable) pseudoHermitian operators with discrete spectrum, and we establish for such a class the equivalence between the pseudoHermiticity property and the existence of an antilinear involutory symmetry. Moreover, we prove that this class actually coincides with the one of (possibly nondiagonalizable) weak pseudoHermitian operators, and that in no case (unless they are diagonalizable and have a real spectrum) they are Hermitian with respect to a definite inner product. Finally, we show that a typical degeneracy of the real eigenvalues (which reduces to the wellknown Kramers degeneracy in the Hermitian case) occurs whenever a fermionic (possibly nondiagonalizable) pseudoHermitian Hamiltonian admits an antilinear symmetry like the timereversal operator Some consequences and applications are briefly discussed.

Potential and field singularity at a surface point charge
View Description Hide DescriptionThe behavior of the magnetic potential near a point charge (fluxon) located at a curved regular boundary surface is shown to be essentially different from that of a volume point charge. In addition to the usual inverse distance singularity, two singular terms are generally present. The first of them, a logarithmic one, is axially symmetric with respect to the boundary normal at the charge location, and proportional to the sum of the two principal curvatures of the boundary surface at this point, that is, to the local mean curvature. The second term is asymmetric and proportional to the difference of the two principal curvatures in question; it is also bounded at the charge location. Both terms vanish, apparently, if the charge is at a planar point of the boundary, and only in this case. The field in the charge vicinity behaves accordingly, featuring generally two singular terms proportional to the inverse distance, in addition to the main inverse distance squared singularity. This result is significant, in particular, for studying the interaction of magnetic vortices in type II superconductors.
 Top

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


Group theory approach to the Dirac equation with a Coulomb plus scalar potential in dimensions
View Description Hide DescriptionWe generalize the Dirac equation to space–time. The conservedangular momentum operators and their quantum numbers are discussed. The eigenfunctions of the total angular momentums are calculated for both odd and even cases. The exact solutions of the dimensional radial equations of the Dirac equation with a Coulomb plus scalar potential are analytically presented by studying the Tricomi equations obtained from a pair of coupled firstorder ones. The eigenvalues are also discussed in some detail.

A quantum weak energy inequality for spinone fields in curved space–time
View Description Hide DescriptionQuantum weak energy inequalities (QWEI) provide stateindependent lower bounds on averages of the renormalized energy density of a quantum field. We derive QWEIs for the electromagnetic and massive spinone fields in globally hyperbolic space–times whose Cauchy surfaces are compact and have trivial first homology group. These inequalities provide lower bounds on weighted averages of the renormalized energy density as “measured” along an arbitrary timelike trajectory, and are valid for arbitrary Hadamard states of the spinone fields. The QWEI bound takes a particularly simple form for averaging along static trajectories in ultrastatic space–times; as specific examples we consider Minkowski space (in which case the topological restrictions may be dispensed with) and the static Einstein universe. A significant part of the paper is devoted to the definition and properties of Hadamard states of spinone fields in curved space–times, particularly with regard to their microlocal behavior.

A direct derivation of polynomial invariants from perturbative Chern–Simons gauge theory
View Description Hide DescriptionThere have been several methods to show that the expectation values of Wilson loop operators in the Chern–Simons gauge theory satisfy the HOMFLY skein relation. We shall give another method from the perturbative method of the Chern–Simons gauge theory in the lightcone gauge, which is more direct than already known methods.

Seiberg–Witten monopole equations on noncommutative
View Description Hide DescriptionIt is well known that, due to vanishing theorems, there are no nontrivial finite action solutions to the Abelian Seiberg–Witten (SW) monopole equations on Euclidean fourdimensional space We show that this is no longer true for the noncommutative version of these equations, i.e., on a noncommutative deformation of there exist smooth solutions to the SW equations having nonzero topological charge. We introduce action functionals for the noncommutative SW equations and construct explicit regular solutions. All our solutions have finite energy. We also suggest a possible interpretation of the obtained solutions as codimension four vortexlike solitons representing  and branes in a brane system in type II superstring theory.

Semiclassical asymptotics for the Maxwell–Dirac system
View Description Hide DescriptionWe study the coupled system of Maxwell and Dirac equations from a semiclassical point of view. A rigorous nonlinear WKBanalysis, locally in time, for solutions of (critical) order is performed, where the small semiclassical parameter denotes the microscopic/macroscopic scale ratio.
 Top

 DYNAMICAL SYSTEMS


Algebro–geometric constructions of the discrete Ablowitz–Ladik flows and applications
View Description Hide DescriptionResorting to the finiteorder expansion of the Lax matrix, the elliptic coordinates are introduced, from which the discrete Ablowitz–Ladik equations and the dimensional Toda lattice are decomposed into solvable ordinary differential equations. The straightening out of the continuous flow and the discrete flow is exactly given through the Abel–Jacobi coordinates. As an application, explicit quasiperiodic solutions for the dimensional Toda lattice are obtained.

Unifying scheme for generating discrete integrable systems including inhomogeneous and hybrid models
View Description Hide DescriptionA unifying scheme based on an ancestor model is proposed for generating a wide range of integrable discrete and continuum as well as inhomogeneous and hybrid models. They include in particular discrete versions of sineGordon, Landau–Lifshitz, nonlinear Schrödinger (NLS), derivative NLS equations, Liouville model, (non)relativistic Toda chain, Ablowitz–Ladik model, etc. Our scheme introduces the possibility of building a novel class of integrable hybrid systems including multicomponent models like massive Thirring, discrete selftrapping, twomode derivative NLS by combining different descendant models. We also construct inhomogeneous systems like Gaudin model including new ones like variable mass sineGordon, variable coefficient NLS, Ablowitz–Ladik, Toda chains, etc. keeping their flows isospectral, as opposed to the standard approach. All our models are generated from the same ancestor Lax operator (or its limit) and satisfy the classical Yang–Baxter equation sharing the same rmatrix. This reveals an inherent universality in these diverse systems, which become explicit at their actionangle level.

General soliton matrices in the Riemann–Hilbert problem for integrable nonlinear equations
View Description Hide DescriptionWe derive the soliton matrices corresponding to an arbitrary number of higherorder normal zeros for the matrix Riemann–Hilbert problem of arbitrary matrix dimension, thus giving the complete solution to the problem of higherorder solitons. Our soliton matrices explicitly give all higherorder multisoliton solutions to the nonlinear partial differential equations integrable through the matrix Riemann–Hilbert problem. We have applied these general results to the threewave interaction system, and derived new classes of higherorder soliton and twosoliton solutions, in complement to those from our previous publication [Stud. Appl. Math. 110, 297 (2003)], where only the elementary higherorder zeros were considered. The higherorder solitons corresponding to nonelementary zeros generically describe the simultaneous breakup of a pumping wave into the other two components and and merger of and waves into the pumping wave. The twosoliton solutions corresponding to two simple zeros generically describe the breakup of the pumping wave into the and components, and the reverse process. In the nongeneric cases, these twosoliton solutions could describe the elastic interaction of the and waves, thus reproducing previous results obtained by Zakharov and Manakov [Zh. Éksp. Teor. Fiz. 69, 1654 (1975)] and Kaup [Stud. Appl. Math. 55, 9 (1976)].

Isomonodromy deformations for the ZSAKNS system with quadratic spectral variables
View Description Hide DescriptionWe solve the monodromy problem and prove the Painleve property for selfsimilar ZSAKNS flows with a quadratic spectral variable in this report. In particular, we obtain meromorphic solutions for the Cauchy problem of the selfsimilar derivative nonlinear Schrödinger equation.
 Top

 CLASSICAL MECHANICS AND CLASSICAL FIELDS


MultiLagrangians, hereditary operators and Lax pairs for the Korteweg–de Vries positive and negative hierarchies
View Description Hide DescriptionWe present an approach to the construction of action principles (the inverse problem of the calculus of variations), for first order (in time derivatives) differential equations, and generalize it to field theory in order to construct systematically, for integrable equations which are based on the existence of a Nijenhuis (or hereditary) operator, a (multiLagrangian) ladder of action principles which is complementary to the wellknown multiHamiltonian formulation. We work out results for the Korteweg–de Vries (KdV) equation, which is a member of the positive hierarchy related to a hereditary operator. Three negative hierarchies of (negative) evolution equations are defined naturally from the hereditary operator as well, in a concise way, suitable for field theory. The Euler–Lagrange equations arising from the action principles are equivalent to deformations of the original evolution equation, and the deformations are obtained explicitly in terms of the positive and negative evolution vectors. We recognize, after appropriate coordinate transformations, the Liouville, Sinh–Gordon, Hunter–Zheng, and Camassa–Holm equations as negative evolution equations. The multiLagrangian ladder for KdV is directly mappable to a ladder for any of these negative equations and other positive evolution equations (e.g., the Harry–Dym and a special case of the Krichever–Novikov equations). For example, several nonequivalent, nonlocal timereparametrization invariant action principles for KdV are constructed, and a new nonlocal action principle for the deformed system Sinh–Gordon+spatial translation vector is presented. Local and nonlocal Hamiltonian operators are obtained in factorized form as the inverses of all the nonequivalent symplectic twoforms in the ladder. Alternative Lax pairs for all negative evolution vectors are constructed, using the negative vectors and the hereditary operator as only input. This result leads us to conclude that, basically, all positive and negative evolution equations in the hierarchies share the same infinitedimensional sets of local and nonlocal constants of the motion for KdV, which are explicitly obtained using symmetries and the local and nonlocal action principles for KdV.

Liénard–Wiechert potentials in even dimensions
View Description Hide DescriptionThe motion of point charged particles is considered in an even dimensional Minkowski space–time. The potential functions corresponding to the massless scalar and the Maxwell fields are derived algorithmically. It is shown that in all even dimensions particles lose energy due to acceleration.
 Top

 STATISTICAL PHYSICS


The Green–Kubo formula and power spectrum of reversible Markov processes
View Description Hide DescriptionAs is known, the entropy production rate of a stationary Markov process vanishes if and only if the process is reversible. In this paper, we discuss the reversibility of a stationary Markov process from a functional analysis point of view. It is shown that the process is reversible if and only if it has a symmetric Markov semigroup, equivalently, a selfadjoint infinitesimal generator. Applying this fact, we prove that the Green–Kubo formula holds for reversible Markov processes. By demonstrating that the power spectrum of each reversible Markov process is Lorentztyped, we show that it is impossible for stochastic resonance to occur in systems with zero entropy production.

Solvable models of Bose–Einstein condensates: A new algebraic Bethe ansatz scheme
View Description Hide DescriptionA new algebraic Bethe ansatz scheme is proposed to diagonalize classes of integrable models relevant to the description of Bose–Einstein condensation in dilute alkali gases. This is achieved by introducing the notion of Zgraded representations of the Yang–Baxter algebra.
 Top

 METHODS OF MATHEMATICAL PHYSICS


On the variational cohomology of ginvariant foliations
View Description Hide DescriptionLet be an integrable Pfaffian system. If it is invariant under a transversally free infinitesimal action of a finite dimensional real Lie algebra we show that the “vertical” variational cohomology of is equal to the Lie algebra cohomology of with values in the space of the “horizontal” cohomology in a maximum dimension. This result, besides giving an effective algorithm for the computation of the variational cohomology of an invariant Pfaffian system, provides a method for detecting obstructions to the existence of infinitesimal actions leaving a given system invariant.

Dirac operators on coset spaces
View Description Hide DescriptionThe Dirac operator for a manifold and its chirality operator when is even dimensional, have a central role in noncommutative geometry. We systematically develop the theory of this operator when where and are compact connected Lie groups and is simple. An elementary discussion of the differential geometric and bundle theoretic aspects of including its projective modules and complex, Kähler and Riemannian structures, is presented for this purpose. An attractive feature of our approach is that it transparently shows obstructions to spin and structures. When a manifold is and not spin, U(1) gauge fields have to be introduced in a particular way to define spinors, as shown by Avis, Isham, Cahen, and Gutt. Likewise, for manifolds like SU(3)/SO(3), which are not even we show that SU(2) and higher rank gauge fields have to be introduced to define spinors. This result has potential consequences for string theories if such manifolds occur as branes. The spectra and eigenstates of the Dirac operator on spheres invariant under are explicitly found. Aspects of our work overlap with the earlier research of Cahen et al.
