Volume 37, Issue 4, April 1996
Index of content:

Deformations of density functions in molecular quantum chemistry
View Description Hide DescriptionWe generalize the use of the local scaling transformation developed by E. S. Kryachko and E. V. Ludeña to molecules in order to deform density functions. The connection with the Jacobian problem is clearly made, and we solve that problem using a formalism introduced by J. Moser. As a consequence, we can control the density information contained in a wave function, in some sense, at the same time as we keep particular regularity and behavior assumptions in the wave function (in particular concerning the symmetries of the wave function). The principal aim of the paper is to develop a correct mathematical background for further utilization in connection with density functional theory.Theoretical implications and numerical aspects are also discussed.

Localization in single Landau bands
View Description Hide DescriptionWe consider a single‐band approximation to the random Schrödinger operator in an external magnetic field. The random potential is taken to be constant on unit squares and i.i.d. on each square with a bounded distribution. We prove that the eigenstates corresponding to energies at the edges of the Landau band are localized. This is an important ingredient in the theory of the Quantum Hall Effect.

Spontaneous symmetry breaking in the SO(3) gauge theory to discrete subgroups
View Description Hide DescriptionIn this paper we give a systematical description of the possible symmetry breakings in the SO(3)‐gauge theory and show an algorithmical method to construct SU(2)‐ or SO(3)‐invariant Higgs potentials in an arbitrary irreducible representation using regular graphs. We close our paper with the explicit construction of the Lagrangian of the simplest SO(3)→A _{4} theory.

A q‐deformation of the Coulomb problem
View Description Hide DescriptionThe algebra of observables of SO_{ q }(3)‐symmetric quantum mechanics is extended to include the inverse 1/R of the radial coordinate and used to obtain eigenvalues and eigenfunctions of a q‐deformed Coulomb Hamiltonian.

Reduced SL(2,R) WZNW quantum mechanics
View Description Hide DescriptionThe SL(2,R) WZNW→Liouville reduction leads to a nontrivial phase space on the classical level both in 0+1 and 1+1 dimensions. To study the consequences in the quantum theory, the quantum mechanics of the 0+1‐dimensional, point particle version of the constrained WZNW model is investigated. The spectrum and the eigenfunctions of the obtained (rather nontrivial) theory are given, and the physical connection between the pieces of the reduced configuration space is discussed in all the possible cases of the constraint parameters.

Commutator expansion. II. Relativistic reduced Green’s functions and the Lamb shift calculation
View Description Hide DescriptionThis is a continuation of an earlier paper [J. Math. Phys. 34, 5509 (1993)]. A projection operator technique is introduced in the Lamb shift calculation in order to manage spurious infrared divergences. The method provides an efficient tool to exhibit factors of the field in the commutator expansion of the Lamb shift. Mass eigenfunction expansion concepts developed elsewhere provide the setting for the new technique. As a by‐product of our method, relativistic reduced Green’s functions are found to appear in a natural way. The method may prove useful, since reduced Green’s functions are simpler than the full Green’s functions, and behave like constant operators as regards the parameter integrals.

Dually charged mesoatom on the space of constant negative curvature
View Description Hide DescriptionThe discrete spectrum solutions corresponding to dually charged mesoatom on the space of constant negative curvature are obtained. The discrete spectrum of energies is finite and vanishes when the magnetic charge of the nucleus exceeds the critical value.

Recursively minimally‐deformed oscillators
View Description Hide DescriptionA recursive deformation of the boson commutation relation is introduced. Each step consists of a minimal deformation of a commutator [a,a ^{°}]=f _{ k }(... ;n̂) into [a,a ^{°}]_{ q } _{ k+1}=f _{ k }(... ;n̂), where ... stands for the set of deformation parameters that f _{ k } depends on, followed by a transformation into the commutator [a,a ^{°}]=f _{ k+1}(... ,q _{ k+1};n̂) to which the deformed commutator is equivalent within the Fock space. Starting from the harmonic oscillator commutation relation [a,a ^{°}]=1 we obtain the Arik–Coon and Macfarlane–Biedenharn oscillators at the first and second steps, respectively, followed by a sequence of multiparameter generalizations. Several other types of deformed commutation relations related to the treatment of integrable models and to parastatistics are also obtained. The ‘‘generic’’ form consists of a linear combination of exponentials of the number operator, and the various recursive families can be classified according to the number of free linear parameters involved, that depends on the form of the initial commutator.

Quantization of a loop extended SU(2) affine Kac–Moody algebra
View Description Hide DescriptionA quantization of a Lie bialgebra structure on a loop extended SU■(2) algebra is considered. An asymptotic expansion for R‐matrix is given.

Quantum scattering theory in light of an exactly solvable model with rearrangement collisions
View Description Hide DescriptionWe present an exactly solvable quantum field theory which allows rearrangement collisions. We solve the model in the relevant sectors and demonstrate the orthonormality and completeness of the solutions, and construct the S‐matrix. In light of the exact solutions constructed, we discuss various issues and assumptions in quantum scattering theory, including the isometry of the Möller wave matrix, the normalization and completeness of asymptotic states, and the nonorthogonality of basis states. We show that these common assertions are not obtained in this model. We suggest a general formalism for scattering theory which overcomes these and other shortcomings and limitations of the existing formalisms in the literature.

Remarks on the quantization of gauge theories
View Description Hide DescriptionThe methods of reduced phase space quantization and Dirac quantization are examined in a simple gauge theory. It is pointed out that care needs to be exercised in implementing the reduced phase space quantization method properly.

Directly interacting massless particles—A twistor approach
View Description Hide DescriptionTwistor phase spaces are used to provide a general description of the dynamics of a finite number of directly interacting massless spinning particles, forming a closed relativistic massive and spinning system with an internal structure. A Poincaré invariant canonical quantization of the so obtained twistor phase space dynamics is performed.

A solvable N‐body problem in the plane. I
View Description Hide DescriptionWe introduce and discuss an n‐body problem in the plane, characterized by equations of motion of Newtonian type, r⃗̈ _{ j }=∑_{ k=1} ^{ n } F⃗ _{ jk }, j=1,..,n, with given ‘‘forces’’ F⃗ _{ jk } having the following characteristics: F⃗ _{ jk } depends only on r⃗ _{ j },r⃗ _{ k },r⃗̇ _{ j },r⃗̇ _{ k } (i.e., only ‘‘one‐body’’ and ‘‘two‐body’’ forces are present); F⃗ _{ jk } behaves as a (two‐dimensional) vector under rotations in the plane (i.e., the model is ‘‘rotation‐invariant’’); for j=k,F⃗ _{ jk } is linear in r⃗ _{ j } and r⃗̇ _{ j }; for j≠k, F⃗ _{ jk }=r⃗ _{ j }−r⃗ _{ k }^{−2} f⃗ _{ jk } with f⃗ _{ jk } a homogeneous polynomial of third degree in r⃗ _{ j },r⃗ _{ k },r⃗̇ _{ j },r⃗̇ _{ k } (hence F⃗ _{ jk } is homogeneous of degree one in r⃗ _{ j },r⃗ _{ k },r⃗̇ _{ j },r⃗̇ _{ k }); F⃗ _{ jk } contains linearly 8 arbitrary (‘‘coupling’’) constants. The n‐body problem is solvable for arbitrary n and for arbitrary values of the 8 coupling constants; its solutions display a rich phenomenology. If the 8 coupling constants are suitably restricted, the model is translation‐invariant, and/or Hamiltonian; of course, when it is Hamiltonian, it is integrable; indeed in some case a Hamiltonian function can be explicitly displayed, as well as the corresponding Lax pair.

On Abelianization of first class constraints
View Description Hide DescriptionA systematic method for the conversion of first class constraints to an equivalent set of the Abelian constraints based on the Dirac equivalence transformation is developed. A representation for the corresponding matrix of this transformation is proposed. This representation allows one to reduce the problem of Abelianization to the solution of a certain system of first order linear differential equations for matrix elements.

The complete Kepler group can be derived by Lie group analysis
View Description Hide DescriptionIt is shown that the complete symmetry group for the Kepler problem, as introduced by Krause, can be derived by Lie group analysis. The same result is true for any autonomous system.

Standard thermal statistics with q‐entropies
View Description Hide DescriptionWe report results on the quantum thermal statistics à la Gibbs–Shannon–Szilard–Jaynes based on q‐entropies S _{ q }[ρ]=(q−1)^{−1} (1−tr(ρ^{ q })) (0<q≠1) and the internal energy functional U[ρ]=tr(ρH) proposed by C. Tsallis [J. Stat. Phys. 52, 479–487 (1988)].

Non‐standard thermal statistics with q‐entropies
View Description Hide DescriptionWe consider the quantum thermal statistics à la Gibbs–Shannon–Szilard–Jaynes based on q‐entropies S _{ q }[ρ]=(q−1)^{−1} (1−tr(ρ^{ q })) (0<q≠1) and the non‐standard ‘‘internal energy’’ functionals U _{ q }[ρ]=tr(ρ^{ qH }) proposed by C. Tsallis [J. Stat. Phys. 52, 479–487 (1988)].

Covariant quantum Markovian evolutions
View Description Hide DescriptionQuantum Markovian master equations with generally unbounded generators, having physically relevant symmetries, such as Weyl, Galilean or boost covariance, are characterized. It is proven in particular that a fully Galilean covariant zero spin Markovian evolution reduces to the free motion perturbed by a covariant stochastic process with independent stationary increments in the classical phase space. A general form of the boost covariant Markovian master equation is discussed and a formal dilation to the Langevin equation driven by quantum Boson noises is described.

On realizations of solutions of the KdV equation by determinants on operator ideals
View Description Hide DescriptionUsing new developments in the theory of traces, determinants, and elementary operators on quasi‐Banach operator ideals we clarify and extend Marchenko’s method for realizing solutions of the KdV equation. Moreover, we point out why abstract traces and determinants on quasi‐Banach operator ideals are appropriate tools for obtaining solutions of the KdV equation. The method we present can also be applied to other nonlinear equations in soliton physics.

Singular and unstable solutions of the Korteweg–de Vries hierarchy
View Description Hide DescriptionThe Korteweg–de Vries (KdV) equation is solved using an inverse scattering transform approach using the continuous part of the spectrum of the Schrödinger equation. It is assumed that the reflection coefficient that corresponds to the initial condition of the KdV equation is a rational function of the wave number. It is shown that the Lyapunov exponent of the associated nonlinear evolution equation can be larger than zero if the reflection coefficient has poles that are close to zero. A positive Lyapunov exponent suggests that corresponding solutions of the KdV equation are unstable. This approach is generalized to the KdV hierarchy.