Continuity of state functions of the Bose gas

In this paper we show that some state functions of the Bose gas, especially the entropy, depend continuously on the energy levels for the free Hamiltonian and on perturbations of the free Hamiltonian by operators of degree 0. The method used here is to introduce an appropriate topology on the density matrices and on the perturbations of the free Hamiltonian.     

On the theories of the interacting perturbations of the Reissner-Nordström black hole

A complete account of the Hamiltonian approach to the coupled perturbations of the Reissner-Nordström black hole, initiated by Moncrief, is given. All Hamiltonian equations are expressed explicitly in suitable forms; the metric and electromagnetic field perturbations are found in terms of Moncrief's gauge invariant canonical variables in the Regge-Wheeler gauge. The basic (both tetrad and coordinate) gauge invariant scalars occurring in the perturbation studies based on the Newman-Penrose formalism are then related to Moncrief's variables. The strikingly simple relations obtained enable us to show that the fundamental pair of decoupled equations, derived recently within the Newman-Penrose formalism by Chandrasekhar, can be cast into gauge invariant form, and that it can be obtained from Moncrief's formalism.It is demonstrated how the fundamental equations, supplemented by another combination of the Newman — Penrose equations, generalize the Bardeen-Press equations for uncoupled electromagnetic and gravitational perturbations of the Schwarzschild black hole.The odd and the even parityl=1 perturbations are also considered in detail. In the Appendix the relations to Zerilli's work on coupled perturbations of the Reissner-Nordström black hole are given.     

Dimensionally hybrid Green's functions for impurity scattering in the presence of interfaces

A previous calculation of Green's function (E-H)^-1 for an interface Hamiltonian which interpolates between two and three dimensions is generalized to include scattering from perturbations both on and off the interface.     

Statistical Dynamics of Solitons in Optical Fibers

The soliton perturbation theory is used to study and analyze the stochastic perturbation of optical solitons in addition to deterministic perturbations of optical solitons that are governed by the nonlinear Schro¨dinger's equation. The Langevin equations are derived and analyzed. The deterministic perturbations that are considered here are of both Hamiltonian as well as of non-Hamiltonian type.     

Gravitational perturbations of spherically symmetric systems. II. Perfect fluid interiors

Methods developed in a previous paper on perturbations of the Schwarzschild metric are here extended to the treatment of perturbations of perfect fluid stellar models. The perturbations of a perfect fluid sphere are explicitly decomposed into their gauge invariant and gauge dependent parts and a variational principle for the perturbation equations is derived. The Hamiltonian for the perturbations is constructed and a sufficient condition for stability against nonradial, radiative perturbations is derived from it. The stability criterion is applied to two interesting classes of stellar models, polytropic white dwarf models and high-density neutron star cores with pressure proportional to energy density.     

Lie algebra structures for four-component Hamiltonian hydrodynamic type systems

Four-component Hamiltonian systems of hydrodynamic type induce, through the Haantjes tensor, a Lie algebra structure on tangent planes in the space of dependent variables. We show that this Lie algebra is either reductive or solvable with a nilpotent three-dimensional subalgebra. We demonstrate how the precise Lie algebraic structure is determined by the Hamiltonian structure of the system. An application to perturbations of the Benney system is presented.     

Optical solitons: Quasi-stationarity versus Lie transform

This paper is a comparitive study of the two approaches namely the quasi-stationarity and the Lie transform that are used for studying the classical optical solitons along with its perturbations governed by the non-linear Schrödinger's equation with Kerr law of non-linearity. The relative advantages and disadvantages of the two methods are enumerated. The study is conducted with both, Hamiltonian as well as non-Hamiltonian type perturbations.     

Gauge Transformations for a Driven Quantum Particle in an Infinite Square Well

Quantum mechanics of a particle in an infinite square well under the influence of a time-dependent electric field is reconsidered. In some gauge, the Hamiltonian depends linearly on the momentum operator, which is symmetric but not self-adjoint when defined on a finite interval. In spite of this symmetric part, the Hamiltonian operator is shown to be self-adjoint. This follows from a theorem by Kato and Rellich which guarantees the stability of a self-adjoint operator under certain symmetric perturbations. The result, which has been assumed tacitly by other authors, is important in order to establish the equivalence of different Hamiltonian operators related to each other by quantum gauge transformations.     

Perturbations Can Enhance Quantum Search

In general, a quantum algorithm wants to avoid decoherence or perturbation, since such factors may cause errors in the algorithm. We show that some perturbations to the generalized quantum search Hamiltonian can reduce the running time and enhance the success probability. We also provide the narrow bound to the perturbation which can be beneficial to quantum search. In addition, we show that the error induced by a perturbation on the Farhi and Gutmann Hamiltonian can be corrected by another perturbation.     

On perturbations of the periodic Toda lattice

A class of Hamiltonian systems including perturbations of the periodic Toda lattice and homogeneous cosmological models is studied. Separatrix approximation of oscillation regimes in these systems connected with Coxeter groups is obtained. Hamiltonian systems connected with simple Lie algebras are pointed out, which generalize the system describing periodic Toda lattice and allow theL -A pair representation.     

Perturbations of unitary representations of finite dimensional Lie groups

In quantum physical theories, interactions in a system of particles are commonly understood as perturbations to certain observables, including the Hamiltonian, of the corresponding interaction-free system. The manner in which observables undergo perturbations is subject to constraints imposed by the overall symmetries that the interacting system is expected to obey. Primary among these are the spacetime symmetries encoded by the unitary representations of the Galilei group and Poincaré group for the non-relativistic and relativistic systems, respectively. In this light, interactions can be more generally viewed as perturbations to unitary representations of connected Lie groups, including the non-compact groups of spacetime symmetry transformations. In this paper, we present a simple systematic procedure for introducing perturbations to (infinite dimensional) unitary representations of finite dimensional connected Lie groups. We discuss applications to relativistic and non-relativistic particle systems.     

Gravitational perturbations of spherically symmetric systems. I. The exterior problem

Gravitational perturbations of the Schwarzschild metric are treated from a point of view which is adapted, in a natural way, to the gauge group of the perturbed Einstein equations. The metric perturbations are explicitly decomposed into their gauge invariant, gauge dependent and constrained parts and a variational principle for the perturbation equations is derived. The Regge-Wheeler and Zerilli equations are rederived and shown to have a gauge invariant significance. The Hamiltonian for the perturbations is constructed and used to discuss the stability properties of the Schwarzschild black hole.     

Stochastic perturbation of Kerr law optical solitons

The soliton perturbation theory is used to study and analyze the stochastic perturbation of optical solitons, due to Kerr law nonlinearity, in addition to deterministic perturbations of optical solitons that is governed by the nonlinear Schrödingers equation. The Langevin equations are derived and analyzed. The deterministic perturbations that are considered here are of both Hamiltonian as well as of non-Hamiltonian type.     

Gravitational collapse with charge and small asymmetries. II. Interacting electromagnetic and gravitational perturbations

Paper I analyzed the evolution of nonspherical scalar-field perturbations of an electrically charged, collapsing star; this paper treats coupled electromagnetic and gravitational perturbations. It employs the results of recent detailed work in which coupled perturbations were studied in a gauge-invariant manner by using the Hamiltonian (Moncrief s) approach and the Newman-Penrose formalism, and the relations between the fundamental quantities of these two methods were obtained.It is shown that scalar-field perturbations are a prototype for coupled perturbations. The collapse produces a Reissner-Nordström black hole, and the perturbations are radiated away completely. Alll-pole parts of the perturbations of the metric and the electromagnetic field decay according to power laws; in the extreme case (e ² =M ²), the interaction causes the quadrupole perturbations to die out more slowly than the dipole perturbations.     

On the stability of topological phases on a lattice

We study the stability of anyonic models on lattices to perturbations. We establish a cluster expansion for the energy of the perturbed models and use it to study the stability of the models to local perturbations. We show that the spectral gap is stable when the model is defined on a sphere, so that there is no ground state degeneracy. We then consider the toric code Hamiltonian on a torus with a class of abelian perturbations and show that it is stable when the torus directions are taken to infinity simultaneously, and is unstable when the thin torus limit is taken.     

Dilation analyticity in constant electric field

We extend the analysis of Paper I from two body dilation analytic systems in constant electric field toN-body systems in constant electric field. Particular attention is paid to what happens to isolated eigenvalues of an atomic or molecular system in zero field when the field is turned on. We prove that the corresponding eigenvalue of the complex scaled Hamiltonian is stable and becomes a resonance. We study analyticity properties of the levels as a function of the field and also Borel summability.Research partially supported by USNSF grant MCS 78-00101Research partially supported by USNSF grant MCS 78-01885     

Sub-Doppler Infrared Spectra of the OH-Stretch Fundamental of (13)C-Methanol

High-resolution infrared spectra of the (13)C-methanol OH stretching band, nu(1), were obtained by slit-jet absorption spectroscopy at a rotational temperature of 12-15 K. Twenty-eight subbands were assigned within the range of upper state quantum numbers, K' = 0 to 2 and J' = 0 to 10. The upper state energy levels are heavily perturbed; about half of the assigned subbands were found to be split by perturbations with matrix elements in the range 1-3 cm(-1). The doubled lines were deperturbed and together with the "unperturbed" lines were fitted to a global torsion-rotation Hamiltonian with root-mean-square deviation of 0.41 cm(-1) to yield a torsional barrier height of approximately 405 cm(-1) in the vibrationally excited state. Copyright 2000 Academic Press.     

A mechanism of magnon-phonon interaction in KCoF3

A new mechanism is proposed for the magnon-phonon interaction in KCoF₃. The interaction Hamiltonian is derived by treating the vibration-modulated crystal field and the spin-orbit interaction as perturbations. The mechanism can well explain the experimental results of the inelastic neutron scattering and the Raman scattering.     

Renormalization invariance of a Hamiltonian system near the saddle point

The rescaling property of a one degree of freedom Hamiltonian system near the saddle point is analytically studied as regards the transformations of time-periodic perturbations. Two kinds of perturbation functions are considered: (a) linear functions and (b) homogeneous polynomial functions of canonical variables with time-periodic coefficients. The simple rescaling law of the phase-space of the Hamiltonian system near the hyperbolic fixed point with respect to a transformation of the amplitude and phase of the time-periodic perturbation is derived.     

Point perturbations in constant curvature spaces

Point perturbations of the free Hamiltonian in two- and three-dimensional spaces of constant curvatures are considered. The study of the spectral properties of perturbed Hamiltonian and various asymptotics for its point levels are presented. It is shown that the binding energy in comparison with the case of zero curvature reduces in the case of Lobachevsky plane and rises in the case of 2D-sphere, when the scattering length is much less than the curvature radius.