Narrow Results

We study the Floquet Hamiltonian -i∂_t + H + V(ωt), acting in L²([0,T],ℋ, dt), as depending on the parameter ω = 2π/T. We assume that the spectrum of H in ℋ is discrete, , but possibly degenerate, and that t ↦ V(t) ∈ ℬ(ℋ) is a 2π-periodic function with values in the space of Hermitian operators on ℋ. Let J > 0 and set . Suppose that for some σ > 0 it holds true that ∑_{hm > hn}M_mM_n (h_m - h_n)^-σ < ∞ where M_m is the multiplicity of h_m. We show that in that case there exist a suitable norm to measure the regularity of V, denoted ∊_V, and positive constants, ∊_⋆ and δ_⋆, with the property: if ∊_V < ∊_⋆ then there exists a measurable subset Ω_∞ ⊂ Ω₀ such that its Lebesgue measure fulfills |Ω_∞| ≥ |Ω₀| - δ_⋆ ∊_V and the Floquet Hamiltonian has a pure point spectrum for all ω∈Ω_∞.

We consider the class of Hamiltonians:

Given a W*-algebra with a W*-dynamics τ, we prove the existence of the perturbed W*-dynamics for a large class of unbounded perturbations. We compute its Liouvillean. If τ has a β-KMS state, and the perturbation satisfies some mild assumptions related to the Golden–Thompson inequality, we prove the existence of a β-KMS state for the perturbed W*-dynamics. These results extend the well known constructions due to Araki valid for bounded perturbations.

The 1/N expansion in quantum field theory is formulated within an algebraic framework. For a scalar field taking values in the N by N hermitian matrices, we rigorously construct the gauge invariant interacting quantum field operators in the sense of power series in 1/N and the 't Hooft coupling parameter as members of an abstract *-algebra. The key advantages of our algebraic formulation over the usual formulation of the 1/N expansion in terms of Green's functions are (i) that it is completely local so that infrared divergencies in massless theories are avoided on the algebraic level and (ii) that it admits a generalization to quantum field theories on globally hypberbolic Lorentzian curved spacetimes. We expect that our constructions are also applicable in models possessing local gauge invariance such as Yang–Mills theories.

The 1/N expansion of the renormalization group flow is constructed on the algebraic level via a family of *-isomorphisms between the algebras of interacting field observables corresponding to different scales. We also consider k-parameter deformations of the interacting field algebras that arise from reducing the symmetry group of the model to a diagonal subgroup with k factors. These parameters smoothly interpolate between situations of different symmetry.

In the framework of perturbative algebraic quantum field theory a local construction of interacting fields in terms of retarded products is performed, based on earlier work of Steinmann [42]. In our formalism the entries of the retarded products are local functionals of the off-shell classical fields, and we prove that the interacting fields depend only on the action and not on terms in the Lagrangian which are total derivatives, thus providing a proof of Stora's "Action Ward Identity" [45]. The theory depends on free parameters which flow under the renormalization group. This flow can be derived in our local framework independently of the infrared behavior, as was first established by Hollands and Wald [32]. We explicitly compute non-trivial examples for the renormalization of the interaction and the field.

We propose additional conditions (beyond those considered in our previous papers) that should be imposed on Wick products and time-ordered products of a free quantum scalar field in curved spacetime. These conditions arise from a simple "Principle of Perturbative Agreement": for interaction Lagrangians L₁ that are such that the interacting field theory can be constructed exactly — as occurs when L₁ is a "pure divergence" or when L₁ is at most quadratic in the field and contains no more than two derivatives — then time-ordered products must be defined so that the perturbative solution for interacting fields obtained from the Bogoliubov formula agrees with the exact solution. The conditions derived from this principle include a version of the Leibniz rule (or "action Ward identity") and a condition on time-ordered products that contain a factor of the free field φ or the free stress-energy tensor T_ab. The main results of our paper are (1) a proof that in spacetime dimensions greater than 2, our new conditions can be consistently imposed in addition to our previously considered conditions and (2) a proof that, if they are imposed, then for any polynomial interaction Lagrangian L₁ (with no restriction on the number of derivatives appearing in L₁), the stress-energy tensor Θ_ab of the interacting theory will be conserved. Our work thereby establishes (in the context of perturbation theory) the conservation of stress-energy for an arbitrary interacting scalar field in curved spacetimes of dimension greater than 2. Our approach requires us to view time-ordered products as maps taking classical field expressions into the quantum field algebra rather than as maps taking Wick polynomials of the quantum field into the quantum field algebra.

We study perturbations of a class of analytic two-dimensional autonomous systems with perturbations depending periodically on time; for instance one can imagine a periodically driven or forced system with one degree of freedom. In the first part of the paper, we revisit a problem considered by Chow and Hale on the existence of subharmonic solutions. In the analytic setting, under more general (weaker) conditions on the perturbation, we prove their results on the bifurcation curves dividing the region of non-existence from the region of existence of subharmonic solutions. In particular our results apply also when one has degeneracy to the first order — i.e. when the subharmonic Melnikov function is identically constant. Moreover we can deal as well with the case in which degeneracy persists to arbitrarily high orders, in the sense that suitable generalizations to higher orders of the subharmonic Melnikov function are also identically constant. The bifurcation curves consist of four branches joining continuously at the origin, where each of them can have a singularity (although generically they do not). The branches can form a cusp at the origin: we say in this case that the curves are degenerate as the corresponding tangent lines coincide. The method we use is completely different from that of Chow and Hale, and it is essentially based on the proof of convergence of the perturbation theory. It also allows us to treat the Melnikov theory in degenerate cases in which the subharmonic Melnikov function is either identically vanishing or has a zero which is not simple. This is investigated at length in the second part of the paper. When the subharmonic Melnikov function has a non-simple zero, we consider explicitly the case where there exist subharmonic solutions, which, although not analytic, still admit a convergent fractional series in the perturbation parameter.

We study a Hamiltonian describing a pendulum coupled with several anisochronous oscillators, giving a simple construction of unstable KAM tori and their stable and unstable manifolds for analytic perturbations.

We extend analytically the solutions of the equations of motion, order by order in the perturbation parameter, to a uniform neighborhood of the time axis.

Devices exhibiting the integer quantum Hall effect can be modeled by one-electron Schrödinger operators describing the planar motion of an electron in a perpendicular, constant magnetic field, and under the influence of an electrostatic potential. The electron motion is confined to unbounded subsets of the plane by confining potential barriers. The edges of the confining potential barrier create edge currents. In this, the first of two papers, we prove explicit lower bounds on the edge currents associated with one-edge, unbounded geometries formed by various confining potentials. This work extends some known results that we review. The edge currents are carried by states with energy localized between any two Landau levels. These one-edge geometries describe the electron confined to certain unbounded regions in the plane obtained by deforming half-plane regions. We prove that the currents are stable under various potential perturbations, provided the perturbations are suitably small relative to the magnetic field strength, including perturbations by random potentials. For these cases of one-edge geometries, the existence of, and the estimates on, the edge currents imply that the corresponding Hamiltonian has intervals of absolutely continuous spectrum. In the second paper of this series, we consider the edge currents associated with two-edge geometries describing bounded, cylinder-like regions, and unbounded, strip-like, regions.

Four point correlation functions for many electrons at finite temperature in periodic lattice of dimension d (≥1) are analyzed by the perturbation theory with respect to the coupling constant. The correlation functions are characterized as a limit of finite dimensional Grassmann integrals. A lower bound on the radius of convergence and an upper bound on the perturbation series are obtained by evaluating the Taylor expansion of logarithm of the finite dimensional Grassmann Gaussian integrals. The perturbation series up to second-order is numerically implemented along with the volume-independent upper bounds on the sum of the higher order terms in the 2-dimensional case.

A plasmon of a bounded domain Ω ⊂ ℝⁿ is a non-trivial bounded harmonic function on ℝⁿ\∂Ω which is continuous at ∂Ω and whose exterior and interior normal derivatives at ∂Ω have a constant ratio. We call this ratio a plasmonic eigenvalue of Ω. Plasmons arise in the description of electromagnetic waves hitting a metallic particle Ω. We investigate these eigenvalues and prove that they form a sequence of numbers converging to one. Also, we prove regularity of plasmons, derive a variational characterization, and prove a second-order perturbation formula. The problem can be reformulated in terms of Dirichlet–Neumann operators, and as a side result, we derive a formula for the shape derivative of these operators.

In this paper, we consider Anderson type operators on a separable Hilbert space where the random perturbations are finite rank and the random variables have full support on $ℝ$. We show that spectral multiplicity has a uniform lower bound whenever the lower bound is given on a set of positive Lebesgue measure on the point spectrum away from the continuous one. We also show a deep connection between the multiplicity of pure point spectrum and local spectral statistics, in particular, we show that spectral multiplicity higher than one always gives non-Poisson local statistics in the framework of Minami theory.

In particular, for higher rank Anderson models with pure point spectrum, with the randomness having support equal to $ℝ$, there is a uniform lower bound on spectral multiplicity and in case this is larger than one, the local statistics is not Poisson.

In this paper, we investigate the spectral and scattering theory for operators acting on topological crystals and on their perturbations. Special attention is paid to perturbations obtained by the addition of an infinite number of edges, and/or by the removal of a finite number of them, but perturbations of the underlying measures and perturbations by the addition of a multiplication operator are also considered. The description of the nature of the spectrum of the resulting operators and the existence and completeness of the wave operators are standard outcomes for these investigations.

Rotators interacting with a pendulum via small, velocity independent, potentials are considered. If the interaction potential does not depend on the pendulum position then the pendulum and the rotators are decoupled and we study the invariant tori of the rotators system at fixed rotation numbers: we exhibit cancellations, to all orders of perturbation theory, that allow proving the stability and analyticity of the diophantine tori. We find in this way a proof of the KAM theorem by direct bounds of the k-th order coefficient of the perturbation expansion of the parametric equations of the tori in terms of their average anomalies: this extends Siegel's approach, from the linearization of analytic maps to the KAM theory; the convergence radius does not depend, in this case, on the twist strength, which could even vanish ("twistless KAM tori"). The same ideas apply to the case in which the potential couples the pendulum and the rotators: in this case the invariant tori with diophantine rotation numbers are unstable and have stable and unstable manifolds ("whiskers"): instead of studying the perturbation theory of the invariant tori we look for the cancellations that must be present because the homoclinic intersections of the whiskers are "quasi flat", if the rotation velocity of the quasi periodic motion on the tori is large. We rederive in this way the result that, under suitable conditions, the homoclinic splitting is smaller than any power in the period of the forcing and find the exact asymptotics in the two dimensional cases (e.g. in the case of a periodically forced pendulum). The technique can be applied to study other quantities: we mention, as another example, the homoclinic scattering phase shifts.

An algorithm and a computer program in Fortran 95 are presented which enumerate the Hugenholtz diagram representation of the many-body perturbation series for the ground state energy with a two-body interaction. The output is in a form suitable for post-processing such as automatic code generation. The result of a particular application, generation of LATEX code to draw the diagrams, is shown.

In view that drivers would pay attention to the variation of headway on roads, an extended optimal velocity model is proposed by considering anticipation driving behavior. A stability criterion is given through linear stability analysis of traffic flows. The mKdV equation is derived with the reductive perturbation method for headway evolution which could be used to describe the stop-and-go traffic phenomenon. The results show a good effect of anticipation driving behavior on the stabilization of car flows and the anticipation driving behavior can improve the numerical stability of the model as well. In addition, the fluctuation of kinetic energy and the consumption of average energy in congested traffic flows are systematically analyzed. The results show that the reasonable level of anticipation driving behavior can save energy consumption in deceleration process effectively and lead to an associated relation like a "bow-tie" between the energy-saving and the value of anticipation factor.

A new automatic C/C++ program which generates all linked and essentially distinct vacuum Hugenholtz-type diagrams of many-body perturbation series up to arbitrary order is presented. The output results are shown as lists of integers which present the sites of suitable vertex of each diagram with its symmetry factor. The analysis results up to 10-order are displayed.

It is shown that Symbolic Computation provides excellent tools for solving quantum mechanical problems by perturbation theory. The method presented herein solves for both the eigenfunctions and eigenenergies as power series in the order parameter where each coefficient of the perturbation series is obtained in closed form. The algorithms are expressed in the Maple symbolic computation system but can be implemented on other systems. This approach avoids the use of an infinite basis set and some of the complications of degenerate perturbation theory. It is general and can, in principle, be applied to many separable systems.

A short review of the present status of computer packages for the high order analytical perturbative calculations is presented. The mathematical algorithm and the quantum field theory methods used are briefly discussed. The most recent computer package HEPLoops for analytical computations in high energy physics up to four-loops is also discussed.

The relativistic correction of the energy eigenvalues of quantum harmonic oscillator (QHO) are calculated using as eigenstates, for different values of the relativistic parameter α ≡ ħω/m₀c².