Absolute continuity of spectral shift

In this paper we develop the method of double operator integrals to prove trace formulae for functions of contractions, dissipative operators, unitary operators and self-adjoint operators. To establish the absolute continuity of spectral shift, we use the Sz.-Nagy theorem on the absolute continuity of the spectral measure of the minimal unitary dilation of a completely nonunitary contraction. We also give a construction of an intermediate contraction for a pair of contractions with trace class difference.     

Spectral concentration and rapidly decaying potentials

We consider the spectral function ϱ(μ) (μ⩾0) for the Sturm-Liouville equation y″ + (λ − q)y = 0 on [0, ∞) with a boundary condition y(0)cosα + y′(0)sinα = 0 and where q ∈ L(0, ∞). A point μ₀ is called a point of spectral concentration if ϱ′ has a local maximum there. We give a new formula for ϱ′ in terms of the Prüfer transformation angle θ and we identify a rapid transitional property of θ in the neighbourhood of points of spectral concentration. This property provides a sensitive test of the existence of such points and lends itself to a detailed computational investigation which reveals such points in greater profusion than shown by, for example, SLEDGE. Our approach also leads to the possibility of a theoretical investigation.     

Zero energy asymptotics of the resolvent for a class of slowly decaying potentials

We prove a limiting absorption principle at zero energy for two-body Schrödinger operators with long-range potentials having a positive virial at infinity. More precisely, we establish a complete asymptotic expansion of the resolvent in weighted spaces when the spectral parameter tends to zero in cones which are adjacent to the positive real axis. The principal tools are absence of eigenvalue at zero, singular Mourre theory and microlocal estimates.in final form: 14 November 2003S. Fournais was supported by a grant from the Carlsberg Foundation (before 31.12.02) and by a Marie Curie Fellowship of the European Community Programme Improving the Human Research Potential and the Socio-Economic Knowledge Base under contract number HPMF-CT-2002-01822 (from 01.01.03).E. Skibsted is (partially) supported by MaPhySto – A Network in Mathematical Physics and Stochastics funded by The Danish National Research Foundation.     

Absolute continuity of positive spectrum for Schrödinger operators with long-range potentials

Absolute continuity in (0, ∞) for Schrödinger operators ? Δ + V(x), with long range potential V = V₁ + V₂ such that $\text{?V}{}_1\text{?}{}_{\mathrm{r}}\text{, V}{}_2\text{= 0(r}{}^{\mathrm{?1??}}\text{)}$, ? > 0, as $\text{¦ x ¦ → ∞}$, is shown by proving estimates on resolvents near the real axis. Completeness of the modified wave operators for a superposition of Coulomb potentials also follows. Singular local behavior of V is allowed.     

Absolutely continuous spectrum for one-dimensional Schrödinger operators with slowly decaying potentials: Some optimal results

The absolutely continuous spectrum of one-dimensional Schrödinger operators is proved to be stable under perturbation by potentials satisfying mild decay conditions. In particular, the absolutely continuous spectra of free and periodic Schrödinger operators are preserved under all perturbations satisfying This result is optimal in the power scale. Slightly more general classes of perturbing potentials are also treated. A general criterion for stability of the absolutely continuous spectrum of one-dimensional Schrödinger operators is established. In all cases analyzed, the main term of the asymptotic behavior of the generalized eigenfunctions is shown to have WKB form for almost all energies. The proofs rely on maximal function and norm estimates, and on almost everywhere convergence results for certain multilinear integral operators.

     

The spectral function for Sturm-Liouville problems where the potential is of Wigner-von Neumann type or slowly decaying

We consider the linear, second-order, differential equation

(∗)     

Absolutely continuous spectrum of Schrödinger operators with potentials slowly decaying inside a cone

For a large class of multi-dimensional Schrödinger operators it is shown that the absolutely continuous spectrum is essentially supported by [0,∞). We require slow decay and mildly oscillatory behavior of the potential in a cone and can allow for arbitrary non-negative bounded potential outside the cone. In particular, we do not require the existence of wave operators. The result and method of proof extends previous work by Laptev, Naboko and Safronov.     

Fast propagation for KPP equations with slowly decaying initial conditions

This paper is devoted to the analysis of the large-time behavior of solutions of one-dimensional Fisher-KPP reaction-diffusion equations. The initial conditions are assumed to be globally front-like and to decay at infinity towards the unstable steady state more slowly than any exponentially decaying function. We prove that all level sets of the solutions move infinitely fast as time goes to infinity. The locations of the level sets are expressed in terms of the decay of the initial condition. Furthermore, the spatial profiles of the solutions become asymptotically uniformly flat at large time. This paper contains the first systematic study of the large-time behavior of solutions of KPP equations with slowly decaying initial conditions. Our results are in sharp contrast with the well-studied case of exponentially bounded initial conditions.     

Absolute continuity theorems for abstract Riemann integration

Absolute continuity for functionals is studied in the context of proper and abstract Riemann integration examining the relation to absolute continuity for finitely additive measures and giving results in both directions: integrals coming from measures and measures induced by integrals. To this end, we look for relations between the corresponding integrable functions of absolutely continuous integrals and we deal with the possibility of preserving absolute continuity when extending the elemental integrals.     

Absolute continuity for Banach space valued mappings

We consider the notion ofp, λ, δ-absolute continuity for Banach space valued mappings introduced in [2] for real valued functions and for λ=1. We investigate the validity of some basic properties that are shared byn, λ-absolutely continuous functions in the sense of Maly and hencl. We introduce the class δ-BV _λ,loc^p and we give a characterization of the functions belonging to this class.     

Absolute continuity for Banach space valued mappings

We consider the notion ofp, λ, δ-absolute continuity for Banach space valued mappings introduced in [2] for real valued functions and for λ=1. We investigate the validity of some basic properties that are shared byn, λ-absolutely continuous functions in the sense of Maly and hencl. We introduce the class δ-BV _λ,loc ^p and we give a characterization of the functions belonging to this class.     

Absolute continuity of information channels

Let [X, v, Y] be an abstract information channel with the input X = (X, ) and the output Y = (Y, ) which are measurable spaces, and denote by L(Y) = L(Y, ) the Banach space of all bounded signed measures with finite total variation as norm. The channel distribution ν(·,·) is considered as a function defined on (X, ) and valued in L(Y). It will be proved that, if the measurable space (Y, ) is countably generated, then the is a strongly measurable function from X into L(Y) if and only if there exists a probability measure μ on (Y, ) which dominates every measure ν(x, ·) (x X). Furthermore, under this condition, the Radon-Nikodym derivative ν(x, dy)/μ(dy) is jointly measurable with respect to the product measure space (X, , m) (Y, , μ) where m is any but fixed probability measure of (X, ). As an application, it will be shown that the channel given as above is uniformly approximated by channels of Hibert-Schmidt type.     

Absolute continuity of Wiener processes

By a Wiener process we mean a countably additive random measure taking independent values on disjoint sets. Given two continuous Wiener processes we give their decomposition into weakly equivalent and mutually singular parts.     

Some properties of slowly decaying oscillatory solutions of KP

In this paper a class of solutions of KP is discussed. Theory of these solutions, referred to here as breathers, is developed and it is shown that a subclass of these solutions can be used to construct more general solutions of KP similarly to how the functions e^i(λx+νy) are used to perform the same task in the theory of Fourier transform. Nonlinear superposition formula for general solutions of KP similar to the Fourier expansion formula is considered.     

Absolute continuity of vector-valued self-affine measures

This paper is devoted to the absolute continuity of (scalar-valued or vector-valued) self-affine measures and their properties on the boundary of an invariant set. We first extend the definition of WSC to self-affine IFS, and then obtain a necessary and sufficient condition for the vector-valued self-affine measures to be absolutely continuous with respect to the Lebesgue measure. In addition, we prove that, for any IFS and any invariant open set V, the corresponding (scalar-valued or vector-valued) invariant measure is supported either in V or in ∂V.     

Absolute continuity of a vector-valued measure

The necessary and sufficient condition is proved for absolute continuity of a vector-valued measure in the sense of Bartle.