Analysis of integral equations attached to skin effect

summary:The paper is a mathematical background of the paper of D. Mayer, B. Ulrych where the mathematical model of the skin effect is established and discussed. It is assumed that the currents passing through parallel conductors are under effect of a variable magnetic field. The phasors of the density of the current are solutions of $f(x)-jq \sum{^k_{i=1}}b_i \int_{Si} f(y) V(|y-x|)dy-c_p=h(x)$ for $x\in S_p, p=1,\dots,k$, $\int_{Si}f(x)dx=I_i, i=1,\dots, k$, where $j$ is the imaginary unit, $b_i,q,I_i$ are given constants, #h(x)$is a given function and$f(x)$is an unknown function and$c_i$ are unknown constants. The first and the second section of this paper are devoted to the problem of existence and unicity of a solution. The third section is devoted to a numerical method

Asymptotic power series of field correlators

We address the problem of ambiguity of a function determined by an asymptotic perturbation expansion. Using a modified form of the Watson lemma recently proved elsewhere, we discuss a large class of functions determined by the same asymptotic power expansion and represented by various forms of integrals of the Laplace-Borel type along a general contour in the Borel complex plane. Some remarks on possible applications in QCD are made.Comment: Presented at the International Conference "Selected Topics in Mathematical and Particle Physics" (Niederlefest), Prague, 5 - 7 May 200

Operator product expansion and analyticity

We discuss the current use of the operator product expansion in QCD calculations. Treating the OPE as an expansion in inverse powers of an energy-squared variable (with possible exponential terms added), approximating the vacuum expectation value of the operator product by several terms and assuming a bound on the remainder along the euclidean region, we observe how the bound varies with increasing deflection from the euclidean ray down to the cut (Minkowski region). We argue that the assumption that the remainder is constant for all angles in the cut complex plane is not justified. Making specific assumptions on the properties of the expanded function, we obtain bounds on the remainder in explicit form and show that they are very sensitive both to the deflection angle and to the class of functions chosen. The results obtained are discussed in connetcion with calculations of the coupling constant \alpha_{s} from the \tau decay.Comment: Preprint PRA-HEP 99/04, 20 page

The operator-product expansion away from euclidean region

The role of the operator-product expansion in QCD calculations is discussed. Approximating the two-point correlation function by several terms and assuming an upper bound on the truncation error along the euclidean ray, we consider two model situations to examine how the bound develops with increasing deflection from the euclidean ray towards the cut. We obtain explicit bounds on the truncation error and show how they worsen with the increasing deflection. The result does not support the believe that the remainder is constant for all angles in the complex energy plane. Further refinements of the formalism are dicussed.Comment: 4 pages, qcd98 conference report, Montpellier, July 199

Asymptotic properties for half-linear difference equations

summary:Asymptotic properties of the half-linear difference equation $$\Delta (a_{n}|\Delta x_{n}|^{\alpha }\mathop {\mathrm sgn}\Delta x_{n} )=b_{n}|x_{n+1}|^{\alpha }\mathop {\mathrm sgn}x_{n+1} \qquad \mathrm{(*)}$$ are investigated by means of some summation criteria. Recessive solutions and the Riccati difference equation associated to $(*)$ are considered too. Our approach is based on a classification of solutions of $(*)$ and on some summation inequalities for double series, which can be used also in other different contexts