109 research outputs found
The Schrödinger operator: Perturbation determinants, the spectral shift function, trace identities, and all that
No full textTranslated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 41, No. 3, pp. 60-83, 2007 Original Russian Text Copyright © by D. R. Yafaev Dedicated to the 100th anniversary of the birth of Mark Grigor'evich KreinInternational audienceWe discuss applications of the M. G. Krein theory of the spectral shift function to the multidimensional Schrödinger operator. Specific properties of this function, for example, its high-energy asymptotics are studied. Trace identities are derive
Exponential decay of eigenfunctions of first order systems
No full textPapers from the International Conference on Transport and Spectral Problems in Quantum Mechanics held in honor of Jean-Michel Combes at the Université de Cergy-Pontoise, Cergy-Pontoise, September 4--6, 2006The author studies exponential decay of the eigenfunctions of first-order (matrix) differential operators of the form It is shown that under certain assumptions, the eigenfunctions obey estimates of the type The author emphasizes that these estimates are valid everywhere off the essential spectrum , not just below the minimum of
Trace-class approach in scattering problems for perturbations of media
No full textProceedings of the 2nd Conference on Operator Algebras and Mathematical Physics held in Sinaia, June 26-July 4, 2003We consider the operators and where and are positively definite bounded matrix-valued functions and is an elliptic differential operator. Our main result is that the wave operators for the pair , exist and are complete if the difference , , as . Our point is that no special assumptions on are required. Similar results are obtained in scattering theory for the wave equation
Spectral theory of differential operators: M. Sh Birman 80th anniversary collection
No full textThis volume is dedicated to Professor M. Sh. Birman in honor of his eightieth birthday. It contains original articles in spectral and scattering theory of differential operators, in particular, Schrodinger operators, and in homogenization theory. All articles are written by members of M. Sh. Birman's research group who are affiliated with different universities all over the world. A specific feature of the majority of the papers is a combination of traditional methods with new modern ideas
High-energy and smoothness asymptotic expansion of the scattering amplitude
No full textAbstractWe find an explicit expression for the kernel of the scattering matrix for the Schrödinger operator containing at high energies all terms of power order. It turns out that the same expression gives a complete description of the diagonal singularities of the kernel in the angular variables. The formula obtained is in some sense universal since it applies both to short- and long-range electric as well as magnetic potentials
Scattering by magnetic fields
No full textAbstract. Consider the scattering amplitude s(ω, ω′;λ), ω, ω ′ ∈ Sd−1, λ> 0, corresponding to an arbitrary short-range magnetic field B(x), x ∈ Rd. This is a smooth function of ω and ω ′ away from the diagonal ω = ω ′ but it may be singular on the diagonal. If d = 2, then the singular part of the scattering amplitude (for example, in the transversal gauge) is a linear combination of the Dirac δ-function and of a singular denominator. Such structure is typical for long-range magnetic scattering. We refer to this phenomenon as to the long-range Aharonov-Bohm effect. On the contrary, for d = 3 scattering is essentially of short-range nature although, for example, the magnetic potential A(tr)(x) such that curlA(tr)(x) = B(x) and 〈A(tr)(x), x 〉 = 0 decays at infinity as |x|−1 only. To be more precise, we show that, up to the diagonal Dirac function (times an explicit function of ω), the scattering amplitude has only a weak singularity in the forward direction ω = ω′. Our approach relies on a construction in the dimension d = 3 of a short-range magnetic potential A(x) corresponding to a given short-range magnetic field B(x). 1
Spectral analysis of Jacobi operators and asymptotic behavior of orthogonal polynomials
No full textInternational audienceWe find and discuss asymptotic formulas for orthonormal polynomials [Formula: see text] with recurrence coefficients [Formula: see text]. Our main goal is to consider the case where off-diagonal elements [Formula: see text] as [Formula: see text]. Formulas obtained are essentially different for relatively small and large diagonal elements [Formula: see text]. Our analysis is intimately linked with spectral theory of Jacobi operators [Formula: see text] with coefficients [Formula: see text] and a study of the corresponding second order difference equations. We introduce the Jost solutions [Formula: see text], [Formula: see text], of such equations by a condition for [Formula: see text] and suggest an Ansatz for them playing the role of the semiclassical Liouville–Green Ansatz for solutions of the Schrödinger equation. This allows us to study the spectral structure of Jacobi operators and their eigenfunctions [Formula: see text] by traditional methods of spectral theory developed for differential equations. In particular, we express all coefficients in asymptotic formulas for [Formula: see text] as [Formula: see text] in terms of the Wronskian of the solutions [Formula: see text] and [Formula: see text]. The formulas obtained for [Formula: see text] generalize the asymptotic formulas for the classical Hermite polynomials where [Formula: see text] and [Formula: see text]. The spectral structure of Jacobi operators [Formula: see text] depends crucially on a rate of growth of the off-diagonal elements [Formula: see text] as [Formula: see text]. If the Carleman condition is satisfied, which, roughly speaking, means that [Formula: see text], and the diagonal elements [Formula: see text] are small compared to [Formula: see text], then [Formula: see text] has the absolutely continuous spectrum covering the whole real axis. We obtain an expression for the corresponding spectral measure in terms of the boundary values [Formula: see text] of the Jost solutions. On the contrary, if the Carleman condition is violated, then the spectrum of [Formula: see text] is discrete. We also review the case of stabilizing recurrence coefficients when [Formula: see text] tend to a positive constant and [Formula: see text] as [Formula: see text]. It turns out that the cases of stabilizing and increasing recurrence coefficients can be treated in an essentially same way
Self-adjoint operators associated with Hankel moment matrices
No full textIn a paper from 2016 D. R. Yafaev initiated a study of closable Hankel forms associated with the moments (mn) of a positive measure with infinite support on the real line. If mn=o(1) Yafaev characterized the closure of the form based on earlier work on quasi-Carleman operators. We give a new proof of the description of the closure based entirely on moment considerations. The main purpose of the present paper is a description of the self-adjoint Hankel operators associated with closed Hankel forms in the Hilbert space of square summable sequences. We do this not only in the case mn=o(1) studied by Yafaev but also in two other cases, where the Hankel form is closable, namely if the moment sequence is indeterminate or if the moment sequence is determinate with finite index of determinacy.</p
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