Dimitri Yafaev

Mathematical Scattering Theory: General Theory

Preliminary facts Basic concepts of scattering theory Further properties of the WO Scattering for relatively smooth perturbations The general setup in stationary scattering theory Scattering for perturbations of trace class type Properties of the scattering matrix (SM) The spectral shift function (SSF) and the trace formula.

Scattering theory : some old and new problems

Part 1. The Schroedinger operator of two-particle systems: Basic notions.- Short-range interactions.- Asymptotic completeness.- Short-range interactions. Miscellaneous.- Long-range interactions. The scheme of smooth perturbations.- The generalized Fourier transform.- Long-range matrix potentials.- Part 2. The scattering matrix: A stationary representation.- The short-range case.- The long-range case.- The relative scattering matrix.- Part 3. The multiparticle Schroedinger operator and related problems: Setting the scattering problem.- Resolvent equations.- Asymptotic completeness.- A sketch of proof.- The scattering matrix for multiparticle systems.- New channels of scattering.- The Heisenberg model.- Infinite obstacle scattering.

Radiation conditions and scattering theory for

The correct form of the angular part of radiation conditions is found in scattering problem forN-particle quantum systems. The estimates obtained allow us to give an elementary proof of asymptotic completeness for such systems in the framework of the theory of smooth perturbations.

N

For the radial Schrödinger equation with a potentialq(x) decreasing at infinity asq0q−α, α∈(0, 2), the low energy asymptotics of spectral and scattering data is found. In particular, it is shown that forq0>0 the spectral function vanishes exponentially as the energyk2 tends to zero. On the contrary, there is always a zero-energy resonance forq0<0. These results determine the local asymptotics of solutions of the time-dependent Schrödinger equation for large timest. Specifically, for positive potentials its solutions decay as exp(−ϑ0t(2−α)/(2+α), ϑ0>0,t→∞. In the case α∈(1, 2) it is shown that for ±q0>0 the phase shift tends to ±∞ ask→0 and its asymptotics is evaluated.

-particle Hamiltonians

We consider the Schrodinger operator H=(i∇+A)2+V in the space L2(Rd) with long-range electrostatic V(x) and magnetic A(x) potentials. Using the scheme of smooth perturbations, we give an elementary proof of the existence and completeness of modified wave operators for the pair H0=−Δ, H. Our main goal is to study spectral properties of the corresponding scattering matrix S(λ). We obtain its stationary representation and show that its singular part (up to compact terms) is a pseudodifferential operator with an oscillating amplitude which is an explicit function of V and A. Finally, we deduce from this result that, in general, for each λ>0 the spectrum of S(λ) covers the whole unit circle.

The low energy scattering for slowly decreasing potentials

We study the inverse scattering problem for electric potentials and magnetic fields in , that are asymptotic sums of homogeneous terms at infinity. The main result is that all these terms can be uniquely reconstructed from the singularities in the forward direction of the scattering amplitude at some positive energy.

The scattering matrix for the Schrödinger operator with a long-range electromagnetic potential

We consider the Schr?dinger operator H = (i? + A)2 in the space L2(2) with a magnetic potential A(x) = a()(?x2, x1) |x|?2, where a is an arbitrary function on the unit circle. Our goal is to study spectral properties of the corresponding scattering matrix S(?), ? > 0. We obtain its stationary representation and show that its singular part (up to compact terms) is a pseudodifferential operator of zero order whose symbol is an explicit function of a. We deduce from this result that the essential spectrum of S(?) does not depend on ? and consists of two complex conjugated and perhaps overlapping closed intervals of the unit circle. Finally, we calculate the diagonal singularity of the scattering amplitude (kernel of S(?) considered as an integral operator). In particular, we show that for all these properties only the behaviour of a potential at infinity is essential. The preceding papers on this subject treated the case a() = const and used the separation of variables in the Schr?dinger equation in the polar coordinates. This technique does not, of course, work for arbitrary a. From an analytical point of view, our paper relies on some modern tools of scattering theory and well-known properties of pseudodifferential operators.

On inverse scattering at a fixed energy for potentials with a regular behaviour at infinity

Our goal is to extend the theory of the spectral shift function to the case where only the difference of some powers of the resolvents of selfadjoint operators belongs to the trace class. As an example, a pair of Dirac operators is considered.

On the mathematical theory of the Aharonov-Bohm effect

We develop the scattering theory for the Schrodinger operator with the Coulomb potential in the space of an arbitrary dimension d. In particular, we calculate the scattering matrix and show that its spectrum covers the whole unit circle. We also compute the differential cross section and show that it coincides with the classical Coulomb scattering cross section in the dimension d = 3 only.

A Trace Formula for the Dirac Operator

Abstract.We consider a class of translationally invariant magnetic fields such that the corresponding potential has a constant direction. Our goal is to study basic spectral properties of the Schrödinger operator