High energy and smoothness asymptotic expansion of the scattering amplitude,
SM.
We find an explicit expression for the kernel of
the scattering matrix for the Schr\"odinger 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.

A particle in a magnetic field of an infinite rectilinear current
Savart.
We consider the Schr\"odinger operator ${\bf H}=(i\nabla+A)^2 $ in the
space $L_2({\R}^3)$ with a magnetic potential
$A $ created by an infinite rectilinear current
and perform its spectral analysis almost explicitly.
In particular, we show that the operator ${\bf H}$ is absolutely continuous,
its spectrum has infinite
multiplicity
and coincides with the positive half-axis. Then we find the large-time behavior of solutions
$\exp(-i{\bf H}t)f$ of the time dependent Schr\"odinger equation. Our main observation is that
a quantum particle has always a preferable (depending on its charge)
direction of propagation along the current. Similar result is true in classical mechanics.

A trace formula for the Dirac operator,
SSF.
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 self-adjoint operators belongs to the trace class. As an example,
a pair of Dirac operators is considered.

Scattering by magnetic fields,
No-lr-Ah-Bohm.
Consider the scattering amplitude $s(\omega,\omega^\prime;\lambda)$,
$\omega,\omega^\prime\in{\Bbb S}^{d-1}$, $\lambda > 0$,
corresponding to an arbitrary short-range
magnetic field $B(x)$, $x\in{\Bbb R}^d$. This is a smooth function of $\omega$ and $\omega^\prime$ away from the diagonal $\omega=\omega^\prime$ 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 $\delta$-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 ${\rm curl}\,A^{(tr)}(x)=B(x)$ and
$ \langle A^{(tr)}(x),x\rangle=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 $\omega$), the scattering amplitude has only a weak singularity
in the forward direction $\omega = \omega^\prime$.
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 $B(x)$.

On Inverse Scattering at a Fixed Energy for Potentials
with a Regular Behaviour at Infinity (with Ricardo Weder),
Inverse.
We study the inverse scattering problem for electric potentials
and magnetic fields in $\ere^d, d\geq 3$, 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.

Lectures on scattering theory (given by the author at the Australian National University in October and November, 2001),
Scatt-Lectures.
The first two lectures are devoted to describing the basic concepts of scattering theory
in a very compressed way. The last two lectures are based on the recent research of the author.

A possible subject for a thesis: The Schroedinger equation with a long-range potential
sujet/these.