Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A constructive approach to stationary scattering theory   In this paper we give a new and constructive approach to stationary
scattering theory for pairs of self-adjoint operators $H_0$ and $H_1$ on a
Hilbert space $\mathcal H$ which satisfy the following conditions: (i) for any
open bounded subset $\Delta$ of $\mathbb R,$ the operators $F E_\Delta^{H_0}$
and $F E_\Delta^{H_1}$ are Hilbert-Schmidt and (ii) $V = H_1- H_0$ is bounded
and admits decomposition $V = F^*JF,$ where $F$ is a bounded operator with
trivial kernel from $\mathcal H$ to another Hilbert space $\mathcal K$ and $J$
is a bounded self-adjoint operator on $\mathcal K.$ An example of a pair of
operators which satisfy these conditions is the Schr\"odinger operator $H_0 =
-\Delta + V_0$ acting on $L^2(\mathbb R^\nu),$ where $V_0$ is a potential of
class $K_\nu$ (see B.\,Simon, {\it Schr\"odinger semigroups,} Bull. AMS 7,
1982, 447--526) and $H_1 = H_0 + V_1,$ where $V_1 \in L^\infty(\mathbb R^\nu)
\cap L^1(\mathbb R^\nu).$ Among results of this paper is a new proof of
existence and completeness of wave operators $W_\pm(H_1,H_0)$ and a new
constructive proof of stationary formula for the scattering matrix. This
approach to scattering theory is based on explicit diagonalization of a
self-adjoint operator $H$ on a sheaf of Hilbert spaces $\EuScript S(H,F)$
associated with the pair $(H,F)$ and with subsequent construction and study of
properties of wave matrices $w_\pm(\lambda; H_1,H_0)$ acting between fibers
$\mathfrak h_\lambda(H_0,F)$ and $\mathfrak h_\lambda(H_1,F)$ of sheaves
$\EuScript S(H_0,F)$ and $\EuScript S(H_1,F)$ respectively. The wave operators
$W_\pm(H_1,H_0)$ are then defined as direct integrals of wave matrices and are
proved to coincide with classical time-dependent definition of wave operators.

21
The categories of this paper are: adap-org
26
The categories of this paper are: bayes-an
17
The categories of this paper are: q-fin
22
The categories of this paper are: chao-dyn
28
The categories of this paper are: alg-geom
1
The categories of this paper are: cs
36
The categories of this paper are: cmp-lg
25
The categories of this paper are: patt-sol
12
The categories of this paper are: nucl-th
10
The categories of this paper are: q-bio
9
The categories of this paper are: nlin
13
The categories of this paper are: quant-ph
3
The categories of this paper are: physics
19
The categories of this paper are: eess
34
The categories of this paper are: chem-ph
27
The categories of this paper are: comp-gas