Preface
Singular perturbations of Schrödinger type operators are of
interest in mathematics, e.g. to study spectral
phenomena, and in applications of mathematics in
various sciences, e.g. in physics, chemistry, biology,
and in technology.
They also often lead to models in quantum theory
which are solvable in the sense that the spectral
characteristics (eigenvalues, eigenfunctions, and
scattering matrix) can be computed. Such models
then allow us to grasp the essential features of
interesting and complicated phenomena and serve
as an orientation in handling more realistic
situations.
In the last ten years two books have appeared on solvable
models in quantum theory built using special singular
perturbations of
Schrödinger operators. The book by S. Albeverio,
F. Gesztesy, R. Hoegh-Krohn and H. Holden
[39] describes the models in rigorous
mathematical terms. It gives a detailed analysis of
perturbations of the Laplacian in $ {\bf R}^d, d=1,2,3, $ by potentials
with support on a discrete finite or infinite set of point
sources (chosen in a deterministic, respectively,
stochastic manner).
Physically these operators describe
the motion of a quantum mechanical particle moving
under the action of a potential supported,
e.g., by the points of a crystal lattice or a random solid.
Such systems and models are also
described in physical terms in the book by Yu.N.Demkov and
V.N.Ostrovsky [255],
which also contains a description of applications in other areas
such as
in optics and electromagnetism.
Let us also remark that a translation of the book [39]
in Russian has been published with additional comments and literature
[40].
Since the appearance of [39,40,255] several important
new developments have taken place.
It is the main aim of the present book to present some
of these new developments
in a unified formalism which also puts some of the basic
results of the preceding books into a new
light. The new developments concern in particular
a systematic study of finite rank perturbations
of (self--adjoint) operators (in particular differential operators),
of generalized (singular) perturbations,
of the corresponding scattering theory as well as
infinite rank perturbations and multiple particles
(many--body) problems in quantum theory.
We also present the theory of point interaction
Hamiltonians,
as a particular case
of a general theory of singular perturbations
of differential operators.
This theory has received steadily increasing attention over the years
also for its many applications in physics (solid state physics,
nuclear physics), electromagnetism (antennas), and technology
(metallurgy, nanophysics).
We hope this
monograph can serve as a basis for orientation
in a rapidly developing area of analysis,
mathematical physics and their applications.
October 1998
S.Albeverio (Bonn), P.Kurasov (Stockholm)