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This monograph presents the newly developed method of rigged
Hilbert spaces as a modern approach in singular perturbation
theory. A key notion of this approach is the Lax-Berezansky triple
of Hilbert spaces embedded one into another, which specifies the
well-known Gelfand topological triple. All kinds of singular
interactions described by potentials supported on small sets (like
the Dirac -potentials, fractals, singular measures, high degree
super-singular expressions) admit a rigorous treatment only in
terms of the equipped spaces and their scales. The main idea of the
method is to use singular perturbations to change inner products in
the starting rigged space, and the construction of the perturbed
operator by the Berezansky canonical isomorphism (which connects
the positive and negative spaces from a new rigged triplet). The
approach combines three powerful tools of functional analysis based
on the Birman-Krein-Vishik theory of self-adjoint extensions of
symmetric operators, the theory of singular quadratic forms, and
the theory of rigged Hilbert spaces. The book will appeal to
researchers in mathematics and mathematical physics studying the
scales of densely embedded Hilbert spaces, the singular
perturbations phenomenon, and singular interaction problems.
This monograph presents the newly developed method of rigged
Hilbert spaces as a modern approach in singular perturbation
theory. A key notion of this approach is the Lax-Berezansky triple
of Hilbert spaces embedded one into another, which specifies the
well-known Gelfand topological triple. All kinds of singular
interactions described by potentials supported on small sets (like
the Dirac -potentials, fractals, singular measures, high degree
super-singular expressions) admit a rigorous treatment only in
terms of the equipped spaces and their scales. The main idea of the
method is to use singular perturbations to change inner products in
the starting rigged space, and the construction of the perturbed
operator by the Berezansky canonical isomorphism (which connects
the positive and negative spaces from a new rigged triplet). The
approach combines three powerful tools of functional analysis based
on the Birman-Krein-Vishik theory of self-adjoint extensions of
symmetric operators, the theory of singular quadratic forms, and
the theory of rigged Hilbert spaces. The book will appeal to
researchers in mathematics and mathematical physics studying the
scales of densely embedded Hilbert spaces, the singular
perturbations phenomenon, and singular interaction problems.
The notion of singular quadratic form appears in mathematical
physics as a tool for the investigation of formal expressions
corresponding to perturbations devoid of operator sense. Numerous
physical models are based on the use of Hamiltonians containing
perturba tion terms with singular properties. Typical examples of
such expressions are Schrodin ger operators with O-potentials (- +
AD) and Hamiltonians in quantum field theory with perturbations
given in terms of operators of creation and annihilation (P("
This monograph is devoted to the systematic presentation of the
method of singular quadratic forms in the perturbation theory of
self-adjoint operators. The concept of a singular (nowhere
closable) quadratic form, a key notion of the present volume, is
treated from different points of view such as definition,
properties, relations with regular (closable) quadratic forms,
operator representation, classification in the scale of Hilbert
spaces and especially as an object carrying a singular perturbation
for Hamiltonians. The main idea is to interpret singular quadratic
form in the role of an abstract boundary condition for self-adjoint
extension. Various aspects of the singularity principle are
investigated, such as the construction of singularly perturbed
operators, higher powers of perturbed operators, the transition to
a new orthogonally extended state space, as well as approximation
and regularization. Furthermore, applications dealing with singular
Wick monomials in the Fock space and mathematical scattering theory
are included. Audience: This book will be of interest to students
and researchers whose work involves functional analysis, operator
theory and quantum field theory.
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