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Singular Perturbations of Differential Operators

Albeverio, Sergio and Kurasov, Pavel LU (2000) In London Mathematical Society: Lecture Notes Series 271.
Abstract
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... (More)
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

"Solvable Models in Quantum Mechanics" describes the models in rigorous mathematical terms. It gives a detailed analysis of perturbations of the Laplacian in 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 "Zero-range Potentials in Atomic Physics", 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 by S.Albeverio, F.Gesztesy, R.Hoegh-Krohn and H.Holden in Russian has been published with additional comments and literature. Since the appearance of these books 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. (Less)
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author
organization
publishing date
type
Book/Report
publication status
published
subject
keywords
extension theory, Singular perturbations, Krein's formula
in
London Mathematical Society: Lecture Notes Series
volume
271
pages
429 pages
publisher
Cambridge University Press
ISBN
052177912X
language
English
LU publication?
yes
id
ad643743-d0c8-4a0f-9407-c86dd8da3bef (old id 699812)
date added to LUP
2007-12-14 11:39:09
date last changed
2016-09-22 13:50:46
@book{ad643743-d0c8-4a0f-9407-c86dd8da3bef,
  abstract     = {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.<br/><br>
<br/><br>
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<br/><br>
"Solvable Models in Quantum Mechanics" describes the models in rigorous mathematical terms. It gives a detailed analysis of perturbations of the Laplacian in 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 "Zero-range Potentials in Atomic Physics", 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 by S.Albeverio, F.Gesztesy, R.Hoegh-Krohn and H.Holden in Russian has been published with additional comments and literature. Since the appearance of these books 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.<br/><br>
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).<br/><br>
<br/><br>
We hope this monograph can serve as a basis for orientation<br/><br>
in a rapidly developing area of analysis, mathematical physics and their applications.},
  author       = {Albeverio, Sergio and Kurasov, Pavel},
  isbn         = {052177912X},
  keyword      = {extension theory,Singular perturbations,Krein's formula},
  language     = {eng},
  pages        = {429},
  publisher    = {Cambridge University Press},
  series       = {London Mathematical Society: Lecture Notes Series},
  title        = {Singular Perturbations of Differential Operators},
  volume       = {271},
  year         = {2000},
}