Overdetermined systems, dissipative singular Schrödinger operator, index theory
Author(s)
Bibliographic Information
Overdetermined systems, dissipative singular Schrödinger operator, index theory
(Encyclopaedia of mathematical sciences / editor-in-chief, R.V. Gamkrelidze, v. 65 . Partial differential equations ; 8)
Springer-Verlag, c1996
- : gw
- : us
- :pbk
- Other Title
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Differentsial'nye uravneniya s chastnymi proizvodnymi
Different︠s︡ialʹnye uravnenii︠a︡ s chastnymi proizvodnymi
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Note
Translation of: "Different︠s︡ialńye uravnenii︠a︡ s chastnymi proizvodnymi 8," which is vol.65 of the serial "Itogi nauki i tekhniki, Sovremennye problemy matematiki, Fundamentalńye napravlenii︠a︡"
Includes bibliographical references, author and subject indexes
Description and Table of Contents
- Volume
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: gw ISBN 9783540570363
Description
This volume contains three articles on linear overdetermined systems of partial differential equations, dissipative Schroedinger operators, and index theorems. Each article presents a survey of its subject, discussing fundamental results, such as the construction of compatibility operators and complexes for elliptic, parabolic and hyperbolic coercive problems, the method of functional models, and the Atiyah-Singer index theorem and its generalizations. Both classical and recent results are explained in detail and illustrated by means of example.
Table of Contents
I. Linear Overdetermined Systems of Partial Differential Equations. Initial and Initial-Boundary Value Problems.- II. Spectral Analysis of a Dissipative Singular Schrodinger Operator in Terms of a Functional Model.- III. Index Theorems.- Author Index.
- Volume
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:pbk ISBN 9783642489464
Description
Consider a linear partial differential operator A that maps a vector-valued function Y = (Yl," Ym) into a vector-valued function I = (h,***, II). We assume at first that all the functions, as well as the coefficients of the differen- tial operator, are defined in an open domain Jl in the n-dimensional Euclidean n space IR , and that they are smooth (infinitely differentiable). A is called an overdetermined operator if there is a non-zero differential operator A' such that the composition A' A is the zero operator (and underdetermined if there is a non-zero operator A" such that AA" = 0). If A is overdetermined, then A'I = 0 is a necessary condition for the solvability of the system Ay = I with an unknown vector-valued function y. 3 A simple example in 1R is the operator grad, which maps a scalar func- tion Y into the vector-valued function (8y/8x!, 8y/8x2, 8y/8x3)' A necessary solvability condition for the system grad y = I has the form curl I = O.
Table of Contents
I. Linear Overdetermined Systems of Partial Differential Equations. Initial and Initial-Boundary Value Problems.- II. Spectral Analysis of a Dissipative Singular Schroedinger Operator in Terms of a Functional Model.- III. Index Theorems.- Author Index.
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