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Spectral analysis of differential operators : interplay between spectral and oscillatory properties

Author: Fedor S Rofe-Beketov; Aleksandr M Kholʹkin; Ognjen Milatovic
Publisher: Hackensack, NJ : World Scientific, ©2005.
Series: World Scientific monograph series in mathematics, v. 7.
Edition/Format:   Print book : EnglishView all editions and formats
Summary:
"This is the first monograph devoted to the Sturm oscillatory theory for infinite systems of differential equations and its relations with the spectral theory. It aims to study a theory of self-adjoint problems for such systems, based on an elegant method of binary relations. Another topic investigated in the book is the behavior of discrete eigenvalues which appear in spectral gaps of the Hill operator and almost  Read more...
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Material Type: Internet resource
Document Type: Book, Internet Resource
All Authors / Contributors: Fedor S Rofe-Beketov; Aleksandr M Kholʹkin; Ognjen Milatovic
ISBN: 9812562761 9789812562760
OCLC Number: 62534995
Description: xxiii, 438 pages ; 24 cm.
Contents: Relation between spectral and oscillatory properties for the matrix Sturm-Liouville problem --
Problem on a finite interval --
problem on the half-line --
Bibliographical comments --
An example of the operator of (1.1)-(1.3) type with the purely absolutely continuous spectrum --
Oscillatory theory of partial differential equations --
Bochner integral --
Fundamental system of solutions for an operator differential equation with a singular boundary condition --
Separated self-adjoint boundary conditions --
A construction for the fundamental solution of the boundary-value problem --
Self-consistency of the fundamental solution of the self-adjoint problem and the evolution of the corresponding Hermitian relation (or Lagrangian plane in H + H) --
A different construction for the fundamental solution of the boundary-value problem --
Bibliographical comments --
Dependence of the spectrum of operator boundary problems on variations of a finite or semi-infinite interval --
Dependence of Eigenvalues and the greatest lower bound of the spectrum for a semi-bounded below differential operator on variations of the interval --
Continuity and monotonicity of the greatest lower bound of the essential spectrum for semi-bounded below differential operators --
Bibliographical comments --
Relation between spectral and oscillatory properties for operator differential equations of arbitrary order --
Comparison and alternation theorems --
Multiplicative representation of positive differential operators and its applications --
Discrete levels in the gaps of the essential spectrum --
Bibliographical comments --
Symplectic interpretation of Sturm-type theorems and their operator-theoretical proofs --
Self-adjoint extensions of systems of differential equations of arbitrary order on an infinite interval in the absolutely indefinite case --
A description of self-adjoint extensions of differential operators of arbitrary order with operator-valued coefficients on an infinite interval --
Parametrization of the characteristic operator function --
Bibliographical comments --
Everitt-Zettl problem, Brusentsev's example and two open questions --
On the deficiency indices of scalar symmetric differential operators of general kind on the half-axis (solved and unsolved questions) --
Kogan-Rofe-Beketov's asymptotic theorems and deficiency indices of symmetric differential perators --
R. C. Gilbert's class of formally self-adjoint ordinary differential operators whose deficiency numbers differ by an arbitrary pre-assigned positive integer --
Deficiency indices of symmetric differential systems of the first and arbitrary orders and some open questions --
Some comments on Hilbert's 21-st problem and Bolibrukh counterexample --
Some comments on section 5.2 --
Characteristic properties of Weyl solutions for the Sturm-Liouville and Dirac equations. V. A. Marchenko's theorems --
Discrete levels in spectral gaps of perturbed Schrodinger and Hill operators --
Factorized phase matrix of the perturbed system and the discrete spectrum in gaps of the continuous spectrum --
Generalization of D'Alembert-Liouville-Ostrogradsky formula and its application to growth estimates for solutions on canonical almost periodic systems --
A reduction of the problem about the number of discrete levels in a finite spectral gap of one-dimensional Schrodinger operator to a similar problem for a semi-infinite gap of the transformed Schrodinger operator --
Applications to perturbations of Hill operators and Schrodinger operators with an almost periodic potential. Kneser-type constants and effective masses --
Discrete levels in spectral gaps of perturbed Hill operators --
Bibliographical comments --
Some comments on section 6.1 --
Some comments on section 6.2 --
Some comments on section 6.3 and differential inequalities --
The spectrum of self-adjoint periodic and almost periodic operators. Marchenko-Ostrovskiy and Pastur-Tkachenko theorems --
The specturm of non-self-adjoint operators. Batchenko-Gesztesy and Rofe-Beketov theorems --
The specturm of perturbed self-adjoint and non-self-adjoint operators --
Matrix operators or differential-algebraic operators and related topics --
The inverse Sturm-Liouville problem for the spectral matrix on the whole axis --
Three open questions --
Appendix A --
Self-adjoint extensions of differential operators on a finite interval in spaces of vector-functions --
Hermitian relations --
Canonical form of Hermitian relations --
Various forms of Hermitian relations. Dissipative and sectorial relations --
Self-adjoint boundary conditions for infinite systems of second order differential equations --
Self-adjoint extensions of differential operators of arbitrary order with operator coefficients --
The expressions of order 2n --
The expression of order 2n-1 --
Continuous dependecne of intial data on solutions of differential equations --
Bibliographical comments --
M. L. Gorbachuk theory of generalized boundary values. Abstract boundary conditions --
Strong resolvent convergence of Hermitian relations and operators with non-dense domains and resolution of the identity. Stability of Eigenvalues --
Characteristic operator and boundary-value problems with separated boundary conditions --
Bibliography --
List of symbols --
Index.
Series Title: World Scientific monograph series in mathematics, v. 7.
Responsibility: Fedor S. Rofe-Beketov, Aleksandr M. Kholkin ; translated by Ognjen Milatovic ; with foreword by Vladimir A. Marchenko.
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Focuses on the Sturm oscillatory theory for infinite systems of differential equations and its relations with the spectral theory. This monograph aims to study a theory of self-adjoint problems for  Read more...

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    schema:description "Relation between spectral and oscillatory properties for the matrix Sturm-Liouville problem -- Problem on a finite interval -- problem on the half-line -- Bibliographical comments -- An example of the operator of (1.1)-(1.3) type with the purely absolutely continuous spectrum -- Oscillatory theory of partial differential equations -- Bochner integral -- Fundamental system of solutions for an operator differential equation with a singular boundary condition -- Separated self-adjoint boundary conditions -- A construction for the fundamental solution of the boundary-value problem -- Self-consistency of the fundamental solution of the self-adjoint problem and the evolution of the corresponding Hermitian relation (or Lagrangian plane in H + H) -- A different construction for the fundamental solution of the boundary-value problem -- Bibliographical comments -- Dependence of the spectrum of operator boundary problems on variations of a finite or semi-infinite interval -- Dependence of Eigenvalues and the greatest lower bound of the spectrum for a semi-bounded below differential operator on variations of the interval -- Continuity and monotonicity of the greatest lower bound of the essential spectrum for semi-bounded below differential operators -- Bibliographical comments -- Relation between spectral and oscillatory properties for operator differential equations of arbitrary order -- Comparison and alternation theorems -- Multiplicative representation of positive differential operators and its applications -- Discrete levels in the gaps of the essential spectrum -- Bibliographical comments -- Symplectic interpretation of Sturm-type theorems and their operator-theoretical proofs -- Self-adjoint extensions of systems of differential equations of arbitrary order on an infinite interval in the absolutely indefinite case -- A description of self-adjoint extensions of differential operators of arbitrary order with operator-valued coefficients on an infinite interval -- Parametrization of the characteristic operator function -- Bibliographical comments -- Everitt-Zettl problem, Brusentsev's example and two open questions -- On the deficiency indices of scalar symmetric differential operators of general kind on the half-axis (solved and unsolved questions) -- Kogan-Rofe-Beketov's asymptotic theorems and deficiency indices of symmetric differential perators -- R. C. Gilbert's class of formally self-adjoint ordinary differential operators whose deficiency numbers differ by an arbitrary pre-assigned positive integer -- Deficiency indices of symmetric differential systems of the first and arbitrary orders and some open questions -- Some comments on Hilbert's 21-st problem and Bolibrukh counterexample -- Some comments on section 5.2 -- Characteristic properties of Weyl solutions for the Sturm-Liouville and Dirac equations. V. A. Marchenko's theorems -- Discrete levels in spectral gaps of perturbed Schrodinger and Hill operators -- Factorized phase matrix of the perturbed system and the discrete spectrum in gaps of the continuous spectrum -- Generalization of D'Alembert-Liouville-Ostrogradsky formula and its application to growth estimates for solutions on canonical almost periodic systems -- A reduction of the problem about the number of discrete levels in a finite spectral gap of one-dimensional Schrodinger operator to a similar problem for a semi-infinite gap of the transformed Schrodinger operator -- Applications to perturbations of Hill operators and Schrodinger operators with an almost periodic potential. Kneser-type constants and effective masses -- Discrete levels in spectral gaps of perturbed Hill operators -- Bibliographical comments -- Some comments on section 6.1 -- Some comments on section 6.2 -- Some comments on section 6.3 and differential inequalities -- The spectrum of self-adjoint periodic and almost periodic operators. Marchenko-Ostrovskiy and Pastur-Tkachenko theorems -- The specturm of non-self-adjoint operators. Batchenko-Gesztesy and Rofe-Beketov theorems -- The specturm of perturbed self-adjoint and non-self-adjoint operators -- Matrix operators or differential-algebraic operators and related topics -- The inverse Sturm-Liouville problem for the spectral matrix on the whole axis -- Three open questions -- Appendix A -- Self-adjoint extensions of differential operators on a finite interval in spaces of vector-functions -- Hermitian relations -- Canonical form of Hermitian relations -- Various forms of Hermitian relations. Dissipative and sectorial relations -- Self-adjoint boundary conditions for infinite systems of second order differential equations -- Self-adjoint extensions of differential operators of arbitrary order with operator coefficients -- The expressions of order 2n -- The expression of order 2n-1 -- Continuous dependecne of intial data on solutions of differential equations -- Bibliographical comments -- M. L. Gorbachuk theory of generalized boundary values. Abstract boundary conditions -- Strong resolvent convergence of Hermitian relations and operators with non-dense domains and resolution of the identity. Stability of Eigenvalues -- Characteristic operator and boundary-value problems with separated boundary conditions -- Bibliography -- List of symbols -- Index."@en ;
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