Spectral analysis of differential operators: interplay between spectral and oscillatory properties
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New Jersey [u.a.]
World Scientific
2005
|
| Series: | World scientific monograph series in mathematics; 7
7 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XXIII, 438 S. |
| ISBN: | 9812562761 |
Staff View
MARC
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| 245 | 1 | 0 | |a Spectral analysis of differential operators |b interplay between spectral and oscillatory properties |c Fedor S. Rofe-Beketov ; Aleksandr M. Kholkin |
| 264 | 1 | |a New Jersey [u.a.] |b World Scientific |c 2005 | |
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Record in the Search Index
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| adam_text | Contents
Foreword
v
Preface
xiii
Introduction
xxi
1.
Relation Between Spectral and Oscillatory Properties for
the Matrix Sturm-Liouville Problem
1
1.1
Problem on a Finite Interval
................. 1
1.2
Problem on the Half-Line
................... 16
1.3
Bibliographical Comments
................... 24
1.3.1
An Example of the Operator of
(І.І)-(І.З)
Type with
the Purely Absolutely Continuous Spectrum
..... 27
1.3.2
Oscillatory Theory of Partial Differential Equations
. 29
1.3.3
Bochner Integral
.................... 33
2.
Fundamental System of Solutions for an Operator Differ¬
ential Equation with a Singular Boundary Condition
35
2.1
Separated Self-Adjoint Boundary Conditions
........ 35
2.2
A Construction for the Fundamental Solution of the
Boundary-Value Problem
................... 39
2.3
Self-Consistency of the Fundamental Solution of the Self-
Adjoint Problem and the Evolution of the Corresponding
Hermitian Relation (or Lagrangian Plane in
Я
θ
Я)
.... 48
2.4
A Different Construction for the Fundamental Solution of
the Boundary-Value Problem
................. 58
-viii Spectral
Analysis of Differential Operators
2.5
Bibliographical Comments
................... 69
3.
Dependence of the Spectrum of Operator Boundary Prob¬
lems on Variations of a Finite or Semi-Infinite Interval
71
3.1
Dependence of Eigenvalues and the Greatest Lower Bound
of the Spectrum for a Semi-Bounded Below Differential
Operator on Variations of the Interval
............ 71
3.2
Continuity and Monotonicity of the Greatest Lower Bound of
the Essential Spectrum for Semi-Bounded Below Differential
Operators
............................ 76
3.3
Bibliographical Comments
................... 84
4.
Relation Between Spectral and Oscillatory Properties for
Operator Differential Equations of Arbitrary Order
89
4.1
Oscillatory Theorem for Operator Differential Equations of
Even Order
........................... 89
4.2
Comparison and Alternation Theorems
............ 95
4.3
Multiplicative Representation of Positive Differential Oper¬
ators and Its Applications
................... 99
4.4
Discrete Levels in the Gaps of the Essential Spectrum
. . . 107
4.5
Bibliographical Comments
................... 113
4.5.1
Symplectic Interpretation of Sturm-Type Theorems
and Their Operator-Theoretical Proofs
........ 114
5.
Self-Adjoint Extensions of Systems of Differential Equa¬
tions of Arbitrary Order on an Infinite Interval in the
Absolutely Indefinite Case
125
5.1
A Description of Self-Adjoint Extensions of Differential Op¬
erators of Arbitrary Order with Operator-Valued Coefficients
on an Infinite Interval
..................... 125
5.2
Parametrization of the Characteristic Operator Function
. . 135
5.3
Bibliographical Comments
................... 140
5.3.1
Everitt-Zettl Problem, Brusentsev s Example and
Two Open Questions
.................. 141
5.3.2
On the Deficiency Indices of Scalar Symmetric Dif¬
ferential Operators of General Kind on the Half-Axis
(Solved and Unsolved Questions)
........... 143
Contents
ix
5.3.3
Kogan-Rofe-Beketov s Asymptotic Theorems and
Deficiency Indices of Symmetric Differential Opera¬
tors
............................ 147
5.3.4
R. C
Gilbert s Class of Formally Self-Adjoint Ordi¬
nary Differential Operators Whose Deficiency Num¬
bers Differ by an Arbitrary
Pre-
Assigned Positive In¬
teger
........................... 156
5.3.5
Deficiency Indices of Symmetric Differential Systems
of the First and Arbitrary Orders and Some Open
Questions
........................ 159
5.3.6
Some Comments on Hubert s 21-st Problem and Boli-
brukh Counterexample
................. 168
5.3.7
Some Comments on Section
5.2............ 174
5.3.8
Characteristic Properties of Weyl Solutions for the
Sturm-Liouville and Dirac Equations.
V. A. Marchenko s Theorems
............. 175
6.
Discrete Levels in Spectral Gaps of Perturbed
Schrödinger
and Hill Operators
183
6.1
Factorized Phase Matrix of the Perturbed System and the
Discrete Spectrum in Gaps of the Continuous Spectrum
. . 183
6.2
Generalization of D Alembert-Liouville-Ostrogradsky For¬
mula and Its Application to Growth Estimates for Solutions
of Canonical Almost Periodic Systems
............ 192
6.3
A Reduction of the Problem about the Number of Discrete
Levels in a Finite Spectral Gap of One-Dimensional
Schrödinger
Operator to a Similar Problem for a Semi-
Infinite Gap of the Transformed
Schrödinger
Operator
. . . 204
6.4
Applications to Perturbations of Hill Operators and
Schrödinger
Operators with an Almost Periodic Potential.
Kneser-Type Constants and Effective Masses
........ 208
6.5
Discrete Levels in Spectral Gaps of Perturbed Hill Operators
223
6.6
Bibliographical Comments
................... 232
6.6.1
Some Comments on Section
6.1............ 232
6.6.2
Some Comments on Section
6.2............ 233
6.6.3
Some Comments on Section
6.3
and Differential In¬
equalities
........................ 234
χ
Spectral
Analysis of Differential Operators
6.6.4
The Spectrum of Self-Adjoint Periodic and Al¬
most Periodic Operators. Marchenko-Ostrovskiy and
Pastur-Tkachenko Theorems
.............. 235
6.6.5
The Spectrum of Non-Self-Adjoint Operators.
Batchenko-Gesztesy and Rofe-Beketov Theorems
. . 249
6.6.6
The Spectrum of Perturbed Self-Adjoint and Non-
Self-Adjoint Operators
................. 266
6.6.7
Matrix Operators or Differential-Algebraic Operators
and Related Topics
................... 272
6.6.8
The Inverse Sturm-Liouville Problem for the Spectral
Matrix on the Whole Axis
............... 275
6.6.9
Three Open Questions
................. 281
Appendix A Seif-Adjoint Extensions of Differential Opera¬
tors on a Finite Interval in Spaces of Vector-Functions
287
A.I Hermitian Relations
...................... 288
A.I.I Canonical Form of Hermitian Relations
........ 288
A.
1.2
Various Forms of Hermitian Relations. Dissipative
and
Sectorial
Relations
................. 292
A.2 Self-Adjoint Boundary Conditions for Infinite Systems of
Second Order Differential Equations
............. 305
A.3 Self-Adjoint Extensions of Differential Operators of Arbi¬
trary Order with Operator Coefficients
............ 311
A.3.1 The Expressions of Order 2n
.............. 311
A.3.2 The Expressions of Order 2ra
- 1........... 314
A.3.3 Continuous Dependence of Initial Data on Solutions
of Differential Equations
................ 327
A.4 Bibliographical Comments
................... 330
A.4.1
M. L.
Gorbachuk Theory of Generalized Boundary
Values. Abstract Boundary Conditions
........ 332
A.4.2 Strong Resolvent Convergence of Hermitian Relations
and Operators with Non-Dense Domains and Resolu¬
tion of the Identity. Stability of Eigenvalues
.....341
A.4.3 Characteristic Operator and Boundary-Value Prob¬
lems with Separated Boundary Conditions
...... 354
Contents xi
Bibliography
359
List of Symbols
431
Index
433
|
| adam_txt |
Contents
Foreword
v
Preface
xiii
Introduction
xxi
1.
Relation Between Spectral and Oscillatory Properties for
the Matrix Sturm-Liouville Problem
1
1.1
Problem on a Finite Interval
. 1
1.2
Problem on the Half-Line
. 16
1.3
Bibliographical Comments
. 24
1.3.1
An Example of the Operator of
(І.І)-(І.З)
Type with
the Purely Absolutely Continuous Spectrum
. 27
1.3.2
Oscillatory Theory of Partial Differential Equations
. 29
1.3.3
Bochner Integral
. 33
2.
Fundamental System of Solutions for an Operator Differ¬
ential Equation with a Singular Boundary Condition
35
2.1
Separated Self-Adjoint Boundary Conditions
. 35
2.2
A Construction for the Fundamental Solution of the
Boundary-Value Problem
. 39
2.3
Self-Consistency of the Fundamental Solution of the Self-
Adjoint Problem and the Evolution of the Corresponding
Hermitian Relation (or Lagrangian Plane in
Я
θ
Я)
. 48
2.4
A Different Construction for the Fundamental Solution of
the Boundary-Value Problem
. 58
-viii Spectral
Analysis of Differential Operators
2.5
Bibliographical Comments
. 69
3.
Dependence of the Spectrum of Operator Boundary Prob¬
lems on Variations of a Finite or Semi-Infinite Interval
71
3.1
Dependence of Eigenvalues and the Greatest Lower Bound
of the Spectrum for a Semi-Bounded Below Differential
Operator on Variations of the Interval
. 71
3.2
Continuity and Monotonicity of the Greatest Lower Bound of
the Essential Spectrum for Semi-Bounded Below Differential
Operators
. 76
3.3
Bibliographical Comments
. 84
4.
Relation Between Spectral and Oscillatory Properties for
Operator Differential Equations of Arbitrary Order
89
4.1
Oscillatory Theorem for Operator Differential Equations of
Even Order
. 89
4.2
Comparison and Alternation Theorems
. 95
4.3
Multiplicative Representation of Positive Differential Oper¬
ators and Its Applications
. 99
4.4
Discrete Levels in the Gaps of the Essential Spectrum
. . . 107
4.5
Bibliographical Comments
. 113
4.5.1
Symplectic Interpretation of Sturm-Type Theorems
and Their Operator-Theoretical Proofs
. 114
5.
Self-Adjoint Extensions of Systems of Differential Equa¬
tions of Arbitrary Order on an Infinite Interval in the
Absolutely Indefinite Case
125
5.1
A Description of Self-Adjoint Extensions of Differential Op¬
erators of Arbitrary Order with Operator-Valued Coefficients
on an Infinite Interval
. 125
5.2
Parametrization of the Characteristic Operator Function
. . 135
5.3
Bibliographical Comments
. 140
5.3.1
Everitt-Zettl Problem, Brusentsev's Example and
Two Open Questions
. 141
5.3.2
On the Deficiency Indices of Scalar Symmetric Dif¬
ferential Operators of General Kind on the Half-Axis
(Solved and Unsolved Questions)
. 143
Contents
ix
5.3.3
Kogan-Rofe-Beketov's Asymptotic Theorems and
Deficiency Indices of Symmetric Differential Opera¬
tors
. 147
5.3.4
R. C
Gilbert's Class of Formally Self-Adjoint Ordi¬
nary Differential Operators Whose Deficiency Num¬
bers Differ by an Arbitrary
Pre-
Assigned Positive In¬
teger
. 156
5.3.5
Deficiency Indices of Symmetric Differential Systems
of the First and Arbitrary Orders and Some Open
Questions
. 159
5.3.6
Some Comments on Hubert's 21-st Problem and Boli-
brukh Counterexample
. 168
5.3.7
Some Comments on Section
5.2. 174
5.3.8
Characteristic Properties of Weyl Solutions for the
Sturm-Liouville and Dirac Equations.
V. A. Marchenko's Theorems
. 175
6.
Discrete Levels in Spectral Gaps of Perturbed
Schrödinger
and Hill Operators
183
6.1
Factorized Phase Matrix of the Perturbed System and the
Discrete Spectrum in Gaps of the Continuous Spectrum
. . 183
6.2
Generalization of D'Alembert-Liouville-Ostrogradsky For¬
mula and Its Application to Growth Estimates for Solutions
of Canonical Almost Periodic Systems
. 192
6.3
A Reduction of the Problem about the Number of Discrete
Levels in a Finite Spectral Gap of One-Dimensional
Schrödinger
Operator to a Similar Problem for a Semi-
Infinite Gap of the Transformed
Schrödinger
Operator
. . . 204
6.4
Applications to Perturbations of Hill Operators and
Schrödinger
Operators with an Almost Periodic Potential.
Kneser-Type Constants and Effective Masses
. 208
6.5
Discrete Levels in Spectral Gaps of Perturbed Hill Operators
223
6.6
Bibliographical Comments
. 232
6.6.1
Some Comments on Section
6.1. 232
6.6.2
Some Comments on Section
6.2. 233
6.6.3
Some Comments on Section
6.3
and Differential In¬
equalities
. 234
χ
Spectral
Analysis of Differential Operators
6.6.4
The Spectrum of Self-Adjoint Periodic and Al¬
most Periodic Operators. Marchenko-Ostrovskiy and
Pastur-Tkachenko Theorems
. 235
6.6.5
The Spectrum of Non-Self-Adjoint Operators.
Batchenko-Gesztesy and Rofe-Beketov Theorems
. . 249
6.6.6
The Spectrum of Perturbed Self-Adjoint and Non-
Self-Adjoint Operators
. 266
6.6.7
Matrix Operators or Differential-Algebraic Operators
and Related Topics
. 272
6.6.8
The Inverse Sturm-Liouville Problem for the Spectral
Matrix on the Whole Axis
. 275
6.6.9
Three Open Questions
. 281
Appendix A Seif-Adjoint Extensions of Differential Opera¬
tors on a Finite Interval in Spaces of Vector-Functions
287
A.I Hermitian Relations
. 288
A.I.I Canonical Form of Hermitian Relations
. 288
A.
1.2
Various Forms of Hermitian Relations. Dissipative
and
Sectorial
Relations
. 292
A.2 Self-Adjoint Boundary Conditions for Infinite Systems of
Second Order Differential Equations
. 305
A.3 Self-Adjoint Extensions of Differential Operators of Arbi¬
trary Order with Operator Coefficients
. 311
A.3.1 The Expressions of Order 2n
. 311
A.3.2 The Expressions of Order 2ra
- 1. 314
A.3.3 Continuous Dependence of Initial Data on Solutions
of Differential Equations
. 327
A.4 Bibliographical Comments
. 330
A.4.1
M. L.
Gorbachuk Theory of Generalized Boundary
Values. Abstract Boundary Conditions
. 332
A.4.2 Strong Resolvent Convergence of Hermitian Relations
and Operators with Non-Dense Domains and Resolu¬
tion of the Identity. Stability of Eigenvalues
.341
A.4.3 Characteristic Operator and Boundary-Value Prob¬
lems with Separated Boundary Conditions
. 354
Contents xi
Bibliography
359
List of Symbols
431
Index
433 |
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| id | DE-604.BV021325479 |
| illustrated | Not Illustrated |
| index_date | 2024-07-02T14:00:14Z |
| indexdate | 2024-07-09T20:35:42Z |
| institution | BVB |
| isbn | 9812562761 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-014645829 |
| oclc_num | 255536340 |
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| physical | XXIII, 438 S. |
| publishDate | 2005 |
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| series | World scientific monograph series in mathematics; 7 |
| series2 | World scientific monograph series in mathematics; 7 |
| spelling | Rofe-Beketov, Fedor S. Verfasser aut Spectral analysis of differential operators interplay between spectral and oscillatory properties Fedor S. Rofe-Beketov ; Aleksandr M. Kholkin New Jersey [u.a.] World Scientific 2005 XXIII, 438 S. txt rdacontent n rdamedia nc rdacarrier World scientific monograph series in mathematics; 7 Nichtlinearer Operator (DE-588)4225824-8 gnd rswk-swf Spektralanalyse Stochastik (DE-588)4056125-2 gnd rswk-swf Spektraltheorie (DE-588)4116561-5 gnd rswk-swf Spektralanalyse Stochastik (DE-588)4056125-2 s Nichtlinearer Operator (DE-588)4225824-8 s DE-604 Spektraltheorie (DE-588)4116561-5 s Kholkin, Aleksandr M. Verfasser aut World scientific monograph series in mathematics; 7 7 (DE-604)BV013389881 7 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014645829&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Rofe-Beketov, Fedor S. Kholkin, Aleksandr M. Spectral analysis of differential operators interplay between spectral and oscillatory properties World scientific monograph series in mathematics; 7 Nichtlinearer Operator (DE-588)4225824-8 gnd Spektralanalyse Stochastik (DE-588)4056125-2 gnd Spektraltheorie (DE-588)4116561-5 gnd |
| subject_GND | (DE-588)4225824-8 (DE-588)4056125-2 (DE-588)4116561-5 |
| title | Spectral analysis of differential operators interplay between spectral and oscillatory properties |
| title_auth | Spectral analysis of differential operators interplay between spectral and oscillatory properties |
| title_exact_search | Spectral analysis of differential operators interplay between spectral and oscillatory properties |
| title_exact_search_txtP | Spectral analysis of differential operators interplay between spectral and oscillatory properties |
| title_full | Spectral analysis of differential operators interplay between spectral and oscillatory properties Fedor S. Rofe-Beketov ; Aleksandr M. Kholkin |
| title_fullStr | Spectral analysis of differential operators interplay between spectral and oscillatory properties Fedor S. Rofe-Beketov ; Aleksandr M. Kholkin |
| title_full_unstemmed | Spectral analysis of differential operators interplay between spectral and oscillatory properties Fedor S. Rofe-Beketov ; Aleksandr M. Kholkin |
| title_short | Spectral analysis of differential operators |
| title_sort | spectral analysis of differential operators interplay between spectral and oscillatory properties |
| title_sub | interplay between spectral and oscillatory properties |
| topic | Nichtlinearer Operator (DE-588)4225824-8 gnd Spektralanalyse Stochastik (DE-588)4056125-2 gnd Spektraltheorie (DE-588)4116561-5 gnd |
| topic_facet | Nichtlinearer Operator Spektralanalyse Stochastik Spektraltheorie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014645829&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV013389881 |
| work_keys_str_mv | AT rofebeketovfedors spectralanalysisofdifferentialoperatorsinterplaybetweenspectralandoscillatoryproperties AT kholkinaleksandrm spectralanalysisofdifferentialoperatorsinterplaybetweenspectralandoscillatoryproperties |