Boundary Value Problems, Weyl Functions, and Differential Operators

Contents......Page 6
Preface......Page 9
1 Introduction......Page 10
1.1
Elementary facts about linear relations......Page 20
1.2
Spectra, resolvent sets, and points of regular type......Page 32
1.3
Adjoint relations......Page 39
1.4
Symmetric relations......Page 51
1.5
Self-adjoint relations......Page 57
1.6
Maximal dissipative and accumulative relations......Page 67
1.7
Intermediate extensions and von Neumann's formulas......Page 74
1.8
Adjoint relations and indefinite inner products......Page 83
1.9
Convergence of sequences of relations......Page 88
1.10
Parametric representations for relations......Page 96
1.11
Resolvent operators with respect to a bounded operator......Page 105
1.12
Nevanlinna families and their representations......Page 109
2.1
Boundary triplets......Page 116
2.2
Boundary value problems......Page 124
2.3 Associated γ-fields and Weyl functions......Page 127
2.4
Existence and construction of boundary triplets......Page 135
2.5
Transformations of boundary triplets......Page 143
2.6
Kreın's formula for intermediate extensions......Page 157
2.7
Kreın's formula for exit space extensions......Page 164
2.8
Perturbation problems......Page 172
3.1
Analytic descriptions of minimal supports of Borel measures......Page 177
3.2
Growth points of finite Borel measures......Page 186
3.3
Spectra of self-adjoint relations......Page 191
3.4
Simple symmetric operators......Page 196
3.5
Eigenvalues and eigenspaces......Page 204
3.6
Spectra and local minimality......Page 211
3.7
Limit properties of Weyl functions......Page 220
3.8
Spectra and local minimality for self-adjoint extensions......Page 226
4.1
Reproducing kernel Hilbert spaces......Page 230
4.2
Realization of uniformly strict Nevanlinna functions......Page 242
4.3
Realization of scalar Nevanlinna functions via L2-space models......Page 259
4.4
Realization of Nevanlinna pairs and generalized resolvents......Page 268
4.5
Kreın's formula for exit space extensions......Page 277
4.6
Orthogonal coupling of boundary triplets......Page 281
6 Boundary Triplets and Boundary Pairs for Semibounded Relations......Page 287
5.1
Closed semibounded forms and their representations......Page 288
5.2
Ordering and monotonicity......Page 306
5.3
Friedrichs extensions of semibounded relations......Page 317
5.4
Semibounded self-adjoint extensions and their lower bounds......Page 325
5.5
Boundary triplets for semibounded relations......Page 338
5.6
Boundary pairs and boundary triplets......Page 349
7 Sturm–Liouville Operators......Page 371
6.1
Sturm–Liouville differential expressions......Page 372
6.2
Maximal and minimal Sturm–Liouville differential operators......Page 386
6.3
Regular and limit-circle endpoints......Page 394
6.4
The case of one limit-point endpoint......Page 403
6.5
The case of two limit-point endpoints and interface conditions......Page 418
6.6
Exit space extensions......Page 427
6.7
Weyl functions and subordinate solutions......Page 431
6.8
Semibounded Sturm–Liouville expressions in the regular case......Page 440
6.9
Closed semibounded forms for Sturm–Liouville equations......Page 448
6.10
Principal and nonprincipal solutions of Sturm–Liouville equations......Page 460
6.11
Semibounded Sturm–Liouville operators and the limit-circle case......Page 475
6.12
Semibounded Sturm–Liouville operators and the limit-point case......Page 483
6.13
Integrable potentials......Page 489
8 Canonical Systems of Differential Equations......Page 505
7.1
Classes of integrable functions......Page 506
7.2
Canonical systems of differential equations......Page 510
7.3
Regular and quasiregular endpoints......Page 516
7.4
Square-integrability of solutions of real canonical systems......Page 519
7.5
Definite canonical systems......Page 526
7.6
Maximal and minimal relations for canonical systems......Page 531
7.7
Boundary triplets for the limit-circle case......Page 540
7.8
Boundary triplets for the limit-point case......Page 549
7.9
Weyl functions and subordinate solutions......Page 565
7.10
Special classes of canonical systems......Page 572
8.1
Rigged Hilbert spaces......Page 583
8.2
Sobolev spaces, C2-domains, and trace operators......Page 587
8.3
Trace maps for the maximal Schrödinger operator......Page 594
8.4
A boundary triplet for the maximal Schrödinger operator......Page 606
8.5
Semibounded Schrödinger operators......Page 617
8.6
Coupling of Schrödinger operators......Page 622
8.7
Bounded Lipschitz domains......Page 630
A.1
Borel transforms and their Stieltjes inversion......Page 637
A.2
Scalar Nevanlinna functions......Page 642
A.3
Operator-valued integrals......Page 651
A.4
Operator-valued Nevanlinna functions......Page 661
A.5
Kac functions......Page 669
A.6
Stieltjes and inverse Stieltjes functions......Page 674
Self-adjoint Operators and Fourier Transforms......Page 683
B.1
The scalar case......Page 684
B.2
The vector case......Page 691
Sums of Closed Subspaces in Hilbert Spaces......Page 696
Factorization of Bounded Linear Operators......Page 704
Notes......Page 710
Bibliography......Page 725
List of Symbols......Page 769
Index......Page 772