Direct Methods of Qualitative Spectral Analysis of Singular Differential Operators

Title......Page 1
Copyright Page......Page 2
Contents......Page 3
Annotation......Page 7
Preface......Page 9
INTRODUCTION......Page 11
1. Classification of the points of the spectrum of a closed linear operator......Page 16
2. Extension spectra and decomposition of closed linear operators.......Page 20
3. Spectrum of a self-adjoint operator and trial manifolds......Page 21
4. Relative complete continuity of symmetrical operators......Page 27
5. Generalizations of H. Weyl's theorem on completely continuous perturbations......Page 31
6. On the square of a closed symmetric operator with a finite defect number......Page 34
7. Decomposition of operators and properties of the spectrum independent of the behavior of the coefficients at finite distances......Page 37
8. Small and relatively small perturbations of differential operations......Page 39
9. Test systems of finite functions and the continuous part of the spectrum of a singular operator as the set of accumulation points of the spectra of regular operators......Page 42
10. The quadratic functional, and general criteria for the discreteness of the negative part of the spectrum......Page 44
11. Localization principles......Page 47
12. The negative part of the spectrum and the oscillatory properties of differential operations......Page 50
13. On differential operations on vector-functions......Page 53
14. A finite-difference analog of the one-dimensional self-adjoint singular differential operator......Page 54
15. Inclusion of multidimensional boundary-value problems in the general theory of symmetric operators......Page 57
16. On the defect index of partial differential operators......Page 64
17. Decomposition in multidimensional domains......Page 69
18. Small and relatively small perturbations in the multi-dimensional case......Page 73
19. The negative part of the spectrum and the quadratic functional......Page 75
20. S. L. Sobolev's spaces and the decomposition principle......Page 80
21. The domain of values of a quadratic functional and the external regularity field of a linear operator......Page 84
22. 1 -symmetric operators and their I-self-adjoint extensions......Page 86
23. Some spectral properties of 1-self-adjoint extensions of I -symmetric operators with a finite defect number......Page 87
24. 1 -symmetric differential operators......Page 89
25. Two-term differential operations......Page 92
26. Some auxiliary inequalities......Page 93
27. Criteria for the boundedness and discreteness of the negative part of the spectrum......Page 95
28. Criteria for the semi-boundedness and discreteness of the whole spectrum......Page 99
29. Spectrum discreteness criteria for the case of the polar differential operation......Page 103
30. Criteria for non-oscillatory and oscillatory behavior......Page 105
31. The positive part of the continuous spectrum......Page 116
32. The spectral alternative for semi-bounded Schroedinger operators......Page 123
33. The case when the spectrum of the Schroedinger operator covers the entire axis......Page 125
34. Examples of the study of the character of the spectrum in the case of a one-term operation......Page 128
35. On the spectrum of the Schroedinger operator with a complex-valued potential energy......Page 131
36. The spectrum of two-term differential operators on vector functions......Page 134
37. A discrete analog of the oscillation theorems......Page 135
39. Criteria for the semi-boundedness of the spectrum......Page 139
40. Domains of non-oscillatory and oscillatory behavior......Page 142
41. On gaps in the continuous part of the spectrum......Page 144
42. Continuity criteria for the positive part of the spectrum......Page 145
43. Birman's theorem......Page 147
44. M. G. Krein's theorem on the multiplicative structure of positive differential operators......Page 150
46. Theorems on the discrete part of the spectrum......Page 156
47. On criteria for the semi -boundedness and discreteness of the spectrum of multidimensional boundary-value problems......Page 162
48. The negative part of the spectrum and the nodal lines of the solutions of elliptic equations......Page 165
49. The character of the continuous part of the spectrum as related to the behavior of the potential and to the form of the infinite domain......Page 169
50. On the discrete part of the spectrum of the Laplace operator in limit-cylindrical domains......Page 177
51. Formulation of the problem......Page 185
52. Some a priori estimates for the solution of the homogeneous elliptic equation......Page 186
53. Behavior of the solutions at points of the discrete part of the spectrum......Page 188
54. An estimate of the distance of a point from the spectrum by the behavior of the solution at this point......Page 191
55. The behavior of the solutions at points of the continuous part of the spectrum......Page 193
56. The spectrum of the Hill differential operator......Page 197
57. The Hartman-Putnam theorem on the gaps in the spectrum of operators with a bounded potential......Page 201
58. On V. A. Steklov's problem in the theory of orthogonal polynomials and its continual analog......Page 205
59. Various generalizations of A. M. Molchanov's criterion.......Page 209
60. The negative part of the spectrum of multidimensional two-term higher-order operators......Page 211
61. Algebraic investigation of the spectrum of the energy operator of a harmonic oscillator......Page 212
62. The Dirac operator......Page 215
63. The Pauli operator......Page 217
64. The spectrum of the energy operator of a many-particle system......Page 219
65. The structure of the continuous part of the spectrum of the Schroedinger operator......Page 221
66. Complex potentials......Page 224
67. The model of the crystal lattice......Page 226
LIST OF REFERENCES......Page 230
INDEX......Page 243