Introduction .................................................... 6
0.1 Short summary ........................................... 6
0.2 Introduction ............................................ 7
1 Preliminaries ............................................... 14
1.1 Notation ............................................... 14
1.2 Functions holomorphic in  + ........................... 14
1.3 Measure theory ......................................... 14
1.3.1 Vitali's theorem ................................ 15
1.3.2 Poisson integral ................................ 15
1.3.3 Fatou's theorem ................................. 16
1.3.4 Privalov's theorem .............................. 16
1.3.5 The set Λ(ƒ) .................................... 17
1.3.6 De la Vallee Poussin decomposition theorem ...... 17
1.3.7 Standard supports of measures ................... 18
1.4 Bounded operators ...................................... 19
1.5 Self-adjoint operators ................................. 20
1.6 Trace-class and Hilbert-Schmidt operators .............. 21
1.6.1 Schatten ideals ................................. 21
1.6.2 Trace-class operators ........................... 23
1.6.3 Hilbert-Schmidt operators ....................... 24
1.6.4 Fredholm determinant ............................ 24
1.6.5 The Birman-Koplienko-Solomyak inequality ........ 25
1.7 Direct integral of Hilbert spaces ...................... 25
1.8 Operator-valued holomorphic functions .................. 27
1.8.1 Operator-valued meromorphic functions ........... 29
1.8.2 Analytic Fredholm alternative ................... 29
1.9 The limiting absorption principle ...................... 29
2 Framed Hilbert space ........................................ 31
2.1 Definition ............................................. 32
2.2 Spectral triple associated with an operator on
a framed Hilbert space ................................. 34
2.3 Non-compact frames ..................................... 34
2.4 The set Λ(H0;F) and the matrix ø(λ) .................... 35
2.5 A core of the singular spectrum  \ Λ(H0;F) ............ 36
2.6 The Hilbert spaces  α(F) .............................. 37
2.6.1 Diamond conjugate ............................... 38
2.7 The trace-class matrix ø(λ+iy) ......................... 38
2.8 The Hilbert-Schmidt matrix η(λ+iy) ..................... 39
2.9 Eigenvalues αj(λ+iy) of η(λ+iy) ........................ 39
2.10 Zero and non-zero type indices ......................... 39
2.11 Vectors ej(λ+iy) ....................................... 39
2.12 Vectors ηj(λ+iy) ....................................... 40
4 N. Azamov
2.13 Unitary matrix e(λ+iy) ................................. 41
2.14 Vectors øj(λ+iy) ....................................... 41
2.15 The operator ελ+iy ...................................... 42
2.16 Vectors bj(λ+iy)  1 .................................. 44
3 The evaluation operator ελ .................................. 45
3.1 Definition of ελ ....................................... 45
3.2 ε is an isometry ....................................... 49
3.3 ε is unitary ........................................... 50
3.4 Diagonality of H0 in  ................................. 52
4 The resonance set R(λ;{Hr},F) ............................... 53
4.1 Resonance points of a path of operators ................ 53
4.2 Essentially regular points ............................. 57
5 Wave matrix ω±(λ;Hr,H0) ..................................... 58
5.1 Operators α±(λ;Hr,H0) .................................. 59
5.2 Definition of the wave matrix ω±(λ;Hr,H0) .............. 61
5.3 Multiplicative property of the wave matrix ............. 62
5.4 The wave operator ...................................... 66
6 Connection with the time-dependent definition of the wave
operator .................................................... 67
6.1. Time-dependent definition of the wave operator ......... 67
7 The scattering matrix ....................................... 71
7.1 Definition of the scattering matrix .................... 71
7.2 Stationary formula for the scattering matrix ........... 72
7.3 Infinitesimal scattering matrix ........................ 76
8 Absolutely continuous and singular spectral shift
functions ................................................... 78
8.1 Infinitesimal spectral flow ............................ 78
8.2 Absolutely continuous and singular spectral shift
functions .............................................. 83
8.3 Non-additivity of the singular spectral shift
function ............................................... 87
9 Pushnitski μ-invariant and singular spectral shift
function .................................................... 88
9.1 Spectral flow for unitary operators .................... 88
9.2 Absolutely continuous part of the Pushnitski
μ-invariant ............................................ 90
9.3 Pushnitski μ-invariant ................................. 91
9.4 M-function ............................................. 92
9.5 Smoothed spectral shift function ....................... 93
9.6 Pushnitski formula ..................................... 94
9.7 Singular part of μ-invariant ........................... 95
Appendix. Chronological exponential ............................ 98
A.l. Definition and main properties ............................ 98
References .................................................... 100
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