Index Absence of positive eigenvalues, 231, 399 Absolutely continuous subspace, 19 Admissible functions, 28 Agmon-H¨ ormander spaces, 235 Analytic Fredholm alternative, 33 Asymptotic completeness, see also Wave op- erators Averaged scattering cross section, 331 high-energy asymptotics, 337 semiclassical asymptotics, 336 universal upper bounds, 331 Birman-Kato-Kre˘ ın theorem, 42 Birman-Kre˘ ın formula, 47 Boundary values of the resolvent of a self- adjoint operator, 44 Boundedness of integral operators, 68 Break-down of completeness, 91 additional channels of scattering, 91 Cauchy integral, 358 Compact operators, 20 singular numbers, 20 Completeness of wave operators, see also Asymptotic completeness Complex interpolation, 22 Hadamard three-line theorem, 22 three-line theorem for operator-valued functions in Sp, 22 Conditions of self-adjointness, 24 Cook’s criterion, 29 Determinant, 21 regularized determinant, 21 Diagonalization of a self-adjoint operator, 20 Differential operators, 53, 136 with constant coefficients, 53, 136 perturbations, 54, 136 spectral analysis, 54, 109 Dirac operator, 106 Direct integral, 19 Eikonal approximation, 319 Exceptional set N , 33 Expansion theorem, 248 a generalized Fourier integral, 255 relation to the wave operators, 248 standing waves, 250 the diagonalization of the Hamiltonian, 245 Free Hamiltonian, 75 resolvent, 78 spectral representation, 75 unitary group, 76 Friedrichs’ extension, 25 Generalized Fourier transform, 170 in the one-dimensional problem, 214 relation with wave operators, 215 on half-line, 170 relation with wave operators, 175 Hamiltonian of a relativistic spinless parti- cle, 104 Hardy-Rellich inequalities, 68 High-energy asymptotic expansion of the re- solvent, 277 away from the spectrum, 277 in the whole complex plane, 361 High-energy asymptotics of scattering data, 188 in the one-dimensional problem, 219 on half-line, 188 High-energy asymptotics of the spectral shift function, see also spectral shift function High-energy estimates of the resolvent, 268 the free case, 268 Hilbert identity, 17 Homogeneous Schr¨ odinger equation, 241 an exhaustive description of all solutions, 241 Integral operators, 133 from Schatten-von Neumann classes, 133 from trace class, 133 Intertwining property, 173 Invariance principle, 29 for perturbations of trace class type, 43 441
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