Basic Notation In this book we consider a pair of self-adjoint operators (Hamiltonians) H0 and H acting in Hilbert spaces H0 and H, respectively. We denote by J a fixed (identification) operator acting from H0 to H, although usually H0 = H and J = I (the identity operator). The perturbation V = HJ JH0 can often be factored as the product V = G∗G0, where G0 : H0 G, G : H G, G is an auxiliary Hilbert space and G∗ is the operator adjoint to G. As a rule, the notation for various objects associated with H0 are endowed with the subindex “0”. Different objects pertaining to the absolutely continuous parts of operators H0 and H are endowed with the superindex “a”. In the notation of various function spaces, we often denote in brackets the set on which the functions considered are defined. In the case of vector-valued functions the space in which the functions have their range is usually indicated. The notation (a1,...,an)t means that we consider the vector (a1,...,an) Cn as a column. If not specified otherwise, a simple contour around some domain in the complex plane is passed in the positive direction it means that the domain remains to the left from this contour. The letters C and c denote various estimation constants whose precise value is immaterial. For a R, we set a+ = max{a, 0} and a− = a+−a. An equality containing the signs ± is always understood as two independent equalities. We usually use the letter x for the coordinate and the letter ξ for the mo- mentum (dual) variable ˆ = Φf is the Fourier transform of f. The operators of multiplication by functions are usually denoted by the same letters (often capital ones). To simplify notation, we sometimes write q1(x)q2(ξ) instead of the operator Q1Φ∗Q2Φ where Q1 and Q2 are multiplications by q1(x) and q2(ξ), respectively. Functions and ln z are always supposed to be analytic in C \ R+ and their branches are fixed by the condition arg z = π for z 0. We also set (−z)β = e−πiβzβ and ln(−z) = ln z πi. In asymptotic expansions, estimates of remainders are usually clear from the context, but often they are specified explicitly. An asymptotic expansion can be differentiated with respect to some parameters, if estimates of the corresponding remainders can be differentiated. We use the following: Abbreviations a.e. almost every LAP limiting absorption principle PD perturbation determinant PDO pseudodifferential operator SM scattering matrix 1
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