Abstract
Let H be a self-adjoint operator in a Hubert space \(H,R(z) = {(H - z)^{ - 1}}\) its resolvent and λ a real number in the spectrum of H. \(Since\left\| {R(\lambda + i\mu } \right\| = {\left| \mu \right|^{ - 1}},R(\lambda + i\mu )\) cannot have limits in B(H) as μ → ±0. However, for certain vectors f ∈ H, the function \(F(z) = \left\langle {f,R(z)f} \right\rangle \), which is defined and holomorphic for z outside the spectrum of H, could have a limit as z converges to λ from the upper or lower half-plane (these two limits will be different in general). If this happens for sufficiently many f, one can infer results on the spectral properties of H which are useful for example in scattering theory. This will be elaborated in Section 7.1. The remainder of the chapter is devoted to a detailed description of a method, called the “conjugate operator method”, for proving the existence of such limits.
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© 1996 Springer Basel AG
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Amrein, W.O., de Monvel, A.B., Georgescu, V. (1996). The Conjugate Operator Method. In: C 0-Groups, Commutator Methods and Spectral Theory of N-Body Hamiltonians. Progress in Mathematics, vol 135. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-7762-6_7
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DOI: https://doi.org/10.1007/978-3-0348-7762-6_7
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-7764-0
Online ISBN: 978-3-0348-7762-6
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