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The limiting absorption principle for partial differential operators. (English) Zbl 0624.35068

Mem. Am. Math. Soc. 364, 70 p. (1987).
From the authors’ abstract: Let H be a self-adjoint operator in a Hilbert space H. It is said to satisfy the “limiting absorption principle” (l.a.p.) in \(U\subset {\mathbb{R}}\) if the limits \(R^{\pm}(\lambda)=\lim_{\epsilon \to 0}(H-\lambda \mp i\epsilon)^{- 1}\), \(\lambda\in U\), exists in some operator topology of B(X,Y), \(X\subset H\), \(Y\subset X\). The paper presents a unified abstract approach to the l.a.p. for operators of the form \(H=H_ 0+V\). The spectral measure associated with \(H_ 0\) is assumed to satisfy certain smoothness assumptions which yields the l.a.p. The perturbation V is assumed to be short-range with respect to \(H_ 0\) and the l.a.p. is proved, along with the discreteness and finite multiplicity of its eigenvalues embedded in U. Various classes of differential operators are studied as special cases, including Schrödinger operators, generalizations of the Stark Hamiltonian and simply characteristic operators.
Reviewer: N.Jacob

MSC:

35P25 Scattering theory for PDEs
47A40 Scattering theory of linear operators
81Q15 Perturbation theories for operators and differential equations in quantum theory
81U05 \(2\)-body potential quantum scattering theory
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