Eigenfunctions, transfer matrices, and absolutely continuous spectrum of one-dimensional Schrödinger operators
The authors give several criteria which identify the support of the absolutely continuous part of the spectrum of discrete Schrödinger operators of the form \(h : u\mapsto u(n+1)+u(n-1)+V(n)u(n)\), \(n\in {\mathbb{N}}\) or \({\mathbb{Z}}\) (with a boundary condition in the former case), in terms of bounds on the transfer matrix for \(hu=Eu\), i.e. the map \(T_E(n)\) which sends \((u(1),u(0))\) to \((u(n+1), u(n))\). A number of variants, and analogues for Schrödinger operators on \({\mathbb{R}}\), are given. \(V\) is generally assumed to be bounded. An application to almost-periodic potentials is treated. Furthermore, an argument by \textit{D. Ruelle} [Publ. Math., Inst. Hautes Etud. Sci. 50, 275-306 (1979)] is used to find a sufficient condition on \(\|T_E(n)\|\) which ensures the existence of a square integrable eigenfunction. For applications to sparse potentials, random potentials, and \(n^{-\alpha}\) potentials, the authors refer to a companion paper with \textit{A. Kiselev} [Commun. Math. Phys. 194, No. 1, 1-45 (1998; Zbl 0912.34074)].
discrete and continuous one-dimensional Schrödinger operators
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