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Eigenfunctions, transfer matrices, and absolutely continuous spectrum of one-dimensional Schrödinger operators. (English) Zbl 0931.34066

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 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 A. Kiselev [Commun. Math. Phys. 194, No. 1, 1-45 (1998; Zbl 0912.34074)].

MSC:

34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
39A70 Difference operators
47B39 Linear difference operators

Citations:

Zbl 0912.34074
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