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

Last, Yoram; Simon, Barry

Invent. Math. 135, No. 2, 329-367 (1999).

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)].

Reviewer: Satyanad Kichenassamy (Leipzig)

MSC:

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

Keywords:

discrete and continuous one-dimensional Schrödinger operators; absolutely continuous spectrum; almost-periodic potentials

Citations:

Zbl 0912.34074