Lyapunov numbers for the almost periodic Schrödinger equation
type: Item
instance of: scholarly article
title: Lyapunov numbers for the almost periodic Schrödinger equation
zbMATH Open document ID: 0558.34029
author: Russell Johnson
published in: Illinois Journal of Mathematics
publication date: 1984-01-01T00:00:00Z
review text: Consider the almost periodic Schrödinger operator \(L=-d^ 2/dt^ 2+q(t)\), where q(t) is continuous and Bohr almost periodic. Let F denote the essential spectrum of L and let \(\beta (\lambda)=\sup_{\phi \neq 0}\{\lim_{t\to \infty}(1/2t)\ell n[\phi^ 2(t)+\phi '{}^ 2(t)]\}\) for \(\lambda\in {\mathbb{C}}\). The sup being taken over all solutions \(\phi\) of the equation \((-d^ 2/dt^ 2+\omega (t))\phi =\lambda \phi\) and all \(\omega\) in the hull of q (definition of hull as given in {\S}2 of the paper). Discussion of some of the properties of \(\beta\) is given. The following two important properties are pointed out specially and proved. ''First, \(\beta\) is harmonic on the resolvent set \({\mathbb{C}}\setminus F\) of the operator L. This fact is then used to prove that, if I is an open interval such that \(F\cap I\neq \emptyset\), then \(F\cap I\) has positive logarithmic capacity. Second, \(\beta\) is one sided continuous at an end point \(\lambda\) of a spectral gap: if \(\lambda_ n\in {\mathbb{R}}\setminus F\) and \(\lambda_ n\to \lambda\) then \(\beta (\lambda_ n)\to \beta (\lambda).''\)
Mathematics Subject Classification ID: 34B30
zbMATH DE Number: 3889940
zbMATH Keywords: almost periodic Schrödinger operator
zbMATH Keywords: essential spectrum
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 3889940