Lyapunov numbers for the almost periodic Schrödinger equation (Q762674)

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 \(\beta (\lambda_ n)\to \beta (\lambda).''\)