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Floquet operators with singular spectrum. I. (English) Zbl 0689.34022

Let H be a positive definite, self-adjoint operator acting in a separable Hilbert space \({\mathcal H}\), having a discrete spectrum consisting of simple eigenvalues \(0<\lambda_ 0<\lambda_ 1<...\); let \(\{\) V(t)\(\}\) be a uniformly bounded measurable family of bounded operators which is periodic with period \(2\pi\), and define the Floquet operator \(K(\beta)=i(d/dt)+H+\beta V(t)\) on \({\mathcal K}=L_ 2[0,2a]\times {\mathcal H}\), with periodic boundary condition \(u(0)=u(2\pi)\). It is proved that if \(\lambda_{n+1}-\lambda_ n\) grows like \(n^{2+\epsilon}\) for some \(\epsilon >0\) then K(\(\beta)\) has a dense pure point spectrum for ‘almost every’ H in an appropriate probabilistic sense and for all \(\beta\).
Reviewer: W.D.Evans

MSC:

34L99 Ordinary differential operators

References:

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