Commutation relations for unitary operators. I
There is considered a class of unitary operators \(U\), called propagating unitary operators, satisfying a certain commutation relation with an auxiliary selfadjoint operator \(A\). One main result (Theorem 2.3) is: If \(\Theta\) is a subset of the torus \(\mathbb{R}/2\pi\mathbb{Z}\), then \(e^{i\Theta}\cup \sigma(U)\) consists only of a finite number of eigenvalues with finite multiplicity. Certain matrix elements of \((\mathbf{1}-z^{-1}U^*)^{-1}\) and \((\mathbf{1}-\overline{z}^{-1}U^*)^{-1}\) are continuously extendable to \(e^{i\Theta}\setminus \sigma_{pp}(U)\). Moreover it is proved that \(e^{i\Theta}\cap \sigma_{sc}(U)=\emptyset\). The proof of this theorem given in Section 4 uses the Virial theorem and Mourre inequalities. These abstract results are applied to Floquet operators associated to AC-Stark perturbations of the harmonic oscillator on \(L^2(\mathbb{R})\). New results are contained in Theorems 3.1 and 3.2. They show the absence of singular continuous spectra for Floquet operators associated to time-dependent Hamiltonians perturbed by certain multiplication operators. The remaining sections treat regularity classes, Weyl commutation relations and \(U\)-smoothness in a little more detail. Part II has appeared in [J. Approx. Theory 199, Article ID 5016, 63--94 (2015; Zbl 1333.81145)].
scientific article; zbMATH DE number 6475680