Spectral properties of unitary weighted translation operators (Q1813421)

The author considers operators in \(L^ 2(0,1)\) of the form \[ (Tf)(x)=e^{iM(x)} f(x+\alpha), \] where \(M\) is a real-valued measurable function, and \(\alpha\) is an irrational number. Necessary conditions for \(T\) to have a discrete (point) spectrum are obtained. These involve analytic properties of the function \(M\) and arithmetic properties of the number \(\alpha\). If \(M\in C^ 1[0,1]\), and if \(\alpha\) satisfies a certain arithmetic property, it is shown that a necessary condition for \(T\) to have an absolutely continuous spectrum is that \(M(1)-M(0)\) is a non-zero integer multiple of \(2\pi\). In the case when \(M(x)=\lambda x\) this necessary condition is also sufficient. Moreover if the linear function \(\lambda x\) is perturbed a little in a sufficiently ``good'' sense then also a necessary and sufficient condition involving \(\alpha\) is obtained for \(T\) to have an absolutely continuous spectrum. The results in this paper are obtained by estimating the Fourier coefficients of the spectral measure associated with the operator \(T\).