The continuous spectrum and the effect of parametric resonance. The case of bounded operators

The paper is concerned with the Mathieu-type differential equation $u''=-A^2 u+\varepsilon B(t)u$ in a Hilbert space $H$. It is assumed that $A$ is a bounded self-adjoint operator which only has an absolutely continuous spectrum and $B(t)$ is almost periodic operator-valued function. Sufficient conditions are obtained under which the Cauchy problem for this equation is stable for small $\varepsilon$ and hence free of parametric resonance.
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