Spectral properties of a coupled system of Schrödinger equations with time-periodic coefficients

Abstract

We consider a coupled system of Schrödinger equations with time-periodic coefficients \begin{eqnarray*} i\, u_t = -\Delta u + V(x,t)u + g(x,t)v \\ i\, v_t = -\Delta v + W(x,t)v + g(x,t)u \end{eqnarray*} on the Hilbert space $ \mathcal H = L^2({\mathbb R}^n) \times L^2({\mathbb R}^n)$, where $g$, $V$ and $W$ are periodic time-dependent potentials, with period $T$. We denote by $(U(t,s))_{(t,s) \in {\mathbb R}\times {\mathbb R}}$ its associated propagator. By using a multiplier method, we rule out the existence of regular eigenvectors of the Floquet operator $U(T,0)$.