Resonances for Schrödinger operators on infinite cylinders and other products

Abstract

We study the resonances of Schr\"odinger operators on the infinite product $X=\mathbb{R}^d\times \mathbb{S}^1$, where $d$ is odd, $\mathbb{S}^1$ is the unit circle, and the potential $V\in L^\infty_c(X)$. This paper shows that at high energy, resonances of the Schr\"odinger operator $-\Delta +V$ on $X=\mathbb{R}^d\times \mathbb{S}^1$ which are near the continuous spectrum are approximated by the resonances of $-\Delta +V_0$ on $X$, where the potential $V_0$ given by averaging $V$ over the unit circle. These resonances are, in turn, given in terms of the resonances of a Schr\"odinger operator on $\mathbb{R}^d$ which lie in a bounded set. If the potential is smooth, we obtain improved localization of the resonances, particularly in the case of simple, rank one poles of the corresponding scattering resolvent on $\mathbb{R}^d$. In that case, we obtain the leading order correction for the location of the corresponding high energy resonances. In addition to direct results about the location of resonances, we show that at high energies away from the resonances, the resolvent of the model operator $-\Delta+V_0$ on $X$ approximates that of $-\Delta+V$ on $X$. If $d=1$, in certain cases this implies the existence of an asymptotic expansion of solutions of the wave equation. Again for the special case of $d=1$, we obtain a resonant rigidity type result for the zero potential among all real-valued potentials.