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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 theunit circle, and the potential $V\in L^\infty_c(X)$. This paper shows that athigh energy, resonances of the Schr\"odinger operator $-\Delta +V$ on$X=\mathbb{R}^d\times \mathbb{S}^1$ which are near the continuous spectrum areapproximated 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, inturn, 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, weobtain improved localization of the resonances, particularly in the case ofsimple, rank one poles of the corresponding scattering resolvent on$\mathbb{R}^d$. In that case, we obtain the leading order correction for thelocation of the corresponding high energy resonances. In addition to directresults about the location of resonances, we show that at high energies awayfrom 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 thisimplies the existence of an asymptotic expansion of solutions of the waveequation. Again for the special case of $d=1$, we obtain a resonant rigiditytype result for the zero potential among all real-valued potentials.

Resonances for Schr\"odinger operators on infinite cylinders and other products