  We consider a general elliptic operator in an infinite multi-dimensional
cylinder with several distant perturbations; this operator is obtained by
``gluing'' several single perturbation operators $\mathcal{H}^{(k)}$,
$k=1,\ldots,n$, at large distances. The coefficients of each operator
$\mathcal{H}^{(k)}$ are periodic in the outlets of the cylinder; the structure
of these periodic parts at different outlets can be different. We consider a
point $\lambda_0 \in\mathds{R}$ in the essential spectrum of the operator with
several distant perturbations and assume that this point is not in the
essential spectra of middle operators $\mathcal{H}^{(k)}$, $k=2,\ldots,n-1$,
but is an eigenvalue of at least one of $\mathcal{H}^{(k)}$, $k=1,\ldots,n$.
Under such assumption we show that the operator with several distant
perturbations possesses finitely many resonances in the vicinity of
$\lambda_0$. We find the leading terms in asymptotics for these resonances,
which turn out to be exponentially small. We also conjecture that the made
assumption selects the only case, when the distant perturbations produce
finitely many resonances in the vicinity of $\lambda_0$. Namely, as $\lambda_0$
is in the essential spectrum of at least one of operators $\mathcal{H}^{(k)}$,
$k=2,\ldots,n-1$, we do expect that infinitely many resonances emerge in the
vicinity of $\lambda_0$.
