  We study the spectrum of one dimensional integral operators in bounded real
intervals of length $2L$, for value of $L$ large. The integral operators are
obtained by linearizing a non local evolution equation for a non conserved
order parameter describing the phases of a fluid. We prove a Perron-Frobenius
theorem showing that there is an isolated, simple minimal eigenvalue strictly
positive for $L$ finite, going to zero exponentially fast in $L$. We lower
bound, uniformly on $L$, the spectral gap by applying a generalization of the
Cheeger's inequality. These results are usefulfor deriving spectral properties
for non local Cahn-Hilliard type of equations in problems of interface
dynamics.
