Spectral properties of integral operators in bounded, large intervals

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 needed for deriving spectral properties for non local Cahn-Hilliard type of equations in problems of interface dynamics, see [16].