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We study spectral properties of convolution operators $\mathcal L$ and theirperturbations $H=\mathcal L+v(x)$ by compactly supported potentials. Resultsare applied to determine the front propagation of a population density governedby operator $H$ with a compactly supported initial density provided that $H$has positive eigenvalues. If there is no positive spectrum, then thestabilization of the population density is proved.

Spectral analysis of non-local Schr\"odinger operators