Spectral analysis of non-local Schr\"odinger operators

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