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The paper deals with jump generators with a convolution kernel. Assuming thatthe kernel decays either exponentially or polynomially we prove a number oflower and upper bounds for the resolvent of such operators. We consider twoapplications of these results. First we obtain pointwise estimates forprincipal eigenfunction of jump generators perturbed by a compactly supportedpotential (so-called nonlocal Schr\"odinger operators). Then we consider theCauchy problem for the corresponding inhomogeneous evolution equations andstudy the behaviour of its solutions.

Resolvent bounds for jump generators and ground state asymptotics for nonlocal Schr\"{o}dinger operators