We consider Dirichlet exterior value problems related to a class of non-local
Schr\"odinger operators, whose kinetic terms are given in terms of Bernstein
functions of the Laplacian. We prove elliptic and parabolic
Aleksandrov-Bakelman-Pucci type estimates, and as an application obtain
existence and uniqueness of weak solutions. Next we prove a refined maximum
principle in the sense of Berestycki-Nirenberg-Varadhan, and a converse. Also,
we prove a weak anti-maximum principle in the sense of Cl\'ement-Peletier,
valid on compact subsets of the domain, and a full anti-maximum principle by
restricting to fractional Schr\"odinger operators. Furthermore, we show a
maximum principle for narrow domains, and a refined elliptic ABP-type estimate.
Finally, we obtain Liouville-type theorems for harmonic solutions and for a
class of semi-linear equations. Our approach is probabilistic, making use of
the properties of subordinate Brownian motion.Comment: 35 pages, Liouville-type theorems for semi-linear equations adde
