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The aim of this paper is to prove a qualitative property, namely the preservation of positivity, for Schrödinger-type operators acting on $L^p$
functions defined on (possibly incomplete) Riemannian manifolds. A key assumption is a control of the behaviour of the potential of the operator near the Cauchy boundary of the manifolds. As a by-product, we establish the essential self-adjointness of such operators, as well as its generalization to the case $p\neq 2$
, i.e. the fact that smooth compactly supported functions are an operator core for the Schrödinger operator in $L^p$
.
