Compactness of Schrödinger semigroups with unbounded below potentials

By using the super Poincaré inequality of a Markov generator L0 on L2(μ) over a σ-finite measure space (E,F,μ), the Schrödinger semigroup generated by L0−V for a class of (unbounded below) potentials V is proved to be L2(μ)-compact provided μ(V⩽N)<∞ for all N>0. This condition is sharp at least in the context of countable Markov chains, and considerably improves known ones on, e.g., Rd under the condition that V(x)→∞ as |x|→∞. Concrete examples are provided to illustrate the main result.