Compactness of Schrödinger semigroups with unbounded below potentials

Abstract By using the super Poincare inequality of a Markov generator L 0 on L 2 ( μ ) over a σ-finite measure space ( E , F , μ ) , the Schrodinger semigroup generated by L 0 − V for a class of (unbounded below) potentials V is proved to be L 2 ( 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., R d under the condition that V ( x ) → ∞ as | x | → ∞ . Concrete examples are provided to illustrate the main result.