A probabilistic approach to bounded/positive solutions for Schrödinger operators with certain classes of potentials

Abstract

Abstract. Consider the equationFor certain classes of potentials V , we use probabilistic tools to study the bounded solutions and the positive solutions for (*). A primary motivation is to offer probabilistic intuition for the results.

A characterization of the finiteness of perpetual integrals of Lévy processes

“…If we denote by L the generator of the Lévy process ξ = (ξ s ) s≥0 then at least formally the function h is harmonic for the operator L − f , which is a more general version of the Schrödinger operator above. This connection has been successfully exploited in the context of Brownian motion in Pinsky, R. (2008).…”

A probabilistic approach to the Liouville property for Schrödinger operators with an application to infinite configurations of balls

“…It is known [11], [2] that the following probabilistic condition is equivalent to the Liouville property: …”

“…For example, recently, it was shown in [6] that if the Liouville property for (1.1) does not hold, and if u solves (1.1) and satisfies 0 < c 1 ≤ u ≤ c 2 , then for any W ≥ 0 and any j ≥ 1, the j-th negative eigenvalue of the operator − [2], [4], [11]). In [11] it was pointed out that results in [3] show that (1.5) is necessary and sufficient for the Liouville property to hold for the class of functions V 0 decaying at least quadratically: 2 , for some c > 0. Furthermore, it was shown that this result is sharp.…”