Finance
June 1978
In a different paper we constructed imaginary time Schrödinger operators Hq=-1/2Δ+V acting on Lq(ℝn, dx). The negative part of typical potential function V was assumed to be in L∞+Lq for some p>max{1, n/2}. Our proofs were based on the evaluation of Kac’s averages over Brownian motion paths. The present paper continues this study: using probabilistic techniques we prove pointwise upper bounds for Lq-Schrödinger eigenstates and pointwise lower bounds for the corresponding groundstate. The potential functions V are assumed to be neither smooth nor bounded below. Consequently, our results generalize Schnol’s and Simon’s ones. Moreover probabilistic proofs seem to be shorter and more informative than existing ones.