Pointwise bounds for Schrödinger eigenstates

In a different paper we constructed imaginary time Schrödinger operators H_q=-1/2Δ+V acting on L^q(ℝⁿ, dx). The negative part of typical potential function V was assumed to be in L^∞+L^q 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 L^q-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.