Abstract
In a different paper we constructed imaginary time Schrödinger operatorsH q=−1/2Δ+V acting onL q(ℝn,dx). The negative part of typical potential functionV was assumed to be inL ∞+L q for somep>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 forL q-Schrödinger eigenstates and pointwise lower bounds for the corresponding groundstate. The potential functionsV 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.
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Communicated by J. Glimm
Laboratoire de Mathématiques de Marseille associé au C.N.R.S. L.A.225
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Carmona, R. Pointwise bounds for Schrödinger eigenstates. Commun. Math. Phys. 62, 97–106 (1978). https://doi.org/10.1007/BF01248665
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DOI: https://doi.org/10.1007/BF01248665