Abstract

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.

Original languageEnglish (US)
Pages (from-to)97-106
Number of pages10
JournalCommunications In Mathematical Physics
Volume62
Issue number2
DOIs
StatePublished - Jun 1 1978

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Mathematical Physics

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