On positivity, criticality, and the spectral radius of the shuttle operator for elliptic operators
Yehuda Pinchover · Duke Mathematical Journal · 1996
In this paper, we study the shuttle operator for a second order linear elliptic operator P on a noncompact manifold X. Zhao has introduced and studied the shuttle operator and its relation to the theory of positive solutions in the case of small perturbations of the Laplacian in IR, n ≥ 3. Zhao was motivated by works of Chung and Varadhan which consider one-dimensional Schrodinger operators. The main purpose of the paper is to extend the above studies. We prove that, in the general case, the spectral radius of the shuttle operator is strictly less, equal, or strictly greater than 1 if and only if the operator P is respectively subcritical, critical, or supercritical in X. We demonstrate the usefulness of the characterization for proving theorems. Our approach is purely analytic. ∗This research was partially supported by THE ISRAEL SCIENCE FOUNDATION administered by THE ISRAEL ACADEMY OF SCIENCES AND HUMANITIES and by the Fund for the Promotion of Research at the Technion