On positivity, criticality, and the spectral radius of the shuttle operator for elliptic operators
On positivity, criticality, and the spectral radius of the shuttle operator for elliptic operators
复制标题
关于椭圆算子的穿梭算子的正性、临界性和谱半径
DOI：
10.1215/s0012-7094-96-08518-x
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发表时间：
1996
影响因子：
2.5
通讯作者：
Y. Pinchover
中科院分区：
文献类型：
--
作者：
Y. Pinchover
本文研究非紧流形X上二阶线性椭圆型算子P的穿梭算子。在IR，n≥3中拉普拉斯算子发生小扰动的情况下，赵引入并研究了穿梭算子及其与正解理论的关系。赵建国的动机是Chung和Varadhan关于一维薛定谔算子的工作。本文的主要目的是对上述研究进行拓展。我们证明了在一般情况下，穿梭算子的谱半径严格小于、等于或严格大于1当且仅当算子P在X中分别是亚临界、临界或超临界的。我们的方法纯粹是分析性的。∗这项研究得到了由以色列科学院和人文科学学院管理的以色列科学基金会和以色列理工学院促进研究基金的部分支持
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