Discreteness of spectrum and positivity criteria for Schrodinger operators

• 成果类型: Article
• 署名作者: Maz'ya, V; Shubin, M
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2005.162.919
• 发表日期: 2005
• 页码: 919-942
• 关键词: differential-operators

摘要

We provide a class of necessary and sufficient conditions for the discreteness of spectrum of Schrodinger operators with scalar potentials which are semibounded below. The classical discreteness of spectrum criterion by A. M. Molchanov (1953) uses a notion of negligible set in a cube as a set whose Wiener capacity is less than a small constant times the capacity of the cube. We prove that this constant can be taken arbitrarily between 0 and 1. This solves a problem formulated by I. M. Gelfand in 1953. Moreover, we extend the notion of negligibility by allowing the constant to depend on the size of the cube. We give a complete description of all negligibility conditions of this kind. The a priori equivalence of our conditions involving different negligibility classes is a nontrivial property of the capacity. We also establish similar strict positivity criteria for the Schrodinger operators with nonnegative potentials.