On spectral estimates for Schr?dinger-type operators: The case of small local dimension

摘要

The behavior of the discrete spectrum of the Schr?dinger operator - Δ -V is determined to a large extent by the behavior of the corresponding heat kernel P(t; x,y) as t → 0 and t→ ∞. If this behavior is power-like, i. e., then it is natural to call the exponents δ and D the local dimension and the dimension at infinity, respectively. The character of spectral estimates depends on a relation between these dimensions. The case where δ < D, which has been insufficiently studied, is analyzed. Applications to operators on combinatorial and metric graphs are considered.