On the spectral estimates for Schrödinger type operators. The case of small local dimension

2010/05/15 by Rozenblum, Grigori, Solomyak, Michael
#47A75 #FOS: Mathematics #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.1005.2690

Abstract

The behavior of the discrete spectrum of the Schrödinger operator -\D - V, in quite a general setting, up to a large extent is determined by the behavior of the corresponding heat kernel P(t;x,y) as t→ 0 and t→∞. If this behavior is powerlike, i.e., ‖P(t;⋅,⋅)‖_L^∞=O(t^-δ/2), t→ 0; ‖P(t;⋅,⋅)‖_L^∞=O(t^-D/2), t→∞, then it is natural to call the exponents δ,D "\it the local dimension" and "\it the dimension at infinity" respectively. The character of spectral estimates depends on the relation between these dimensions. In the paper we analyze the case where δ

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