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

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

2010-05-15

arxiv.org/abs/1005.2690v3

Mathematics

Spectral Theory

16 pages, version 2

Scientific paper

The behavior of the discrete spectrum of the Schr\"odinger 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 $t\to 0$ and $t\to\infty$. If this behavior is powerlike, i.e., \[\|P(t;\cdot,\cdot)\|_{L^\infty}=O(t^{-\delta/2}),\ t\to 0;\qquad \|P(t;\cdot,\cdot)\|_{L^\infty}=O(t^{-D/2}),\ t\to\infty,\] then it is natural to call the exponents $\delta,D$ "{\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 $\delta

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