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

The behavior of the discrete spectrum of the Schr&quot;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 $\delta<D$ that was insufficiently studied before. Our applications concern the combinatorial and the metric graphs.


    • EPFL-REPORT-175663

    Record created on 2012-03-13, modified on 2016-08-09

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