On the spectral estimates for Schr\"odinger type operators. The case of small local dimension

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 $t\to \mathrm{\infty }$. If this behavior is powerlike, i.e., $$\mathrm{\parallel }P(t;\cdot ,\cdot ){\mathrm{\parallel }}_{L^{\mathrm{\infty }}}=O(t^{-\delta \mathrm{/}2}),t\to 0;\mathrm{\parallel }P(t;\cdot ,\cdot ){\mathrm{\parallel }}_{L^{\mathrm{\infty }}}=O(t^{-D\mathrm{/}2}),t\to \mathrm{\infty },$$ 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.