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 $\delta<D$ that was insufficiently studied before. Our applications concern the combinatorial and the metric graphs.
Categories: math.SP