On Spectral Estimates for Schrödinger-Type Operators: The Case of Small Local Dimension

The behavior of the discrete spectrum of the Schrödinger operator $-\Delta-V$ is determined to a large extent by the behavior of the corresponding heat kernel $P(t;x,y)$ as $t\to 0$ and $t\to\infty$. If this behavior is power-like, i.e.,
$$ \|P(t;\cdot,\cdot)\|_{L^\infty}=O(t^{-\delta/2}),\quad t\to 0,\qquad \|P(t;\cdot,\cdot)\|_{L^\infty}=O(t^{-D/2}),\quad t\to\infty, $$
then it is natural to call the exponents $\delta$ and $D$ the local dimension and the dimension at infinity, respectively. The character of spectral estimates depends on a relation between these dimensions. The case where $\delta<D$, which has been insufficiently studied, is analyzed. Applications to operators on combinatorial and metric graphs are considered.