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On spectral estimates for Schrödinger-type operators: the case of small local dimension. (English. Russian original) Zbl 1272.47055
Funct. Anal. Appl. 44, No. 4, 259-269 (2010); translation from Funkts. Anal. Prilozh. 44, No. 4, 21-33 (2010).
Summary: 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., \begin{aligned}\| P(t;\cdot,\cdot)\|_{L^\infty}=O(t^{-\delta/2}),\quad t\to 0,\quad\| P(t;\cdot,\cdot)\|_{L^\infty}=O(t^{-D/2}),\quad t\to\infty,\end{aligned} 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.

##### MSC:
 47F05 General theory of partial differential operators (should also be assigned at least one other classification number in Section 47-XX) 35J10 Schrödinger operator, Schrödinger equation 35P20 Asymptotic distributions of eigenvalues in context of PDEs 35R02 PDEs on graphs and networks (ramified or polygonal spaces)
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