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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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