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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.

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)
Full Text: DOI
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