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# On spectral estimates for Schrödinger-type operators: The case of small local dimension

Grigori Rozenblioum (Institutionen för matematiska vetenskaper, matematik) ; Michael Solomyak
Functional Analysis and Its Applications (0016-2663). Vol. 44 (2010), 4, p. 259-269.

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

Nyckelord: Schrödinger operator, quantum graphs, eigenvalue estimates

Translated from Funktsional´nyi Analiz i Ego Prilozheniya, Vol. 44, No. 4, pp. 21–33, 2010

CPL Pubid: 132538

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Institutionen för matematiska vetenskaper, matematik (2005-2016)