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On spectral estimates for SchrÃ¶dinger-type operators: The case of small local dimension
Journal article, 2010

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

Schrödinger operator

quantum graphs

eigenvalue estimates

## Author

### Grigori Rozenblioum

Chalmers, Mathematical Sciences, Mathematics

University of Gothenburg

### Michael Solomyak

#### Functional Analysis and its Applications

0016-2663 (ISSN) 1573-8485 (eISSN)

Vol. 44 4 259-269### Subject Categories

Mathematical Analysis