Till sidans topp

Sidansvarig: Webbredaktion

On spectral estimates for… - Göteborgs universitet Till startsida
Webbkarta
Till innehåll Läs mer om hur kakor används på gu.se

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

Artikel i vetenskaplig tidskrift
Författare Grigori Rozenblioum Michael Solomyak Functional Analysis and Its Applications 44 4 259-269 0016-2663 2010 Institutionen för matematiska vetenskaper, matematik 259-269 en springerlink.metapress.com/content/... Schrödinger operator, quantum graphs, eigenvalue estimates Matematisk analys

## Sammanfattning

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