On Spectral Estimates for Schr\"odinger-Type Operators: The Case of Small Local Dimension

G. V. Rozenblum ; M. Z. Solomyak

Funkcionalʹnyj analiz i ego priloženiâ, Tome 44 (2010) no. 4, pp. 21-33

Résumé

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., $$ \|P(t;\cdot,\cdot)\|_{L^\infty}=O(t^{-\delta/2}),\quad t\to 0,\qquad \|P(t;\cdot,\cdot)\|_{L^\infty}=O(t^{-D/2}),\quad t\to\infty, $$ 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 $\delta$, which has been insufficiently studied, is analyzed. Applications to operators on combinatorial and metric graphs are considered.

Détail

Keywords: eigenvalue estimates, Schrödinger operator, metric graph, dimension at infinity.
Mots-clés : local dimension