Spectral estimates for the Schrödinger operators with sparse potentials on graphs

Details Spectral estimates for the Schrödinger operators with sparse potentials on graphs Spectral estimates for the Schrödinger operators with sparse potentials on graphs

2011-04-18

arxiv.org/abs/1104.3455v1

Mathematics

Spectral Theory

Scientific paper

The construction of "sparse potentials", suggested in \cite{RS09} for the lattice $\Z^d,\ d>2$, is extended to a wide class of combinatorial and metric graphs whose global dimension is a number $D>2$. For the Schr\"odinger operator $-\D-\a V$ on such graphs, with a sparse potential $V$, we study the behavior (as $\a\to\infty$) of the number $N_-(-\D-\a V)$ of negative eigenvalues of $-\D-\a V$. We show that by means of sparse potentials one can realize any prescribed asymptotic behavior of $N_-(-\D-\a V)$ under very mild regularity assumptions. A similar construction works also for the lattice $\Z^2$, where D=2.

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