Spectral estimates for the Schr\"odinger operators with sparse potentials on graphs

TL;DR

This paper extends the concept of sparse potentials to a broad class of graphs with global dimension greater than two, analyzing the asymptotic behavior of negative eigenvalues of Schrödinger operators as the potential strength increases.

Contribution

It generalizes the construction of sparse potentials to various graphs and characterizes their impact on the spectral properties of Schrödinger operators.

Findings

Any prescribed asymptotic behavior of negative eigenvalues can be realized with sparse potentials.

The results apply to both combinatorial and metric graphs with dimension D>2.

A similar approach is valid for the two-dimensional lattice .

Abstract

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.