On the spectral estimates for Schr\"odinger type operators. The case of small local dimension

TL;DR

This paper investigates spectral estimates for Schrödinger operators on graphs, focusing on the less-studied case where local dimension is less than the dimension at infinity, revealing how heat kernel behavior influences the discrete spectrum.

Contribution

It analyzes the spectral behavior of Schrödinger operators in the case where local dimension is less than the dimension at infinity, extending previous studies.

Findings

Spectral estimates depend on the relation between local and global dimensions.

The case d4 was previously underexplored.

Applications to combinatorial and metric graphs.

Abstract

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 $\delta<D$ that was insufficiently studied before. Our applications concern the combinatorial and the metric graphs.