Schroedinger Operators on Regular Metric Trees with Long Range Potentials: Weak Coupling Behavior

TL;DR

This paper investigates the spectral behavior of Schrödinger operators on regular metric trees with long-range potentials, focusing on how the spectrum's bottom behaves under weak coupling conditions for specific decay rates.

Contribution

It provides a detailed analysis of the weak coupling asymptotics of the spectrum's bottom for Schrödinger operators on regular metric trees with potentials decaying as x^{-eta} for certain eta.

Findings

Characterizes the asymptotic behavior of the lowest spectral point as coupling tends to zero.

Identifies the influence of decay rate eta on spectral properties.

Extends understanding of spectral theory on metric trees with long-range potentials.

Abstract

Consider a regular $d$-dimensional metric tree $\Gamma$ with root $o$. Define the Schroedinger operator $-\Delta - V$, where $V$ is a non-negative, symmetric potential, on $\Gamma$, with Neumann boundary conditions at $o$. Provided that $V$ decays like $x^{-\gamma}$ at infinity, where $1 < \gamma \leq d \leq 2, \gamma \neq 2$, we will determine the weak coupling behavior of the bottom of the spectrum of $-\Delta - V$. In other words, we will describe the asymptotical behavior of $\inf \sigma(-\Delta - \alpha V)$ as $\alpha \to 0+$