Schrodinger operators on regular metric trees with long range potentials: Weak coupling behavior

Consider a regular d-dimensional metric tree Gamma with root o. Define the Schrodinger 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 <= d <= 2, gamma not equal 2, we will determine the weak coupling behavior of the bottom of the spectrum of -Delta -V. In other words. we will describe the asymptotic behavior of inf sigma(-Delta - alpha V) as alpha -> 0+. (C) 2009 Elsevier Inc. All rights reserved.