Schroedinger 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 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 $1 < gamma leq d leq 2, gamma eq 2$, 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 $alpha to 0+$