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 $1 < \gamma \leq d \leq 2, \gamma \neq 2$, 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 $\alpha \to 0+$ 
Tomas Ekholm, Andreas Enblom, Hynek Kovarik