Schrödinger Operator on Homogeneous Metric Trees : Spectrum in Gaps

The paper studies the spectral properties of the Schrödinger operator A gV = A 0 + gV on a homogeneous rooted metric tree, with a decaying real-valued potential V and a coupling constant g ≥ 0. The spectrum of the free Laplacian A 0 = −∆ has a band-gap structure with a single eigenvalue of infinite multiplicity in the middle of each finite gap. The perturbation gV gives rise to extra eigenvalues in the gaps. These eigenvalues are monotone functions of g if the potential V has a fixed sign. Assuming that the latter condition is satisfied and that V is symmetric, i.e. depends on the distance to the root of the tree, we carry out a detailed asymptotic analysis of the counting function of the discrete eigenvalues in the limit g → ∞. Depending on the sign and decay of V , this asymptotics is either of the Weyl type or is completely determined by the behaviour of V at infinity.