Schrödinger operator on homogeneous metric trees: spectrum in gaps

Michael Solomyak The Weizmann Institute of Science

We study 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} = −Δ 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.

The results reported were obtained in cooperation with A.V.Sobolev, see A.V. Sobolev and M. Solomyak, Schrödinger operator on homogeneous metric trees: spectrum in gaps, Preprint 2001 (http://xxx.lanl.gov/ps/math/0109016).