Quasiperiodic surface Maryland models on quantum graphs

We study quantum graphs corresponding to isotropic lattices with quasiperiodic coupling constants given by the same expressions as the coefficients of the discrete surface Maryland model. The absolutely continuous and the pure point spectra are described. It is shown that the transition between them is governed by the Hill operator corresponding to the edge potential.Comment: 12 page

On the discrete spectrum of Robin Laplacians in conical domains

We discuss several geometric conditions guaranteeing the finiteness or the infiniteness of the discrete spectrum for Robin Laplacians on conical domains.Comment: 12 page

Localization on quantum graphs with random vertex couplings

We consider Schr\"odinger operators on a class of periodic quantum graphs with randomly distributed Kirchhoff coupling constants at all vertices. Using the technique of self-adjoint extensions we obtain conditions for localization on quantum graphs in terms of finite volume criteria for some energy-dependent discrete Hamiltonians. These conditions hold in the strong disorder limit and at the spectral edges

Approximation by point potentials in a magnetic field

We discuss magnetic Schrodinger operators perturbed by measures from the generalized Kato class. Using an explicit Krein-like formula for their resolvent, we prove that these operators can be approximated in the strong resolvent sense by magnetic Schrodinger operators with point potentials. Since the spectral problem of the latter operators is solvable, one in fact gets an alternative way to calculate discrete spectra; we illustrate it by numerical calculations in the case when the potential is supported by a circle.Comment: 16 pages, 2 eps figures, submitted to J. Phys.