We study the spectral properties of Dirichlet Laplacian on the conical layer of the opening angle 2 and thickness equal to . We demonstrate that below the continuum threshold, which is equal to 1, there is an infinite sequence of isolated eigenvalues and analyse properties of these geometrically induced bound states. By numerical computation we find examples of the eigenfunctions.

We consider rectangular graph superlattices of sides l 1, l 2 with the wave-function coupling at the junctions either of the type, when they are continuous and the sum of their derivatives is proportional to the common value at the junction with a coupling constant , or the s type with the roles of functions and derivatives reversed; the latter corresponds to the situations where the junctions are realized by complicated geometric scatterers. We show that the band spectra have a hidden fractal structure with respect to the ratio := l 1/l 2. If the latter is an irrational badly approximable by rationals, lattices have no gaps in the weak-coupling case. We show that there is a quantization for the asymptotic critical values of at which new gap series open, and explain it in terms of number-theoretic properties of . We also show how the irregularity is manifested in terms of Fermi-surface dependence on energy, and possible

We study Schrdinger operators on an infinite quantum graph of a chain form which consists of identical rings connected at the touching points by -couplings with a parameter\alpha\in {\bb R}. If the graph is' straight', ie periodic with respect to ring shifts, its Hamiltonian has a band spectrum with all the gaps open whenever 0. We consider a'bending'deformation of the chain consisting of changing one position at a single ring and show that it gives rise to eigenvalues in the open spectral gaps. We analyze dependence of these eigenvalues on the coupling and the'bending angle'as well as resonances of the system coming from the bending. We also discuss the behaviour of the eigenvalues and resonances at the edges of the spectral bands.

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