Complete asymptotic expansion of the integrated density of states of multidimensional almost-periodic pseudo-differential operators

We obtain a complete asymptotic expansion of the integrated density of states of operators of the form H =(-\Delta)^w +B in R^d. Here w >0, and B belongs to a wide class of almost-periodic self-adjoint pseudo-differential operators of order less than 2w. In particular, we obtain such an expansion for magnetic Schr\"odinger operators with either smooth periodic or generic almost-periodic coefficients.Comment: 47 pages. arXiv admin note: text overlap with arXiv:1004.293

Band spectra of rectangular graph superlattices

We consider rectangular graph superlattices of sides l1, l2 with the wavefunction coupling at the junctions either of the delta type, when they are continuous and the sum of their derivatives is proportional to the common value at the junction with a coupling constant alpha, or the "delta-prime-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 theta := l1/l2. If the latter is an irrational badly approximable by rationals, delta lattices have no gaps in the weak-coupling case. We show that there is a quantization for the asymptotic critical values of alpha at which new gap series open, and explain it in terms of number-theoretic properties of theta. We also show how the irregularity is manifested in terms of Fermi-surface dependence on energy, and possible localization properties under influence of an external electric field. KEYWORDS: Schroedinger operators, graphs, band spectra, fractals, quasiperiodic systems, number-theoretic properties, contact interactions, delta coupling, delta-prime coupling.Comment: 16 pages, LaTe

Zero-Range Model of p-scattering by a Potential Well, Preprint Forschungsinstitut für Mathematik

Abstract. A well-known method of zero-range potentials consists of replacing a deep potential well of a small radius by a boundary condition at the point of the centre of the well. However, in passing to the limit from a deep and narrow potential well to the zero-range model, information, concerning pscattering and scatterings of higher orders, disappears. Traditional zero-range model describes only bound states and s-scattering. The principal mathematical difficulty, which arises in the mathematical construction of a zero-range model, describing p-scattering, is that p-scattered waves have a square nonintegrable singularity at the point, where the well should be located. It is not possible to construct the corresponding energy operator in L2(R 3 ). We construct the energy operator in some Hilbert space, which naturally arises from the problem and includes L2(R 3 ). We explicitly construct the complete system of generalized eigenfunctions in this space