Bethe-Sommerfeld conjecture for periodic operators with strong perturbations

We consider a periodic self-adjoint pseudo-differential operator $H=(-\Delta)^m+B$, $m>0$, in $\R^d$ which satisfies the following conditions: (i) the symbol of $B$ is smooth in \bx, and (ii) the perturbation $B$ has order less than $2m$. Under these assumptions, we prove that the spectrum of $H$ contains a half-line. This, in particular implies the Bethe-Sommerfeld Conjecture for the Schr\"odinger operator with a periodic magnetic potential in all dimensions.Comment: 61 page

Cantor and band spectra for periodic quantum graphs with magnetic fields

We provide an exhaustive spectral analysis of the two-dimensional periodic square graph lattice with a magnetic field. We show that the spectrum consists of the Dirichlet eigenvalues of the edges and of the preimage of the spectrum of a certain discrete operator under the discriminant (Lyapunov function) of a suitable Kronig-Penney Hamiltonian. In particular, between any two Dirichlet eigenvalues the spectrum is a Cantor set for an irrational flux, and is absolutely continuous and has a band structure for a rational flux. The Dirichlet eigenvalues can be isolated or embedded, subject to the choice of parameters. Conditions for both possibilities are given. We show that generically there are infinitely many gaps in the spectrum, and the Bethe-Sommerfeld conjecture fails in this case.Comment: Misprints correcte

Lithospheric mantle beneath the south-eastern Siberian craton: Petrology of peridotite xenoliths in basalts from the Tokinsky Stanovik

We provide petrographic, major and trace element data for over 30 spinel peridotite xenoliths from the Tokinsky Stanovik (Tok) volcanic field on the Aldan shield to characterize the lithospheric mantle beneath the south-eastern margin of the Siberian craton, which formed in the Mesoproterozoic. High equilibration temperatures (870-1,010°C) of the xenoliths and the absence of garnet-bearing peridotites indicate a much thinner lithosphere than in the central craton. Most common among the xenoliths are clinopyroxene-poor lherzolites and harzburgites with Al2O3 and CaO contents nearly as low as in refractory xenoliths from kimberlite pipes (Mir, Udachnaya) in the central and northern Siberian craton. By contrast, the Tok peridotites have higher FeO, lower Mg-numbers and lower modal orthopyroxene and are apparently formed by shallow partial melting (<3 GPa). Nearly all Tok xenoliths yield petrographic and chemical evidence for metasomatism: Accessory phlogopite, amphibole, phosphates, feldspar and Ti-rich oxides, very high Na2O (2-3.1%) in clinopyroxene, LREE enrichments in whole-rocks. © Springer-Verlag 2005.SCOPUS: ar.jinfo:eu-repo/semantics/publishe