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

Multiscale Analysis in Momentum Space for Quasi-periodic Potential in Dimension Two

We consider a polyharmonic operator H=(-\Delta)^l+V(\x) in dimension two with $l\geq 2$, $l$ being an integer, and a quasi-periodic potential V(\x). We prove that the absolutely continuous spectrum of $H$ contains a semiaxis and there is a family of generalized eigenfunctions at every point of this semiaxis with the following properties. First, the eigenfunctions are close to plane waves $e^{i}$ at the high energy region. Second, the isoenergetic curves in the space of momenta \k corresponding to these eigenfunctions have a form of slightly distorted circles with holes (Cantor type structure). A new method of multiscale analysis in the momentum space is developed to prove these results.Comment: 125 pages, 4 figures. arXiv admin note: incorporates arXiv:1205.118

Yang-Mills action from minimally coupled bosons on R^4 and on the 4D Moyal plane

We consider bosons on Euclidean R^4 that are minimally coupled to an external Yang-Mills field. We compute the logarithmically divergent part of the cut-off regularized quantum effective action of this system. We confirm the known result that this term is proportional to the Yang-Mills action. We use pseudodifferential operator methods throughout to prepare the ground for a generalization of our calculation to the noncommutative four-dimensional Moyal plane (also known as noncommutative flat space). We also include a detailed comparison of our cut-off regularization to heat kernel techniques. In the case of the noncommutative space, we complement the usual technique of asymptotic expansion in the momentum variable with operator theoretic arguments in order to keep separated quantum from noncommutativity effects. We show that the result from the commutative space R^4 still holds if one replaces all pointwise products by the noncommutative Moyal product.Comment: 37 pages, v2 contains an improved treatment of the theta function in Appendix A.

On the Fredholm property of bisingular pseudodifferential operators

For operators belonging either to a class of global bisingular pseudodifferential operators on $R^m \times R^n$ or to a class of bisingular pseudodifferential operators on a product $M \times N$ of two closed smooth manifolds, we show the equivalence of their ellipticity (defined by the invertibility of certain associated homogeneous principal symbols) and their Fredholm mapping property in associated scales of Sobolev spaces. We also prove the spectral invariance of these operator classes and then extend these results to the even larger classes of Toeplitz type operators.Comment: 21 pages. Expanded sections 3 and 4. Corrected typos. Added reference

The Gabor wave front set of compactly supported distributions

We show that the Gabor wave front set of a compactly supported distribution equals zero times the projection on the second variable of the classical wave front set

Analytic and Reidemeister torsion for representations in finite type Hilbert modules

For a closed Riemannian manifold we extend the definition of analytic and Reidemeister torsion associated to an orthogonal representation of fundamental group on a Hilbert module of finite type over a finite von Neumann algebra. If the representation is of determinant class we prove, generalizing the Cheeger-M\"uller theorem, that the analytic and Reidemeister torsion are equal. In particular, this proves the conjecture that for closed Riemannian manifolds with positive Novikov-Shubin invariants, the L2 analytic and Reidemeister torsions are equal.Comment: 78 pages, AMSTe

Generation of proxy GIM-TEC for extreme storms before the Era of GNSS observations

For the first time, we reconstructed global distribution of both the total electron content disturbance W index and TEC values for eight extreme storms (Dst < -250 nT) occurred before the epoch of GNSS observations in solar cycle 22. We created a model based on superposed epoch analysis of the training set of GIM-W maps of nine SC23 extreme storms. Global GIM-W index maps are calculated from 15-min UPC GIM-TEC (UQRG) as the logarithmic deviation of instantaneous TEC from the monthly median GIMMTEC empirical model. We introduced the storm phase metrics for main and recovery phases of the positive ionosphere disturbance (the WU-index), the negative disturbance (the WL-index) and the ring current (the Dst-index). The probabilistic forecasting model (Pmodel) for SC22 GIM-Wx maps is developed based on GIM-W maps of the SC23 training set. The storm phase distribution Fx for the eight SC22 extreme storms is calculated from the proxy time shift (lag) of peak WUmax and WLmin relative to Dstmin. Proxy GIM-TECx maps are calculated by adjusting the GIM-MTEC median to the GIM-Wx prediction. Validation of the technique based on data of UPC and JPL for four intense ionospheric storms showed a root-mean-square error less than 3 TECU. The proposed technique can be applied for both the past and future forecasting of GIM-W index and GIM-TEC maps.Peer ReviewedPostprint (published version

Extended States for Polyharmonic Operators with Quasi-periodic Potentials in Dimension Two

We consider a polyharmonic operator H=(-\Delta)^l+V(\x) in dimension two with $l\geq 2$, $l$ being an integer, and a quasi-periodic potential V(\x). We prove that the spectrum of $H$ contains a semiaxis and there is a family of generalized eigenfunctions at every point of this semiaxis with the following properties. First, the eigenfunctions are close to plane waves $e^{i}$ at the high energy region. Second, the isoenergetic curves in the space of momenta \k corresponding to these eigenfunctions have a form of slightly distorted circles with holes (Cantor type structure). A new method of multiscale analysis in the momentum space is developed to prove these results.Comment: This is an announcement only. Text with the detailed proof is under preparation. 11 pages, 4 figures. arXiv admin note: text overlap with arXiv:math-ph/0601008, arXiv:0711.4404, arXiv:1008.463