Quasi-classical versus non-classical spectral asymptotics for magnetic Schroedinger operators with decreasing electric potentials

We consider the Schroedinger operator H on L^2(R^2) or L^2(R^3) with constant magnetic field and electric potential V which typically decays at infinity exponentially fast or has a compact support. We investigate the asymptotic behaviour of the discrete spectrum of H near the boundary points of its essential spectrum. If the decay of V is Gaussian or faster, this behaviour is non-classical in the sense that it is not described by the quasi-classical formulas known for the case where V admits a power-like decay.Comment: Corrected versio

On the semi-classical analysis of the groundstate energy of the Dirichlet Pauli operator in non-simply connected domains

We consider the Dirichlet Pauli operator in bounded connected domains in the plane, with a semi-classical parameter. We show, in particular, that the ground state energy of this Pauli operator will be exponentially small as the semi-classical parameter tends to zero and estimate this decay rate. This extends our results, discussing the results of a recent paper by Ekholm--Kova\v{r}\'ik--Portmann, to include also non-simply connected domains.Comment: 15 pages, 4 figure

A trace formula and high energy spectral asymptotics for the perturbed Landau Hamiltonian

A two-dimensional Schr\"odinger operator with a constant magnetic field perturbed by a smooth compactly supported potential is considered. The spectrum of this operator consists of eigenvalues which accumulate to the Landau levels. We call the set of eigenvalues near the $n$'th Landau level an $n$'th eigenvalue cluster, and study the distribution of eigenvalues in the $n$'th cluster as $n\to\infty$. A complete asymptotic expansion for the eigenvalue moments in the $n$'th cluster is obtained and some coefficients of this expansion are computed. A trace formula involving the first eigenvalue moments is obtained.Comment: 23 page

The Choice of a Design of the Device for Production of Compound Feed

In the work the analysis of the equipment and technology of its production for production of the granulated forages is carried out, it is shown that the most responsible element is the matrix. Ways of increase in its resource are planned

Distant foreground and the Planck-derived Hubble constant

It is possible to reduce the discrepancy between the local measurement of the cosmological parameter $H_0$ and the value derived from the $Planck$ measurements of the Cosmic Microwave Background (CMB) by considering contamination of the CMB by emission from some medium around distant extragalactic sources, such as extremely cold coarse-grain dust. Though being distant, such a medium would still be in the foreground with respect to the CMB, and, as any other foreground, it would alter the CMB power spectrum. This could contribute to the dispersion of CMB temperature fluctuations. By generating a few random samples of CMB with different dispersions, we have checked that the increased dispersion leads to a smaller estimated value of $H_0$, the rest of the cosmological model parameters remaining fixed. This might explain the reduced value of the $Planck$-derived parameter $H_0$ with respect to the local measurements. The signature of the distant foreground in the CMB traced by SNe was previously reported by the authors of this paper -- we found a correlation between the SN redshifts, $z_{\rm SN}$, and CMB temperature fluctuations at the SNe locations, $T_{\rm SN}$. Here we have used the slopes of the regression lines $T_{\rm SN}\,/\,z_{\rm SN}$ corresponding to different {\it Planck} wave bands in order to estimate the possible temperature of the distant extragalactic medium, which turns out to be very low, about 5\,K. The most likely ingredient of this medium is coarse-grain ($grey$) dust, which is known to be almost undetectable, except for the effect of dimming remote extragalactic sources.Comment: 5 pages, 4 figures, 1 tabl

Asymptotic Density of Eigenvalue Clusters for the Perturbed Landau Hamiltonian

We consider the Landau Hamiltonian (i.e. the 2D Schroedinger operator with constant magnetic field) perturbed by an electric potential V which decays sufficiently fast at infinity. The spectrum of the perturbed Hamiltonian consists of clusters of eigenvalues which accumulate to the Landau levels. Applying a suitable version of the anti-Wick quantization, we investigate the asymptotic distribution of the eigenvalues within a given cluster as the number of the cluster tends to infinity. We obtain an explicit description of the asymptotic density of the eigenvalues in terms of the Radon transform of the perturbation potential V.Comment: 30 pages. The explicit dependence on B and V in Theorem 1.6 (i) - (ii) indicated. Typos corrected. To appear in Communications in Mathematical Physic