On a factorization of second order elliptic operators and applications

We show that given a nonvanishing particular solution of the equation (divpgrad+q)u=0 (1) the corresponding differential operator can be factorized into a product of two first order operators. The factorization allows us to reduce the equation (1) to a first order equation which in a two-dimensional case is the Vekua equation of a special form. Under quite general conditions on the coefficients p and q we obtain an algorithm which allows us to construct in explicit form the positive formal powers (solutions of the Vekua equation generalizing the usual powers of the variable z). This result means that under quite general conditions one can construct an infinite system of exact solutions of (1) explicitly, and moreover, at least when p and q are real valued this system will be complete in ker(divpgrad+q) in the sense that any solution of (1) in a simply connected domain can be represented as an infinite series of obtained exact solutions which converges uniformly on any compact subset of . Finally we give a similar factorization of the operator (divpgrad+q) in a multidimensional case and obtain a natural generalization of the Vekua equation which is related to second order operators in a similar way as its two-dimensional prototype does

Influence of spin polarization on resistivity of a two-dimensional electron gas in Si MOSFET at metallic densities

Positive magnetoresistance (PMR) of a silicon MOSFET in parallel magnetic fields B has been measured at high electron densities n >> n_c where n_c is the critical density of the metal-insulator transition (MIT). It turns out that the normalized PMR curves, R(B)/R(0), merge together when the field is scaled according to B/B_c(n) where B_c is the field in which electrons become fully spin polarized. The values of B_c have been calculated from the simple equality between the Zeeman splitting energy and the Fermi energy taking into account the experimentally measured dependence of the spin susceptibility on the electron density. This extends the range of validity of the scaling all the way to a deeply metallic regime far away from MIT. The subsequent analysis of PMR for low n >~ n_c demonstrated that the merging of the initial parts of curves can bee achieved only with taking into account the temperature dependence of B_c. It is also shown that the shape of the PMR curves at strong magnetic fields is affected by a crossover from a purely two-dimensional (2D) electron transport to a regime where out-of-plane carrier motion becomes important (quasi-three-dimensional regime).Comment: 5 pages, including 6 figures; misprints corrected; Europhys. Lett. (in press

Thermodynamic Signature of a Two-Dimensional Metal-Insulator Transition

We present a study of the compressibility, K, of a two-dimensional hole system which exhibits a metal-insulator phase transition at zero magnetic field. It has been observed that dK/dp changes sign at the critical density for the metal-insulator transition. Measurements also indicate that the insulating phase is incompressible for all values of B. Finally, we show how the phase transition evolves as the magnetic field is varied and construct a phase diagram in the density-magnetic field plane for this system.Comment: 4 pages, 4 figures, submitted to Physical Review Letters; version 1 is identical to version 2 but didn't compile properl

Mesoscopic Behavior Near a Two-Dimensional Metal-Insulator Transition

We study conductance fluctuations in a two-dimensional electron gas as a function of chemical potential (or gate voltage) from the strongly insulating to the metallic regime. Power spectra of the fluctuations decay with two distinct exponents (1/v_l and 1/v_h). For conductivity $\sigma\sim 0.1 e^{2}/h$, we find a third exponent (1/v_i) in the shortest samples, and non-monotonic dependence of v_i and v_l on \sigma. We study the dependence of v_i, v_l, v_h, and the variances of corresponding fluctuations on \sigma, sample size, and temperature. The anomalies near $\sigma\simeq 0.1 e^{2}/h$ indicate that the dielectric response and screening length are critically behaved, i.e. that Coulomb correlations dominate the physics.Comment: Revised according to referee remark

Heavy Hadron Spectroscopy

I review recent theoretical advances in heavy hadron spectroscopy.Comment: Plenary talk at the XXXIII International Conference on High Energy Physics (ICHEP 06), Moscow, Russia, July 26 - August 2, 2006; 11 page

IceCube-Plus: An Ultra-High Energy Neutrino Telescope

While the first kilometer-scale neutrino telescope, IceCube, is under construction, alternative plans exist to build even larger detectors that will, however, b e limited by a much higher neutrino energy threshold of 10 PeV or higher rather than 10 to 100 GeV. These future projects detect radio and acoustic pulses as w ell as air showers initiated by ultra-high energy neutrinos. As an alternative, we here propose an expansion of IceCube, using the same strings, placed on a gri d with a spacing of order 500 m. Unlike other proposals, the expanded detector uses methods that are understood and calibrated on atmospheric neutrinos. Atmosp heric neutrinos represent the only background at the energies under consideratio n and is totally negligible. Also, the cost of such a detector is understood. We conclude that supplementing the 81 IceCube strings with a modest number of addi tional strings spaced at large distances can almost double the effective volume of the detector. Doubling the number of strings on a 800 m grid can deliver a d etector that this a factor of 5 larger for horizontal muons at modest cost.Comment: Version to be published in JCA

On Lebesgue measure of integral self-affine sets

Let $A$ be an expanding integer $n\times n$ matrix and $D$ be a finite subset of $Z^n$. The self-affine set $T=T(A,D)$ is the unique compact set satisfying the equality $A(T)=\cup_{d\in D} (T+d)$. We present an effective algorithm to compute the Lebesgue measure of the self-affine set $T$, the measure of intersection $T\cap (T+u)$ for $u\in Z^n$, and the measure of intersection of self-affine sets $T(A,D_1)\cap T(A,D_2)$ for different sets $D_1,D_2\subset Z^n$.Comment: 5 pages, 1 figur

Coulomb Drag at the Onset of Anderson Insulators

It is shown that the Coulomb drag between two identical layers in the Anderson insulting state indicates a striking difference between the Mott and Efros-Shklovskii (ES) insulators. In the former, the trans-resistance $\rho_t$ is monotonically increasing with the localization length $\xi$; in the latter, the presence of a Coulomb gap leads to an opposite result: $\rho_t$ is enhanced with a decreasing $\xi$, with the same exponential factor as the single layer resistivity. This distinction reflects the relatively pronounced role of excited density fluctuations in the ES state, implied by the enhancement in the rate of hopping processes at low frequencies. The magnitude of drag is estimated for typical experimental parameters in the different cases. It is concluded that a measurement of drag can be used to distinguish between interacting and non-interacting insulating state.Comment: 15 pages, revte