On the Negative Spectrum of the Two-Dimensional Schrödinger Operator with Radial Potential

For a two-dimensional Schrödinger operator H _{α V}  = ?Δ ?αV with the radial potential V(x) = F(|x|), F(r) ≥ 0, we study the behavior of the number N _?(H _{α V} ) of its negative eigenvalues, as the coupling parameter α tends to infinity. We obtain the necessary and sufficient conditions for the semi-classical growth N _?(H _{α V} ) = O(α) and for the validity of the Weyl asymptotic law.     

Maslov index for power-law potentials

The Maslov index in the semiclassical Bohr–Sommerfeld quantization rule is calculated for one-dimensional power-law potentials V (x) = ?V ₀/x ^s, x > 0, 0 < s < 2 The result for the potential V(x)=-V ₀/x ^1/2 is compared with the recently reported exact solution. The case of a spherically symmetric power-law potential is also considered.     

On a Semiclassical Formula for Non-Diagonal Matrix Elements

Let H(?)=?? ²d²/dx ²+V(x) be a Schrödinger operator on the real line, W(x) be a bounded observable depending only on the coordinate and k be a fixed integer. Suppose that an energy level E intersects the potential V(x) in exactly two turning points and lies below V _∞=lim?inf?_|x|→∞ V(x). We consider the semiclassical limit n→∞, ?=? _n →0 and E _n =E where E _n is the nth eigenenergy of H(?). An asymptotic formula for 〈n|W(x)|n+k〉, the non-diagonal matrix elements of W(x) in the eigenbasis of H(?), has been known in the theoretical physics for a long time. Here it is proved in a mathematically rigorous manner.     

Effect of the Lattice and Spin-Phonon Contributions on the Temperature Behavior of the Ground State Splitting of Gd<Superscript>3+</Superscript> in SrMoO<Subscript>4</Subscript>

The temperature behavior of the EPR spectra of the Gd³⁺ impurity center in single crystals of SrMoO₄ in the temperature range T = 99–375 K is studied. The analysis of the temperature dependences of the spin Hamiltonian b ₂ ⁰ (T) = b₂(F) + b₂(L) and P ₂ ⁰ (T) = P₂(F) + P₂(L) (for Gd¹⁵⁷) describing the EPR spectrum and contributing to the Gd³⁺ ground state splitting ΔE is carried out. In terms of the Newman model, the values of b₂(L) and P₂(L) depending on the thermal expansion of the static lattice are estimated; the b₂(F) and P₂(F) spin-phonon contributions determined by the lattice ion oscillations are separated. The analysis of b ₂ ⁰ (T) and P ₂ ⁰ (T) is evidence of the positive contribution of the spin-phonon interaction; the model of the local oscillations of the impurity cluster with close frequencies ω describes well the temperature behavior of b₂(F) and P₂(F).     

Mechanism for the appearance of a high-efficiency Brownian motor with fluctuating potential

A version of a Brownian motor (system generating a unidirectional motion of a Brownian particle in an asymmetric fluctuating potential) is considered for the case where the potential consists of an asymmetric periodic component undergoing random shifts by a half-period L with a certain frequency and the potential of an external force F. The high efficiency of such a motor (ratio of the useful work against the load force F to the energy imparted to the particle due to the potential shifts) is due to a high and narrow barrier, as well as to a smooth arbitrarily shaped potential relief repeated with an energy shift on both half-periods L. Simple analytic expressions are obtained for the flux and efficiency as functions of the load force over a wide frequency range.     

Growth of (GaAs)<Subscript>1 − <Emphasis Type="Italic">x</Emphasis></Subscript>(ZnSe)<Subscript><Emphasis Type="Italic">x</Emphasis></Subscript> solid solution films and investigation of their structural and some photoelectric properties

Single-crystal films of the substitutional solid solution (GaAs)_{1 ? x} (ZnSe) _x (0 ≤ x ≤ 0.80) on GaAs substrates have been grown using liquid phase epitaxy. The X-ray diffraction patterns, photoluminescence spectra, and current-voltage characteristics of the n-(GaAs)-p-(GaAs)_{1 ? x} (ZnSe) _x (0 ≤ x ≤ 0.80) heterostructures prepared have been investigated. The lattice parameters of the film a _f = 5.6544 Å and the substrate a _s = 5.6465 Å have been determined, and the profile of the molecular distribution of the solid solution components has been obtained. The photoluminescence spectrum of the (GaAs)_{1 ? x} (ZnSe) _x (0 ≤ x ≤ 0.80) films exhibits a narrow peak (against the background of the broad luminescence band) with a maximum in the luminescence intensity at a photon energy of 2.67 eV due to the presence of Zn-Se bonds in the structure (ZnSe is covalently bonded to the tetrahedral lattice of the GaAs matrix). It has been shown that the direct branch of the current-voltage characteristics of the structures under investigation is described by an exponential dependence I = I ₀exp(qV/ckT) at low voltages (V > 0.3 V) and by a power-law dependence I ～ V ^α with exponents α = 4 at V = 0.4–0.8 V, α = 2 at V = 0.8–1.4 V, and α = 1.5 at V > 2 V. The experimental results have been explained in the framework of the double-injection model for the n-p-p ⁺ structure under the condition that the concentration distribution of nonequilibrium charge carriers has a minimum.     

The analysis of the densities of <Emphasis Type="Italic">s</Emphasis>-wave neutron resonances separated with respect to spin

The density ratio of s-wave neutron resonances z=ρ(J₁)/ρ(J₂) was analyzed on the basis of the experimental data for 22 atomic nuclei and the Gilbert-Cameron formula for ρ(J). Here, J₁=I_x—1/2 and J₂=I_x+1/2, where I_x denotes the spin of the target nucleus in the ground state. Our aim was to verify whether the factor η(I_x), as a multiplier, can be applied in the expression describing ρ(J₁), with the assumption that ρ(J₂) values remain unchanged, or whether the factor 1η(I_x) can be applied, as a multiplier with ρ(J₂), while the ρ(J₁) values remain unchanged. The final conclusions, e.g., the confirmation or the negation of the fact that it may be necessary to apply the η(I_x) factor, depend on the values of “real” errors Δz of the z variable, which can be calculated if the optimal values of Δρ(J₁) and Δρ(J₂) are known.     

Radiative transitions to localized Eu states in Pb<Subscript>1−<Emphasis Type="Italic">x</Emphasis></Subscript>Eu<Subscript><Emphasis Type="Italic">x</Emphasis></Subscript>Te solid solutions

Two types of radiative transitions were observed in the low-temperature photoluminescence spectra of Pb_1?x Eu _x Te (0 ≤ x ≤ 0.09) solid solutions: an intense line corresponding to the transitions from the conduction to the valence band and a series (up to 15) of narrow lines corresponding to the transitions from the hybridized conduction band to the split Eu³⁺ levels. The intensity of the discrete lines increases with x, and their energies (and intensities) weakly depend on the temperature. The absorption and emission of the discrete lines are caused by the 4f ⁷(⁸ S _7/2) ? 4f ⁶(⁷ F _J )[L ₆ ^? + 5dt _2g ] transitions.     

Weyl’s Type Estimates on the Eigenvalues of Critical Schrödinger Operators

We consider on a bounded domain \(\Omega \subset {\mathbb{R}}^N\) , the Schrödinger operator ? Δ ? V supplemented with Dirichlet boundary solutions. The potential V is either the critical inverse square potential V(x) = (N ? 2)²/4|x|^?2 or the critical borderline potential V(x) =  (1/4)dist(x, ?Ω)^?2. We present explicit asymptotic estimates on the eigenvalues of the critical Schrödinger operator in each case, based on recent results on improved Hardy–Sobolev type inequalities.     

Percolation and the electron–electron interaction in an array of antidots

A square lattice of microcontacts with a period of 1 μm in a dense low-mobility two-dimensional electron gas is studied experimentally and numerically. At the variation of the gate voltage V _g , the conductivity of the array varies by five orders of magnitude in the temperature range T from 1.4 to 77 K in good agreement with the formula σ(V _g ) = (V _g ?V _g ^* (T))^β with β = 4. The saturation of σ(T) at low temperatures is absent because of the electron–electron interaction. A random-lattice model with a phenomenological potential in microcontacts reproduces the dependence σ(T, V _g ) and makes it possible to determine the fraction of microcontacts x(V _g , T) with conductances higher than σ. It is found that the dependence x(V _g ) is nonlinear and the critical exponent in the formula σ ∝ ? (x - 1/2) ^t in the range 1.3 < t(T, V _g ) < β.     

Screening and antiscreening effects in <Emphasis Type="Italic">J</Emphasis>/ψ production on nuclei

Nuclear effects in J/ψ hadro-and electroproduction on nuclei are considered in the framework of the reggeon approach. It is shown that the screening regime which holds for electroproduction at x _F ≥0.7 and for hadroproduction at x _F ≥?(0.2–0.4) changes with the antiscreening regime for smaller x _F values.     

A statistical model of a metallic inclusion in semiconducting media

The atomic dynamics of the binary Al_100–xCu_x system is simulated at a temperature T = 973 K, a pressure p = 1.0 bar, and various copper concentrations x. These conditions (temperature, pressure) make it possible to cover the equilibrium liquid Al_100–xCu_x phase at copper concentrations 0 ≤ x ≤ 40% and the supercooled melt in the concentration range 40% ≤ x ≤ 100%. The calculated spectral densities of the time correlation functions of the longitudinal \({\tilde C_L}\)(k, ω) and transverse \({\tilde C_T}\)(k, ω) currents in the Al_100–xCu_x melt at a temperature T = 973 K reveal propagating collective excitations of longitudinal and transverse polarizations in a wide wavenumber range. It is shown that the maximum sound velocity in the v_L(x) concentration dependence takes place for the equilibrium melt at an atomic copper concentration x = 10 ± 5%, whereas the supercooled Al_100–xCu_x melt saturated with copper atoms (x ≥ 40%) is characterized by the minimum sound velocity. In the case of the supercooled melt, the concentration dependence of the kinematic viscosity ν(x) is found to be interpolated by a linear dependence, and a deviation from the linear dependence is observed in the case of equilibrium melt at x < 40%. An insignificant shoulder in the ν(x) dependence is observed at low copper concentrations (x < 20%), and it is supported by the experimental data. This shoulder is caused by the specific features in the concentration dependence of the density ρ(x).     

Inflation of the early cold Universe filled with a nonlinear scalar field and a nonideal relativistic Fermi gas

We consider a possible scenario for the evolution of the early cold Universe born from a fairly large quantum fluctuation in a vacuum with a size a ₀ ? l _P (where l _P is the Planck length) and filled with both a nonlinear scalar field φ, whose potential energy density U(φ) determines the vacuum energy density λ, and a nonideal Fermi gas with short-range repulsion between particles, whose equation of state is characterized by the ratio of pressure P(n _F ) to energy density ε(n _F ) dependent on the number density of fermions n _F . As the early Universe expands, the dimensionless quantity ν(n _F ) = P(n _F )/ε(n _F ) decreases with decreasing n _F from its maximum value ν_max = 1 for n _F → ∞ to zero for n _F → 0. The interaction of the scalar and gravitational fields, which is characterized by a dimensionless constant ξ, is proportional to the scalar curvature of four-dimensional space R = κ[3P(n _F )–ε(n _F )–4λ] (where κ is Einstein’s gravitational constant), and contains terms both quadratic and linear in φ. As a result, the expanding early Universe reaches the point of first-order phase transition in a finite time interval at critical values of the scalar curvature R = R _c =–μ²/ξ and radius a _c ? a ₀. Thereafter, the early closed Universe “rolls down” from the flat inflection point of the potential U(φ) to the zero potential minimum in a finite time. The release of the total potential energy of the scalar field in the entire volume of the expanding Universe as it “rolls down” must be accompanied by the production of a large number of massive particles and antiparticles of various kinds, whose annihilation plays the role of the Big Bang. We also discuss the fundamental nature of Newton’ gravitational constant G _N .     

Survival of Near-Critical Branching Brownian Motion

Consider a system of particles performing branching Brownian motion with negative drift \(\mu= \sqrt{2 - \varepsilon}\) and killed upon hitting zero. Initially there is one particle at x>0. Kesten (Stoch. Process. Appl. 7:9–47, 1978) showed that the process survives with positive probability if and only if ε>0. Here we are interested in the asymptotics as ε→0 of the survival probability Q _μ (x). It is proved that if \(L=\pi/\sqrt{\varepsilon}\) then for all x∈?, lim? _ε→0 Q _μ (L+x)=θ(x)∈(0,1) exists and is a traveling wave solution of the Fisher-KPP equation. Furthermore, we obtain sharp asymptotics of the survival probability when x<L and L?x→∞. The proofs rely on probabilistic methods developed by the authors in (Berestycki et al. in arXiv:1001.2337, 2010). This completes earlier work by Harris, Harris and Kyprianou (Ann. Inst. Henri Poincaré Probab. Stat. 42:125–145, 2006) and confirms predictions made by Derrida and Simon (Europhys. Lett. 78:60006, 2007), which were obtained using nonrigorous PDE methods.     

Ferroelectric phase transitions in ionic conductors based on bismuth vadanate

Ceramic solid solutions (Bi_{1 ? x} La _x )₄V₂O_{11 ? z} (I), Bi₄(V_{1 ? x} Fe _x )₂O_{11 ? y} (II), and (Bi_{1 ? x} La _x )₄(V_0.96Fe_0.04)₂O_{11 ? y} (III) (x = 0–0.3, step Δx = 0.02) are prepared using solid-phase synthesis. The concentration and temperature ranges of stabilization of different polymorphic modifications, including the ranges of concentrations x corresponding to the stabilization of the ferroelectric phase, are established. It is revealed that an increase in the concentration x in the region of existence of the pseudoorthorhombic phase α of the solid solutions studied leads to a decrease in the transition temperature, smearing of the transition, and an increase in the width of the thermal hysteresis of the ferroelectric phase transition. The effect of compressing of the domain walls by oxygen vacancies was revealed in the samples from the region of existence of the ferroelectric α phase, and the effect of dielectric relaxation was detected in the samples from the region of existence of the orthorhombic phase β.     

Extremal Theory for Spectrum of Random Discrete Schrödinger Operator. II. Distributions with Heavy Tails

We study the asymptotic structure of the first K largest eigenvalues λ _k,V and the corresponding eigenfunctions ψ(?;λ _k,V ) of a finite-volume Anderson model (discrete Schrödinger operator) \(\mathcal{H}_{V}= \kappa \Delta_{V}+\xi(\cdot)\) on the multidimensional lattice torus V increasing to the whole of lattice ? ^ν , provided the distribution function F(?) of i.i.d. potential ξ(?) satisfies condition ?log(1?F(t))=o(t ³) and some additional regularity conditions as t→∞. For z∈V, denote by λ ⁰(z) the principal eigenvalue of the “single-peak” Hamiltonian κΔ _V +ξ(z)δ _z in l ²(V), and let \(\lambda^{0}_{k,V}\) be the kth largest value of the sample λ ⁰(?) in V. We first show that the eigenvalues λ _k,V are asymptotically close to \(\lambda^{0}_{k,V}\). We then prove extremal type limit theorems (i.e., Poisson statistics) for the normalized eigenvalues (λ _k,V ?B _V )a _V , where the normalizing constants a _V >0 and B _V are chosen the same as in the corresponding limit theorems for \(\lambda^{0}_{k,V}\). The eigenfunction ψ(?;λ _k,V ) is shown to be asymptotically completely localized (as V↑?) at the sites z _k,V ∈V defined by \(\lambda^{0}(z_{k,V})=\lambda^{0}_{k,V}\). Proofs are based on the finite-rank (in particular, rank one) perturbation arguments for discrete Schrödinger operator when potential peaks are sparse.     

Threshold perturbations in current-carrying superconducting bridges with a finite length near the critical temperature

Near the critical temperature of a superconducting transition, the energy of the threshold perturbation δF_thr that transfers a superconducting bridge to a resistive state at a current below the critical current I_c has been determined. It has been shown that δF_thr increases with a decrease in the length of a bridge for short bridges with lengths L < ξ (where ξ is the coherence length) and is saturated for long bridges with L ? ξ. At certain geometrical parameters of banks and bridge, the function δF_thr(L) at the current I → 0 has a minimum at L ~ (2–3)ξ. These results indicate that the effect of fluctuations on Josephson junctions made in the form of short superconducting bridges is reduced and that the effect of fluctuations on bridges with lengths ~(2–3)ξ is enhanced.