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.     

Properties of the φ and f′ produced in the channel Kp → KpKK̄ at 10 GeV/c

Data are presented and cross sections given for ф and f′ production in the channel K⁺p → K⁺pKK? at 10 Gev/c. The resonance parameters obtained from a fit to the KK? effective mass distribution are M = (1020.4 ± 0.5) MeV/c² and Γ = (5.0 ± 1.3) MeV/c² for the ф meson and M = (1514 ± 4) MeV/c² and Γ = (28 ± 15) MeV/c² for the f′. The resonance widths are corrected for experimental resolution. The branching fraction for the rate (ф → K⁰₁K⁰₂)/(ф → K⁺K⁻) is determined to be 1.15 ± 0.15. The angular distributions for the decay of the f′ have unnormalized moments _H(2,0) and H(4,0) different from the values for the nearby background, indicating spin 2 or greater for the f′.     

Analytical solution of the double-logarithmic integral equation with QCD running coupling

The analytical solution of the double-logarithmic integral equation with QCD running coupling describing small-x behaviour of the non-singlet structure function ? _NS(x,Q ²) has been found for any cut-off parameter μ. Analytical properties of the solution and a position of the right-most singularity in the complex ρ-plane which determines the asymptotics of ? _NS(x,Q ²) at small x have been studied. The asymptotical formula ? _NS(x,Q ²) = C ₁ x ^-λ1{ln^κ1(Q ²/Λ²) —ln^κ1 (μ ²/Λ²) + κ ₁ ln^κ1-1(Q ²/Λ²)[ψ(1) - ψ(λ₁)]} valid if x ? 1 and ln(Q ²/Λ²) ? 1 has been obtained where C ₁, λ₁ are constants, κ ₁ = g/λ₁, λ₁ < g = 8/(33 - 2gh _f), gh _f is a number of active flavours and ψ(ξ) denotes the digamma function.     

On boundary integral operators for diffraction problems on graphs with finitely many exits at infinity

We consider a diffraction problem in a multi-connected domain ?² \ Γ, where Γ is an oriented graph with finitely many edges some of which are infinite. The problem is described by the Helmholtz equation (1) $\mathcal{H}u(x) = \rho (x)\nabla \cdot \rho ^{ - 1} (x)\nabla u(x) + k^2 (x)u(x) = 0,x \in \mathbb{R}^2 \backslash \Gamma ,$ where ρ and k are functions bounded together with all derivatives, and by the transmission conditions (2) $u_ + (t) - u_ - (t) = 0,t \in \Gamma \backslash \mathcal{V},$ (3) $a_ + (t)(\partial u/\partial n_t )_ + (t) - a_ - (t)(\partial u/\partial n_t )_ - (t) + a_0 (t)u(t) = f(t),t \in \Gamma \backslash \mathcal{V},$ where V is the set of vertices, a _± and a ₀ are functions bounded on Γ, slowly oscillating discontinuous at the vertices in V, and slowly oscillating at infinity, and f ∈ L ²(Γ). Using Green’s function for the Helmholtz operator H, we introduce simple- and double-layer potentials and reduce the diffraction problem (1)–(3) to a boundary integral equation. The main objective of the paper is to study the essential spectrum, the Fredholm property, and the index of boundary operators on Γ associated with the problem (1)–(3).     

Perturbation of the Laplacian supported by null sets,with applications to polymer measures and quantum fields

We discuss hamiltonians in L²(R^d, dx) of the form H = ?Δ + V, with V a potential supported by a zero measure set C. In particular if C is a path of a brownian motion b such that V(x) = ∫₀¹λ(x, ω)δ(x-b(s, ω)) ds, we show that H exists as a nontrivial, self-adjoint, lower bounded perturbation of ?Δ when d ?5. We must choose λ to be an infinitesimal, negative function for d = 4,5, but for d ? 3 any bounded real-valued function λ will do. The connection with Edward's model of polymers as well as with quantum fields of the ?_d⁴-type is also discussed. The proofs use methods of nonstandard analysis.     

The generalized exponential-integral V(x,y)=∫1∞ exp(−xt)ln(t+y)dtt and computer algorithms for y = 0, 1

The generalized exponential-integral function V(x, y) defined here includes as special cases the function E⁽²⁾₁(x) = V(x, 0) introduced by van de Hulst and functions M₀(x) = V(x, 1) and N₀(x) = V(x, -1) introduced by Kourganoff in connection with integrals of the form ∫ E_n)t)E_m(t±x), which play an important role in the theory of monochromatic radiative transfer. Series and asymptotic expressions are derived and, for the most important special cases, y = 0 and y = 1, Chebyshev expansions and rational approximations are obtained that permit the function to be evaluated to at least 10 sf on 0<x<∞ using 16 sf arithmetic.     

A multi-channel scattering theory for some time dependent Hamiltonians,charge transfer problem

Scattering theory for time dependent HamiltonianH(t)=?(1/2) Δ+ΣV _j (x?q _j (t)) is discussed. The existence, asymptotic orthogonality and the asymptotic completeness of the multi-channel wave operators are obtained under the conditions that the potentials are short range: |V _j (x)|≦C _j (1+|x|)^?2?ε, roughly spoken; and the trajectoriesq _j (t) are straight lines at remote past and far future, and |q _j (t)?q _k (t)| → ∞ ast → ± ∞ (j≠k).     

McMillan-Rowell tunneling spectroscopy in the Nb-Zr system

We report the first full application of tunneling spectroscopy to a superconducting transition metal alloy: Nb_1?xZr_x at x = 0.25, corresponding to the maximum T_c in the Nb-Zr system. The spectral function α²F(ω) and related parameters, when compared to those for the Nb, confirm that the increase in T_c from 9.22 K (x = 0) to 10.8 K(x = 0.25) arises largely by softening of the effective phonon spectrum.     

La–Zn Substituted Hexaferrites Prepared by Chemical Method

La–Zn substituted M-type Ba hexaferrite powders were prepared by sol-gel (Mx) and organometallic precursor (Sk) methods with Fe/Ba ratio of 11.6 and 10.8, respectively. The compositions (LaZn) _x Ba_{1 ? x} Fe_{12 ? x} O₁₉ with 0.0 ≤ x ≤ 0.6 were annealed at 975°C/2 h. The cationic site preferences of nonmagnetic La³⁺ instead of Ba²⁺ ions and Zn²⁺ instead of Fe³⁺ ions were determined by Mössbauer spectroscopy. The La³⁺ ions substitute the large Ba²⁺ ions at 2a site and for x ≥ 0.4 also at 4f₂ site. The nearly all Zn²⁺ ions are placed at the 4f₁ sites. The thermomagnetic analysis of χ(?) confirms that only the small substitutions for x ≤ 0.4 can be taken as a single-phase hexaferrites. The coercivity H _c almost does not change at x = 0.2 for (Mx) samples and further decrease up to x = 0.6. For (Sk) samples at substitution x = 0.2 the values of H _c are decreasing and at higher x the values nearly do not change. The Curie points, T _c, slowly decrease with x for both (Mx) and (Sk) samples.     

The green function of an infinite,fluid loaded membrane

In this paper the response of a fluid loaded plane structure (a membrane) to a concentrated line force excitation is considered in great detail. The normalized velocity response—here called the Green function G—depends upon a dimensionless range x₀=k_m|x|, where k_m is the free wavenumber on the membrane in a vacuum, on the Mach number $\text{M=}\text{k}{}_0\text{k}{}_{\mathrm{m}}$, the ratio of wave phase speed ω/k_m on the unloaded membrane to the sound speed ω/k₀, and on a parameter ? which can be regarded as a measure of fluid loading at the “coincidence” condition M=1. In the analogous problem involving a thin elastic plate, the corresponding parameter is independent of frequency and plate thickness and may be regarded as an intrinsic measure of fluid loading; moreover, in cases of common interest (steel in water, aluminium in air) that parameter is small. In the present paper, the asymptotic structure of G(x₀, M, ?) is therefore sought in the limit ? → 0. Naturally, no single asymptotic expansion can be expected to be valid throughout the (x₀, M) plane, and the programme therefore involves the delineation of regions of that plane in which distinct asymptotic results apply, the construction and discussion of those results, and the asymptotic matching (according to the procedures of the method of matched asymptotic expansions) of results holding in adjoining regions. The Fourier integral for G is broken into surface wave and acoustic components, and the asymptotic structure obtained for each. Previously obtained results for the behaviour at large distances are recovered, with a demonstration that very large distances indeed (x₀ ? ?^?2) may be needed for their validity for some ranges of M; and the drive point behaviour, of G(x₀=0, M, ?) as ? → 0 qua function of M, is shown to correspond to that already discussed in the literature. Elsewhere, in the covering of the whole (x₀, M) plane by different asymptotic expressions, a wide variety of analytical results is found, reflecting the achievement in different regions of different balances among the five competing physical mechanisms represented in the model: namely, structural stiffness, structural inertia, fluid pressures, fluid compressibility and fluid inertia. These different balances give rise to a wide variety of expressions for the phase and amplitude of the surface wave and acoustic components which can now be used to isolate the dominant structural and acoustic mechanisms at any point in the (x₀, M) plane.     

Quantum transport in a time-dependent random potential with a finite correlation time

Using an earlier density matrix formalism in momentum space we study the motion of a particle in a time-dependent random potential with a finite correlation time τ, for 0 < t ? τ. Within this domain we consider two subdomains bounded by kinetic time scales (t _c ² = 2m? ^-1 c ², c ² = σ ², ξ ², σξ, with 2σ the width of an initial wavepacket and the correlation length of the gaussian potential fluctuations), where we obtain power law scaling laws for the effect of the random potential in the mean squared displacement 〈x ²〉 and in the mean kinetic energy 〈E _kin〉. At short times, ? min (t _σ ², 1/2t _ξ ²), 〈x ²〉 and 〈E _kin〉 scale classically as t ⁴ and t ², respectively. At intermediate times, t _σξ ? t ? 2t _σ ² and 1/2t _ξ ² ? t ? t _σξ, these quantities scale quantum mechanically as t ^3/2 and as √t, respectively. These results lie in the perspective of recent studies of the existence of (fractional) power law behavior of 〈x ²〉 and 〈E _kin〉 at intermediate times. We also briefly discuss the scaling laws for 〈x ²〉 and 〈E _kin〉 at short times in the case of spatially uncorrelated potential.     

Pressure experiment determination of the direct-indirect transition in the quarternary In1−xGaxP1−zAsz

Spontaneous and laser emission from In_1-xGa_xP_1-zAs_z double heterojunction diodes near the direct-indirect crossover (E_Γ = E_X, x ≡ x_c, z ≡ z_c) are studied at 77°K as a function of hydrostatic pressure up to 6 kbar. The pressure coefficients of the spontaneous emission peaks and of the laser modes are ～- 10.5 × 10^-6 eV/bar which is characteristics of the Γ band edge in III–V semiconductors. Laser threshold current is found to rise rapidly as pressure is applied owing to the decreasing Γ-X separation and the resultant carrier transfer to the X minima. Experimental lower limits for the direct-indirect crossover at three points in the In_1-xGa_xP_1-zAs_z quaternary system are determined. These three points and the established crossover in GaAs_1-yP_y (y_c ≈ 0.46, 77°K) give for the quaternary crossover (77°K) x_c ? 0.52z_c = 0.72. and the value x_c ≈ 0.72 for the limiting case of In_1-xGa_xP. Band edge bowing effects along the direct-indirect crossover in the In_1-xGa_xP_1-zAs_z system are discussed. The highest energy laser (77°K) for this quaternary system is estimated from pressure measurements to be ～ 2.155 eV (5752 Å).     

Phase transitions in manganites with substitution of divalent ions for manganese

Experimental data are presented for the position of phase boundaries “orthorhombic-rhombohedral structure” and “semiconductor-metal” in manganites of La_{1 ? c + x} Sr _{c ? x} ²⁺ Mn_{1 ? x} Me _x ²⁺ O_{3 + γ} (Me = Mg, Zn, Ni) systems depending on the concentration of substituting divalent cations (0.15 ≤ c ≤ 0.35; 0.025 ≤ x ≤ 0.100).     

Stability of bipolarons in the presence of a magnetic field

The stability of large Fröhlich bipolarons in the presence of a static magnetic field is investigated with the path integral formalism. We find that the application of a magnetic field (characterized by the cyclotron frequence ω _c) favors bipolaron formation: (i) the critical electronphonon coupling parameter α _c (above which the bipolaron is stable) decreases with increasing ω _c and (ii) the critical Coulomb repulsion strength U _c (below which the bipolaron is stable) increases with increasing ω _c. The binding energy and the corresponding variational parameters are calculated as a function of α, U and ω _c. Analytical results are obtained in various limiting cases. In the limit of strong electron-phonon coupling (α ? 1) we obtain for ω _c ? 1 that E _estim ? E _estim(ω _c = 0) + c(u)ω _c/α ⁴ with c(u) an explicitly calculated constant, dependent on the ratio u = U/α where U is the strength of the Coulomb repulsion. This relation applies both in 2D and in 3D, but with a different expression for c(u). For ω _c ? α ²? 1 we find in 3D E _estim ? ω _c - α ² A(u) ln²(ω _c/α ²), (also with an explicit analytical expression for A(u)) whereas in 2D E _estim ^2D ? ω _c - α√ω _cπ(u-2-√2)/2. The validity region of the Feynman-Jensen inequality for the present problem, bipolarons in a magnetic field, remains to be examined.     

Upper critical fields of superconducting Gd and Tm doped LaSn3: Effects of crystalline electric fields

The temperature dependence of the upper critical fields, H_c²(T), are presented for (La_1-xGd_x)Sn₃ and (La_1-xTm_x)Sn₃. For samples with nearly the same T_c, H_c²(T) of the Tm-doped LaSn₃ samples are always larger than those for the Gd-doped samples. The results are interpreted in terms of crystalline electric field splitting of magnetic levels of the Tm³⁺. Pure LaSn₃ is found to be a Type I superconductor.     

Electrical conductivity of (Bi,Pb)2MO4 (M = Pd,Pt) linear chain compounds

Partial oxidation of Pd in Bi₂PdO₄ is achieved by substitution of Pb²⁺ for Bi³⁺ up to Bi₁₉₁Pb₀₀₉PdO₄, partial oxidation is necessary to stabilize the isostructural Pt compound, Bi_1?xPb_xPtO₄ within the range 0.33 ? x ? 0.52. In both cases, the tetragonal cell c parameter, therefore metal-metal distance $\text{(d}{}_{\mathrm{<mtext>M?M</mtext>}}\text{=}\text{c}\text{2}\text{)}$, decreases linearly with increasing mean oxidation degree (MOD) of transition metal atom For the insulator B1₂CuO₄, no substitution occurs Powder electrical conductivity measurements of the partially oxidized compounds show that these materials are semiconductors Platinum compounds exhibit relatively high conductivities $\text{(σ?10 (Ω}\text{cm}\text{)}{}^{\mathrm{?1}}\text{)}$ and low activation energies (?0 02 eV) with small variations with x Palladium compounds exhibit lower conductivities which linearly increases with MOD These electronic properties are comparable with those of the most one-dimensional Pt or Pd chain conductors.     

Dynamical Correlation Functions for an Impenetrable Bose Gas with Neumann or Dirichlet Boundary Conditions

Abstract

We study the time and temperature dependent correlation functions for an impenetrable Bose gas with Neumann or Dirichlet boundary conditions ψ(x ₁, 0)ψ ^?(x ₂ , t) _±,T . We derive the Fredholm determinant formulae for the correlation functions, by means of the Bethe Ansatz. For the special case x ₁ = 0, we express correlation functions with Neumann boundary conditions ψ(0, 0)ψ ^?(x ₂ , t) _+,T , in terms of solutions of nonlinear partial differential equations which were introduced in [1] as a generalization of the nonlinear Schrödinger equations. We generalize the Fredholm minor determinant formulae of ground state correlation functions ψ(x ₁)ψ ^?(x ₂) _±,0 in [2], to the Fredholm determinant formulae for the time and temperature dependent correlation functions ψ(x ₁, 0)ψ ^?(x ₂ , t) _±,T , t ∈ R, T ≥ 0.