Analysis of the spin splitting of maxima of quantum oscillations of the resistivity of n-Bi-Sb semiconductor alloys in a magnetic field parallel to the bisector axis

In samples of semiconductor alloys n-Bi_0.93Sb_0.07 with different electron concentrations (n ₁ = 8 × 10¹⁵ cm^?3, n ₂ = 1.2 × 10¹⁷ cm^?3, and n ₃ = 1.9 × 10¹⁸ cm^?3), dependences of the electrical resistivity on magnetic fields up to 45 T parallel to the current and the bisector axis (H ‖ C ₁ ‖ j) have been measured at temperatures of 1.5, 4.5, and 10 K. The obtained dependences ρ₂₂(H) demonstrate quantum oscillations of the resistivity (Shubnikov-de Haas effect), and, in high magnetic fields, there is a resistivity maximum far away from other maxima. On assumption that this maximum is related to the spin-split Landau level N = 0^? for electrons of the main ellipsoid, the spin-splitting parameters are calculated for electrons of the main ellipsoid: γ₁ = 0.87, γ₂ = 0.8, and γ₃ = 0.73. Using these values, the oscillation maxima can be reliably related to the numbers of split Landau levels for electrons of the main and secondary ellipsoids. The dependences of the resistivity ρ₁₁ and the Hall coefficient R _31.2 on magnetic field have been measured in a transverse magnetic field at H ‖ C ₁ and j ‖ C ₂ on the sample with the electron concentration n ₄ = 1.4 × 10¹⁷ cm^?3. Using similar analysis, the spin-splitting parameter is found to be γ₄ = 0.85, which is close to the value of γ₂ = 0.8 obtained for the sample with close electron concentration (n ₂ = 1.2 × 10¹⁷ cm^?3) during the measurements in a longitudinal magnetic field. The quantum oscillation maxima of Hall coefficient R _31.2 are shifted to the range of high magnetic fields as compared to the quantum oscillation maxima of resistivity ρ₁₁.     

Bloch electron dynamics

New equations of motion for a Bloch electron [momentum p=h k,energy ε _n(p),zone number n, charge -e]: $$m_j \frac{{dv_j }}{{dt}} = - e(E + v \times B)_j $$ are proposed, where v≡?ε_n(p)/?p is the velocity, and {m_j}are the principal masses m _j ^? 1=?²ε_n/?p _j ² along the normal and the two principal axes of curvatures at each point of the constant-energy surface represented by ε=ε_n(p).Their advantages over the prevalent equations of motion where the left-hand-side is replaced by hk _j are demonstrated by examining de Haas-van Alphen oscillations and orientation-dependent cyclotron resonance peaks.     

The spectral class of the quantum-mechanical harmonic oscillator

The purpose of this paper is to study the so-calledspectral class Q of anharmonic oscillatorsQ=?D ²+q having the same spectrum λ _n =2n (n≧0) as the harmonic oscillatorQ ⁰=?D ²+x ²?1. Thenorming constants \(t_n = \mathop {\lim }\limits_{x \uparrow \infty } \ell g[( - 1)^n {{e_n (x)} \mathord{\left/ {\vphantom {{e_n (x)} {e_n }}} \right. \kern-0em} {e_n }}( - x)]\) of the eigenfunctions ofQ form a complete set of coordinates inQ in terms of which the potential may be expressed asq=x ²?1?2D ² ?g? with $$\theta = \det \left[ {\delta _{ij} + (e^{ti} - 1)\int\limits_x^\infty {e_i^0 e_j^0 :0 \leqq i,j,< \infty } } \right],$$ e _n ⁰ being then ^th eigenfunctionQ ⁰. The spectrum and norming constants are canonically conjugate relative to the bracket [F, G]=∫ΔFDΔGdx,to wit: [λ _i , λj=0, [t _i, 2λ _j ]=1 or 0 according to whetheri=j or not, and [t _i,t _j]=0. This prompts an investigation of the symplectic geometry ofQ. The function ? is related to the theta function of a singular algebraic curve. Numerical results are also presented.     

Eine scheinbare Abschwächung der Lokalitätsbedingung

For a hermitean, scalar, tempered fieldA(x) the locality axiom can be replaced by the following condition: For any two natural numbersn andj with 1≦j<n and for any configurationX(n, j):X ₁, ...,X _j?1,X _j ,X _j+1,X _j+2, ...X _n that is totally space-like in both orders: 1, ...,j?1,j, j+1,j+2, ...,n and 1, ...,j?,j+1,j,j + 2, ...,n there exist constants α(n,j)>2,C(X(n, j))>0,h(X(n, j))>0 such that with \(x_k = X_k \sqrt { - x^2 } \) : $$\begin{gathered} |\langle A(x_1 ) \ldots A(x_{j - 1} )[A(x_j ), A(x_{j + 1} )] A(x_{j + 2} ) \ldots A(x_n )\rangle |< \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,< C(X(n, j)) exp\{ - h(X(n, j))\sqrt { - x^2 } ^{\alpha (n, j)} \} \hfill \\ \end{gathered} $$ for ?x ²>1.     

Finite-N fluctuation formulas for random matrices

For the Gaussian and Laguerre random matrix ensembles, the probability density function (p.d.f.) for the linear statistic Σ _j ^N =1 (x _j ? 〈x〉) is computed exactly and shown to satisfy a central limit theorem asN → ∞. For the circular random matrix ensemble the p.d.f.’s for the statistics ½Σ _j ^N =1 (θ _j ?π) and ? Σ _j ^N =1 log 2 |sinθ _j/2| are calculated exactly by using a constant term identity from the theory of the Selberg integral, and are also shown to satisfy a central limit theorem asN → ∞.     

Reconstruction of singularities for solutions of Schrödinger's equation

We determine the behavior in time of singularities of solutions to some Schrödinger equations onR ⁿ . We assume the Hamiltonians are of the formH ₀+V, where \(H_0 = 1/2\Delta + 1/2 \sum\limits_{k = 1}^n { \omega _k^2 x_k^2 } \) , and whereV is bounded and smooth with decaying derivatives. When all ω _k =0, the kernelk(t,x,y) of exp (?itH) is smooth inx for every fixed (t,y). When all ω₁ are equal but non-zero, the initial singularity “reconstructs” at times \(t = \frac{{m\pi }}{{\omega _1 }}\) and positionsx=(?1) ^m y, just as ifV=0;k is otherwise regular. In the general case, the singular support is shown to be contained in the union of the hyperplanes \(\{ x|x_{js} = ( - 1)^l js_{y_{js} } \} \) , when ω _j t/π=l _j forj=j ₁,...,j _r .     

The ground state energy of a Bose gas with Coulomb interaction

LetH _N be the 2N particle Hamiltonian $$\begin{array}{*{20}c} {H_N = \sum\limits_{i = 1}^{2N} {( - \Delta _\iota ) + \sum\limits_{i< j = 1}^N {\left| {x_i - x_j } \right|^{ - 1} + } \sum\limits_{i< j = 1}^N {\left| {x_{i + N} - x_{j + N} } \right|^{ - 1} } } } \\ { - \sum\limits_{i,j< j = 1}^N {\left| {x_i - x_{j + N} } \right|^{ - 1} ,} } \\ \end{array} $$ whereΔ _i is the Laplacian in the variablex _i ∈?³, 1≦i≦2N. The operatorH _N is assumed to act on wave functionsΨ(x ₁, ...,x _N ;x _N+1, ...,x _2N ) which are symmetric in the variables (x ₁, ...,x _N ) and (x _N+1, ...,x _2N ). SupposeΨ is supported in a setΛ ^2N , whereΛ is a cube in ?³. It is shown that if a normalized wave functionΨ can be written as a product of two wave functions $$\psi (x_1 ,...,x_N ;x_{N + 1} ,...,x_{2N} ) = \psi _1 (x_2 ,...,x_N )\psi _2 (x_{N + 1} ,...,x_{2N} ),$$ and the density of particles inΛ is constant, then 〈Ψ|H _N |Ψ〉≧?CN ^7/5 for some universal constantC.     

Transverse spin correlation function of the one-dimensional spin-12 XY model

The transverse spin pair correlation function p^x_n=<S^x_mS^x_m+n>=<S^x_mS^x_m+n> is calculated exactly in the thermodynamic limit of the system described by the one-dimensional, isotropic, spin-$\text{1}\text{2}$, XY Hamiltonian

$\text{H=?2J}\text{∑}\text{l=1}\text{N}\text{(S}{}^{\mathrm{x}}{}_{\mathrm{l}}\text{S}{}^{\mathrm{x}}{}_{\mathrm{l+1}}\text{+S}{}^{\mathrm{y}}{}_{\mathrm{l}}\text{S}{}^{\mathrm{y}}{}_{\mathrm{l+1}}\text{)}$

. It is found that at absolute zero temperature (T = 0), the correlation function ρ^x_n for n ≥ 0 is given by

$\text{ρ}{}^{\mathrm{x}}{}_{\mathrm{2p}}\text{=}\text{1}\text{4}\text{2}\text{π}{}^{\mathrm{2p}}\text{Π}\text{j=1}\text{p?1}\text{4j}{}^2\text{4j}{}^2\text{?1}{}^{\mathrm{2p?2j}}\text{if}\text{n=2p}$

,

$\text{ρ}{}^{\mathrm{x}}{}_{\mathrm{2p+1}}\text{=±}\text{1}\text{4}\text{2}\text{π}{}^{\mathrm{2p+1}}\text{Π}\text{j=1}\text{p}\text{4j}{}^2\text{4j}{}^2\text{?1}{}^{\mathrm{2p+2j}}\text{if}\text{n=2p+1}$

, where the plus sign applies when J is positive and the minus sign applies when J is negative. From these the asymptotic behavior as n → ∞ of |?^x_n| at T = 0 is derived to be $\text{|ρ}{}^{\mathrm{x}}{}_{\mathrm{n}}\text{| ～}\text{a}\text{n}$ with a = 0.147088?. For finite temperatures, ρ^x_n is calculated numerically. By using the results for ?^x_n, the transverse inverse correlation length and the wavenumber dependent transverse spin pair correlation function are also calculated exactly.     

On-shell T-matrices in multiple scattering

The transition operator T for the scattering of a particle from N potentials V_j(x) can be expanded into a series featuring the transition operators t_j associated with the individual potentials. For V_j(x) both absolutely and square integrable in x, we show, using an analytic continuation argument, that if T is on-shell, i.e. in 〈k|T(k₀²±i0)|k′〉, |k| = |k′| = k₀, then each t_j is also on-shell.     

Creating Very True Quantum Algorithms for Quantum Energy Based Computing

An interpretation of quantum mechanics is discussed. It is assumed that quantum is energy. An algorithm by means of the energy interpretation is discussed. An algorithm, based on the energy interpretation, for fast determining a homogeneous linear function f(x) := s.x = s ₁ x ₁ + s ₂ x ₂ + ? + s _N x _N is proposed. Here x = (x ₁, … , x _N ), x _j ∈ R and the coefficients s = (s ₁, … , s _N ), s _j ∈ N. Given the interpolation values \((f(1), f(2),...,f(N))=\vec {y}\), the unknown coefficients \(s = (s_{1}(\vec {y}),\dots , s_{N}(\vec {y}))\) of the linear function shall be determined, simultaneously. The speed of determining the values is shown to outperform the classical case by a factor of N. Our method is based on the generalized Bernstein-Vazirani algorithm to qudit systems. Next, by using M parallel quantum systems, M homogeneous linear functions are determined, simultaneously. The speed of obtaining the set of M homogeneous linear functions is shown to outperform the classical case by a factor of N × M.     

Schrödinger operator with a nonlocal potential whose absolutely continuous and point spectra coexist

We consider the Schrödinger-like operatorH in which the role of a potential is played by the lattice sum of rank 1 operators \(|\left. {v_n } \right\rangle \left\langle {v_n |} \right.\) multiplied by g tan π[(α,n)+ω],g>0, α∈? ^d ,n∈? ^d , ω∈[0, 1]. We show that if the vector α satisfies the Diophantine condition and the Fourier transform support of the functionsv _n (x)=v(x-n),x∈? ^d ,n∈? ^d , is small then the spectrum ofH consists of a dense point component coinciding with? and an absolutely continuous component coinciding with [?, ∞), where ? is the radius of the mentioned support. Besides, we find the integrated density of statesN(λ) (it has a jump at λ=?) and zero temperature a.c. conductivityσ _λ (v), that also has a jump at λ=? and vanishes faster than any power of the external field frequency ν as ν→0 and λ≠?.     

Derivation of relativistic charge-monopole and monopole-monopole potentials from field theory

Relativistic potentials (generalizing the Breit-potential of quantum electrodynamics) between spin $\text{1}\text{2}\text{-}\text{electric}$ and magnetic charges are presented, each monopole g_j having its own singularity-string along some direction n_j. The monopole potentials involve integrations along the singularities. By using suitable gauge transformations and limiting procedures a simple form of the potential independent of n_j is derived, if the string connects two monopoles.     

Secondary ion emission under the bombardment of Si by multiply charged Si q+ ions

The spectra of secondary ion emission under the bombardment of a B-doped Si target by multiply charged Si ^q+ ions (q = 1?C5) have been studied in the energy range of 1 to 10 keV per unit of charge. A multifold increase in the yield of secondary cluster Sk _n ⁺ ions, multiply charged Si ^q/+ ion (q = 1?C3), and H⁺, C⁺, B⁺, Si₂N⁺, Si₂O⁺ is observed as the charge of the multiply charged ions increases. The increase in the yield of secondary ions with increasing charge of the multiply charged-ion charge is most significant for ions with relatively high ionization potentials.     

Minimal number of operators for observability ofN-level quantum systems

This paper discusses the minimal numbern _min of operatorsA ₁,...,A _n , whose expectation values at some instants determine the statistical state of anN-level quantum system. We assume that the macroscopic information about the system in question is given by the mean values Tr[ρ(t _j )A _i ]=m _i (t _j ) ofn self-adjoint operatorsA ₁,...,A _n at some instantst ₁<t ₂t _s , wheres ²?1.     

Analytic study of θ vacua on the lattice

A new definition of the topological charge density for four-dimensional lattice gauge theory is given. Using a systematic expansion we find a cusp in the vacuum energy at θ = π signaling the spontaneous breaking of CP there. Unlike its two-dimensional analogue (QED₂), QCD confines at θ = π. As a by-product an expression for the topological mass term for (2+1)-dimensional lattice gauge theory is obtained.     

Operator quantization of non-abelian gauge theories in a completely fixed axial gauge

A consistent quantization of chromodynamics in a completely fixed axial gauge is carried out by using the Dirac bracket quantization procedure. The main results are: The translation of Dirac brackets into equal-time commutators is possible, without ambiguities, because of the absence of ordering problems. All equal-time commutators are compatible with constraints and gauge conditions holding as strong operator relations. All equal-time commutators are compatible with chromoelectric, chromomagnetic, and fermionic fields vanishing at spatial infinity. The colored gauge potentials A^0,a, A^1,a, and A^2,a are seen to develop a physically significant, although pure gauge, behavior at x³ = ± ∞, as required by the presence of a nontrivial topological content. Poincaré invariance is satisfied without introducing in the Hamiltonian “extra” quantum mechanical potentials. The determinant of the Faddeev-Popov matrix does not depend upon the field variables.     

Density functional study of $$\hbox {AgScO}_2$$: Electronic and optical properties

Dielectric relaxation studies of binary (jk) polar mixtures of tetrahydrofuran with N-methyl acetamide, N,N-dimethyl acetamide, N-methyl formamide and N,N-dimethyl formamide dissolved in benzene(i) for different weight fractions (w _{j k} ’s) of the polar solutes and mole fractions (x _j ’s) of tetrahydrofuran at 25 ^°C are attempted by measuring the conductivity of the solution under 9.90 GHz electric field using Debye theory. The estimated relaxation time (τ _{j k} ’s) and dipole moment (μ _{j k} ’s) agree well with the reported values signifying the validity of the proposed methods. Structural and associational aspects are predicted from the plot of τ _{j k} and μ _{j k} against x _j of tetrahydrofuran to arrive at solute–solute (dimer) molecular association upto x _j =0.3 of tetrahydrofuran and thereafter solute–solvent (monomer) molecular association upto x _j =1.0 for all systems except tetrahydrofuran + N,N-dimethyl acetamide.     

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.     

Supersymmetric flipped SU(5) revitalized

We describe a simple N = 1 supersymmetric GUT based on the group SU(5)×U(1) which has the following virtues: the gauge group is broken down to the SU(3)_C×SU(2)_L×U(1)_Y of the standard model using just 10, $\text{10}$ Higgs representations, and doublet-triplet mass splitting problem is solved naturally by a very simple missing-partner mechanism. The successful supersymmetric GUT prediction for sin²θ_w can be maintained, whilst there are no fermion mass relations. The gauge group and representation structure of the model may be obtainable from the superstring.     

Degree Complexity of Birational Maps Related to Matrix Inversion

For a q × q matrix x = (x _{i, j} ) we let ${J(x)=(x_{i,j}^{-1})}For a q × q matrix x = (x _{i, j} ) we let J(x)=(x_i,j^-1){J(x)=(x_{i,j}^{-1})} be the Hadamard inverse, which takes the reciprocal of the elements of x. We let I(x)=(x_i,j)^-1{I(x)=(x_{i,j})^{-1}} denote the matrix inverse, and we define K=I°J{K=I\circ J} to be the birational map obtained from the composition of these two involutions. We consider the iterates Kⁿ=K°?°K{K^n=K\circ\cdots\circ K} and determine the degree complexity of K, which is the exponential rate of degree growth d(K)=lim_n?￥( deg(Kⁿ) )^1/n{\delta(K)=\lim_{n\to\infty}\left( deg(K^n) \right)^{1/n}} of the degrees of the iterates. Earlier studies of this map were restricted to cyclic matrices, in which case K may be represented by a simpler map. Here we show that for general matrices the value of δ(K) is equal to the value conjectured by Anglès d’Auriac, Maillard and Viallet.