The low energy scattering for slowly decreasing potentials

For the radial Schrödinger equation with a potentialq(x) decreasing at infinity asq ₀ q ^–, (0, 2), the low energy asymptotics of spectral and scattering data is found. In particular, it is shown that forq ₀>0 the spectral function vanishes exponentially as the energyk ² tends to zero. On the contrary, there is always a zero-energy resonance forq ₀<0. These results determine the local asymptotics of solutions of the time-dependent Schrödinger equation for large timest. Specifically, for positive potentials its solutions decay as exp(–₀ t ^(2–)/(2+), ₀>0,t. In the case (1, 2) it is shown that for ±q ₀>0 the phase shift tends to ± ask0 and its asymptotics is evaluated.     

Ideals and solutions of nonlinear field equations

Families of horizontal ideals of contact manifolds of finite order are studied. Each horizontal ideal is shown to admit ann-dimensional module of Cauchy characteristic vectors that is also a module of annihilators (in the sense of Cartan) of the contact ideal. Since horizontal ideals are generated by 1-forms, any completely integrable horizontal ideal in the family leads to a foliation of the contact manifold by submanifolds of dimensionn on which the horizontal ideal vanishes. Explicit conditions are obtained under which an open subset of a leaf of this foliation is the graph of a solution map of the fundamental ideal that characterizes a given system of partial differential equations of finite order withn independent variables. The solution maps are obtained by sequential integration of systems of autonomous ordinary differential equations that are determined by the Cauchy characteristic vector fields for the problem. We show that every smooth solution map can be obtained in this manner. Let {_Vi¦1in} be a basis for the module of Cauchy characteristic vector fields that are in Jacobi normal form. If a subsidiary balance ideal admits each of then vector fieldsV _i as a smooth isovector field, then certain leaves of the foliation generated by the corresponding closed horizontal ideal are shown to be graphs of solution maps of the fundamental ideal. A subclass of these constructions agree with those of the Cartan-Kähler theorem. Conditions are also obtained under which every leaf of the foliation is the graph of a solution map. Solving a given system ofr partial differential equations withn independent variables on a first-order contact manifold is shown to be equivalent to the problem of constructing a complete system of independent first integrals. Properties of systems of first integrals are analyzed by studying the collection ISO[A _ij ] of all isovectors of the horizontal ideal. We show that ISO[A _ij ] admits the direct sum decomposition ^*[A _ij ]W[A _ij ] as a vector space, where ^*[A _ij ] is the module of Cauchy characteristics of the horizontal ideal. ISO[A _ij ] also forms a Lie algebra under the standard Lie product,^*[A _ij ] andW[A _ij ] are Lie subalgebras of ISO[A _ij ], and [A _ij ] is an ideal. A change of coordinates that resolves (straightens out) the canonical basis for ^*[A _ij ] is constructed. This change of coordinates is used to reduce the problem of solving the given system of PDE to the problem of root extraction of a system ofr functions ofn variables, and to establish the existence of solutions to a second-order system of overdetermined PDE that generate the subspaceW[A _ij ]. Similar results are obtained for second-order contact manifolds. Extended canonical transformations are studied. They are shown to provide algorithms for calculating large classes of closed horizontal ideals and a partial analog of classical Hamilton-Jacobi theory.     

Intrinsic point defects in niobium: Trapping at100Pd and111In impurities observed by PAC

Radiation-induced lattice defects in high-purity niobium have been investigated in the temperature range of 30K to 540 K by means of - perturbed angular correlation (PAC) measurements using the radioactive probes¹⁰⁰Pd/¹⁰⁰Rh and¹¹¹In/¹¹¹Cd. Both probes were produced within the niobium samples by means of heavy-ion nuclear reactions. At the Pd impurities trapping of defects occurred during heavy-ion irradiation at about 30 K in two defined configurations: defect 1(Pd) withv _Q1=e ² qQ/h=42(±2) MHz, ₁=0 and defect 2 (Pd) withv _Q2=(±2) MHz, ₂=1. Two defects were observed at the In impurities in annealing stage III (around 250 K) after heavy-ion as well as electron irradiations: defect 1(In) withv _Q1=87(±1) MHz, ₁=0 and defect 2(In) withv _Q2=105(±2) MHz, ₂=0.65(±0.02). A third defect (defect 3(In):v _Q3=177(±2) MHz, ₃0.2) appeared above 260 K after heavy-ion irradiation only. The data are interpreted in terms of interstitial trapping at the Pd impurities and vacancy trapping at the In impurities. Information on the microscopic structure of defect 1(In) and 2(In) is obtained from a PAC-single-crystal experiment. For defect 1(In) axial 111-symmetry is found, which leads us to identify this defect with a monovacancy as nearest neighbor with respect to the In probe. Defect 2(In) is the trapped divacancy for which an orientation is found that is consistent with both vacancies being nearest neighbor to the probe but second nearest neighbors to each other.     

Nonlinear Optical Characteristics of BSO and BGO Photorefractive Crystals in Visible and Infrared Ranges

Nonlinear refraction, nonlinear absorption and optical limiting in photorefractive crystals Bi₁₂SiO₂₀(BSO) and Bi₁₂GeO₂₀(BGO) at the wavelengths of 1064 and 532 nm were investigated. It was shown that both BSO and BGO crystals possess by positive nonlinear refraction in two investigated spectral ranges (n ₂ ^BSO=(2.5 ± 0.5)× 10^–12 esu, n ₂ ^BGO=(6.3 ± 1.3)× 10^–12 esu at equals 1064 nm; n ₂ ^BSO=(4.4 plusmn; 0.9)× 10^–12 esu, n ₂ ^BGO=(7.4 ± 1.5)× 10^–12 esu at = 532 nm). The nonlinear absorption was due to three-photon absorption at the wavelength of 1064 nm ( ⁽³⁾ _BSO=(2.5 ± 0.8)× 10^–20cm³W^–2, ⁽³⁾ _BSO=(4.4 ± 1.3)× 10^–20cm³W^–2) and two-photon absorption at the wavelength of 532 nm ( ⁽²⁾ _BSO=(2 ± 0.4)× 10^–9cm W^–1, ⁽²⁾ _BGO=(3.7 ± 0.7)× 10^–9cm W^–1).     

Gauge and Space-Time Symmetry Unification

Unification ideas suggest an integral treatment of fermion and boson spin andgauge-group degrees of freedom. Hence, a generalized quantum field equation,based on Dirac's, is proposed and investigated which contains gauge and flavorsymmetries, determines vector gauge field and fermion solution representations,and fixes their mode of interaction. The simplest extension of the theory with a6-dimensional Clifford algebra has an SU(2) _L × U(1) symmetry, which isassociated with the isospin and the hypercharge, their vector carriers, two-flavorcharged and chargeless leptons, and scalar particles. A mass term producesbreaking of the symmetry to an electromagnetic U(1), and a Weinberg's angle_W with sin ²(_W) = 0.25. A more realistic 8D extension gives coupling constantsof the respective groups g = 1/2 .707 and g = 1/6 .408, with thesame _W.     

The dependence of the neutron temperature on the geometric parameter for water and loose diphenyl

The dependence of the neutron temperatureT _n on the geometric parameterB ² was measured by the pulse method in water and loose diphenyl. The measurements were made on a moderator poisoned by cadmium sulphate, a substance whose absorption cross-section is non 1/v.The following results were obtained: For waterT _n [eV]=–(0·00391±0·00045)B ² [cm^–2]+(0·02537±0·00035) for loose diphenyl:T _n [eV]=–(0·01014±0·00152)B ² [cm^–2]+(0·02518±0·00054).We are indebted to J. Jirou and J. Jadavan for their assistance in the measurements, the accelerator operation and electronic apparatus maintenance.     

Complex temperature plane zeros in the mean-field approximation

We derive asymptotic expressions for the complex temperature plane zeros of the infinite-range Ising model in the scaling regime. The results also apply to high-dimensional, short-range Ising systems. For thenth zero in a system ofN spins, the leading asymptotic result ist _n (n/N)^1/2(–1 ±i).     

Nonlinear lagrangians and Einstein spaces

It is shown that, for a Riemannian spaceV _d of dimensiond, solutions of the equation ((–g)^1/2 R ⁿ)/g^ab = 0 forn = (1/4)(d+2) may be interpreted as (d + 1)-dimensional Einstein spaces.     

Temperature characteristics of positron trapping at defects in electron-irradiated silicon

Positron-annihilation lifetime and Doppler-broadening measurements are used to investigate defects in silicon irradiated at 373 K with 6 MeV electrons to a dose of 1×l0¹⁹e/cm². In the unirradiated silicon sample (p type) a temperature-independent behaviour of the bulk-lifetime is observed in the temperature interval 110–500 K with a constant value of 220±1 ps. The slight effect observed on the S-parameter evolution is explained taking into account the thermal expansion of the lattice. The lifetime results obtained at 80 K and at 300 K after isochronal annealing as well as the behaviour of the intensity of the second lifetime componentI ₂ during lifetime measurements below the irradiation temperature in the irradiated silicon sample (n type), clearly indicate the temperature dependent characteristics of the positron trapping cross section _t(T) T ⁿ withn= –1.905±0.016. From isochronal annealing results, an annealing stage is observed in which di-vacancies agglomerate into quadri-vacancies. The mean positron lifetime in those quadri-vacancies is 350 ps.A.B.O.S., on leave from University of Kinshasa, Zaïre     

On a possible algebra morphism of U q [osp(1/2n)] onto the deformed oscillator algebra W q (n)

We formulate a conjecture stating that the algebra ofn pairs of deformed Bose creation and annihilation operators is a factor algebra of U _q [osp(1/2n)], considered as a Hopf algebra, and prove it for then = 2 case. To this end, we show that for any value ofq, U _q [osp(1/4)] can be viewed as a superalgebra freely generated by two pairsB ₁ ^± ,B ₂ ^± of deformed para-Bose operators. We write down all Hopf algebra relations, an analogue of the Cartan-Weyl basis, the commutation relations between the generators and a basis in U _q [osp(1/2n)] entirely in terms ofB ₁ ^± ,B ₂ ^± .     

Low temperature specific heats of very dilute alloys of Ce in Al

Measurements of the specific heat of two alloys of Ce (0.00275 and 0.0121 at % Ce) in Al have been made in the region 0.3<T<3.8 K. Within the limits of detection (<1 ppm), the alloys contained no metallic impurities other than Ce and their homogeneity was checked by neutron activation analysis. For the alloys in their normal electronic states, the change/c was found to be 6±3 and 3.5±1.5mJ/at% K². The behavior of the alloys in their superconducting states (e.g.: the shift ofT _c with the concentration of Ce; the correlation betweenT _C/T_C0 andC/C ₀) indicates the existence of short-lived local moments associated with the Ce impurities.     

Isomeric transitions between shape coexisting states in iodine isotopes withA=115, 119 and 121

Applying delayed- r.f. coincidences and generalized centroid-shift analysis, nanosecond isomers in some odd-A iodine isotopes are studied in-beam. The reactions¹⁰⁶Cd(¹²C,p2n) and^116,118Sn(⁶Li,3n) have been used. The following half-lives were determined:T _1/2(565 keV)=0.50±0.10 ns andT _1/2(837 keV) =0.90±0.20 ns in¹¹⁵I,T _1/2(687 keV)=5.50±0.50 ns andT _1/2(2418 keV)=4.0±0.50 ns in¹¹⁹I as well asT _1/2(812 keV)=0.35±0.20 ns andT _1/2(445 keV)<0.10 ns=">¹²¹I. A systematics of the absolute probabilities for the transitions deexciting theg _9/2 9/2⁺ and _11/2 11/2 ₁ ^– band heads in^115–125I is presented and discussed.Partly supported by the Bulgarian National Research Foundation under contracts PH14, and PH31 and Kernforschungszentrum Karlsruhe, Stabsabteilung Internationale Beziehungen     

Realizability of a model in infinite statistics

Following Greenberg and others, we study a space with a collection of operatorsa(k) satisfying the q-mutator relationsa(l)a (k)a(l)= _k,l (corresponding forq=±1 to classical Bose and Fermi statistics). We show that then!×n! matrixA _n (q) representing the scalar products ofn-particle states is positive definite for alln ifq lies between –1 and +1, so that the commutator relations have a Hilbert space representation in this case (this has also been proved by Fivel and by Bozejko and Speicher). We also give an explicit factorization ofA _n (q) as a product of matrices of the form(1–q ^jT)^±1 with 1jn andT a permutation matrix. In particular,A _n (q) is singular if and only ifq ^M=1 for some integerM of the formk ²–k, 2kn.     

Scaling of particle trajectories on a lattice

The scaling behavior of the closed trajectories of a moving particle generated by randomly placed rotators or mirrors on a square or triangular lattice is studied numerically. On both lattices, for most concentrations of the scatterers the trajectories close exponentially fast. For special critical concentrations infinitely extended trajectories can occur which exhibit a scaling behavior similar to that of the perimeters of percolation clusters.At criticality, in addition to the two critical exponents =15/7 andd _f=7/4 found before, the critical exponent =3/7 appears. This exponent determines structural scaling properties of closed trajectories of finite size when they approach infinity. New scaling behavior was found for the square lattice partially occupied by rotators, indicating a different universality class than that of percolation clusters.Near criticality, in the critical region, two scaling functions were determined numerically:f(x), related to the trajectory length (S) distributionn _s, andh(x), related to the trajectory sizeR _s (gyration radius) distribution, respectively. The scaling functionf(x) is in most cases found to be a symmetric double Gaussian with the same characteristic size exponent =0.433/7 as at criticality, leading to a stretched exponential dependence ofn _S onS, n_Sexp(–S ^6/7). However, for the rotator model on the partially occupied square lattice an alternative scaling function is found, leading to a new exponent =1.6±0.3 and a superexponential dependence ofn _S onS.h(x) is essentially a constant, which depends on the type of lattice and the concentration of the scatterers. The appearance of the same exponent =3/7 at and near a critical point is discussed.     

Computer simulation of the nonequilibrium critical behavior of disordered two-dimensional Ising systems

We report the results of a computer simulation of the critical relaxation of the magnetization in the two-dimensional Ising model with nonmagnetic impurity atoms frozen at the lattice sites. We assume a square lattice of dimension 400² with spin concentrationsp=1.0, 0.95, 0.9, 0.85, 0.8, 0.75, 0.7. The Monte Carlo and dynamic renormalization group methods are used to determine the dynamical critical indexz as a function ofp: z(p): z(1)=2.24±0.07,z(0.95)=2.24±0.06,z(0.85)=2.38±0.05,z(0.8)=2.51±0.06,z(0.75)=2.66±0.07,z(0.7)=2.88±0.06. It is shown thatz(p) obeys a singular scaling law of the formz=A | ln (p–p _c) |+B withA=0.56±0.07,B=1.62±0.07.Omsk State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 8, pp. 83–88, August, 1994.     

Solution of the Semi-Infinite Toda Lattice for Unbounded Sequences

The semi-infinite Toda lattice is the system of differential equations d _n (t)/dt = _n (t)(b _n+1(t) – b _n (t)), db _n (t)/dt = 2( _n ²(t) – _n–1 ²(t)), n = 1, 2, ..., t > 0. The solution of this system (if it exists) is a pair of real sequences _n (t), b _n (t) which satisfy the conditions _n (0) = _n ,, b _n (0) = b _n , where _n > 0 and b _n are given sequences of real numbers. It is well known that the system has a unique solution provided that both sequences _n and b _n are bounded. When at least one of the known sequences _n and b _n is unbounded, many difficulties arise and, to the best of our knowledge, there are few results concerning the solution of the system. In this letter we find a class of unbounded sequences _n and b _n such that the system has a unique solution. The results are illustrated with a typical example where the sequences _i (t), b _i (t), i = 1, 2, ... can be exactly determined. The connection of the Toda lattice with the semi-infinite differential-difference equation d²/dt ² log h _n = h _n+1 + h _n–1 – 2h _n , n = 1, 2, ... is also discussed and the above results are translated to analogous results for the last equation.     

The semileptonic decay fraction ofB-mesons in the light of interfering amplitudes

Consequences of the interference between spectator amplitudes for the lifetimes and semileptonic decay fractions ofB ⁰ andB ⁺ mesons are discussed. Extracting these amplitudes from a fit to 11 exclusive hadronicB decay fractions we finda ₁=1.05±0.03±0.10,a ₂=+0.227±0.012±0.022, an inclusive semileptonic decay fraction of (11.2±0.5±1.7)%, and a lifetime ratio (B ⁺)/(B ⁰)=0.83±0.01±0.01.Supported under DOE grant number DE-FG02-91-ER40690     

Determination of the nuclear parameters of155Gd excited states using the Mössbauer effect

From Mössbauer spectra of GdAlO₃ and GdVO₄ above and below the Néel temperature and fitted using a transmission integral, we have determined the following parameters of the 86.5 keV and 105 keV levels:g(86)/g(0)=+1.217±0.005,Q(86)/Q(0)= +0.10±0.02,g(105)/g(0)=–0.55±0.02,Q(105)/Q(0)=+0.74±0.02, r ²₁₀₅/ r ²₈₆=+1.30±0.05. The linewidth observed for the 105 keV transition is less than the calculated natural linewidth.     

Effective masses and conformal mappings

LetG _n ,NN, denote the set of gaps of the Hill operator. We solve the following problems: 1) find the effective massesM _n ^± , 2) compare the effective massM _n ^± with the length of the gapG _n , and with the height of the corresponding slit on the quasimomentum plane (both with fixed numbern and their sums), 3) consider the problems 1), 2) for more general cases (the Dirac operator with periodic coefficients, the Schrödinger operator with a limit periodic potential). To obtain 1)–3) we use a conformal mapping corresponding to the quasimomentum of the Hill operator or the Dirac operator.Partially supported by Russian Fund of Fundamental Research (93-011-1697)     

Nonlocality of Equivariant Star Products on T*( RPn)

Lecomte and Ovsienko constructed SL _n+1(R)-equivariant quantization maps Q for symbols of differential operators on -densities on RP ⁿ . We derive some formulas for the associated graded equivariant star products on the symbol algebra Pol(T* RP ⁿ ). These give some measure of the failure of locality. Our main result expresses (for n odd) the coefficients C _p (·,·) of when = in terms of some new SL _n+1(C)-invariant algebraic bidifferential operators Z _p (·,·) on T* CP ⁿ and the operators (E + n/2 ± s)^–1 where E is the fiberwise Euler vector field and s {1, 2, ..., [p/2]}.