Novel perturbation expansion for the Langevin equation

We discuss the randomly driven systemdx/dt= -W(x) +f(t), wheref(t) is a Gaussian random function or stirring force withf(t)f(t)=2(t–t), andW(x) is of the formgx ¹⁺². The parameter is a measure of the nonlinearity of the equation. We show how to obtain the correlation functionsx(t)f(t)···x(t( n)) _f as a power series in. We obtain three terms in the expansion and show how to use Padé approximants to analytically continue the answer in the variable. By using scaling relations, we show how to get a uniform approximation to the equal-time correlation functions valid for allg and.     

Mosaic Structure of (001)YBa2Cu3O7-δ/(10.2)Al2O3 Epitaxial Films and Millimeter Waveband Properties of Microstrip Resonators

The c-axis single-phase YBa₂Cu₃O_7- films ( = 0-0.15) on sapphire substrates prepared by the laser ablation technique and the band-pass stripline resonators for 34 GHz-range have been investigated. Increasing disorientation of mosaic block structure of YBa₂Cu₃O_7- films is related to increasing surface resistance R_s at 135 GHz-range and decreasing unloaded quality factor Q_o of linear stripline resonators. The linear dependence between the YBa₂Cu₃O_7- film mosaicity (M) and half-width B₁ of 00.13 reflecting component is determined. The reflection spreading is due to microstrains resulted mainly from the coherent adjustment of the YBa₂Cu₃O_7- film lattice to GdBa₂Cu₃O_7- sub-layer and sapphire substrate. Increasing number of the block characterized by a weak radial lattice adjustment () is demonstrated by spreading of 00.13 reflection. It is found out that depends on the Bragg angle of reflection due to inhomogenity of YBa₂Cu₃O_7- mosaic structure, which resulted from the gradual mosaicity decreasing with the depth.     

Derivations commuting with abelian gauge actions on lattice systems

Let be an action of a compact abelian groupG on aC*-algebraA, and assume that the fixed-point subalgebraA is an AF-algebra. We show that if is a closed *-derivation onA commuting with , and the restriction of toA generates a one-parameter group of *-automorphisms, then itself is a generator. In particular, the result applies if is an infinite product action ofG on a UHF algebra. Furthermore, if in this situation ₁ and ₂ are two derivations both satisfying the hypotheses on , and ₁ and ₂ have the same restriction toA , then there exists a one-parameter subgroup of the action with generator ₀ such thatD(₁)D(₂)D(₀) is a joint core for the three derivations, and ₂=₁+₀ on this core.     

Small perturbations ofC*-dynamical systems

It is shown that if is the generator of a strongly continuous oneparameter group of *-automorphisms of aC*-algebraA and is an unbounded *-derivation ofA with the same domain as , then + is also a generator for all sufficiently small real numbers .     

Direction-Dependent Free Energy Singularity of the Antiferroelectric Asymmetric Six-Vertex Model

The transition from the ordered commensurate phase to the incommensurate Gaussian phase of the antiferroelectric asymmetric six-vertex model is investigated by keeping the temperature constant below the roughening point and varying the external fields (h, v). In the (h, v) plane, the phase boundary is approached along straight lines v = k h, where (h, v) measures the displacement from the phase boundary. It is found that the free energy singularity displays the exponent 3/2 typical of the Pokrovski–Talapov transition f const(h)^3/2 for any direction other than the tangential one. In the latter case f shows a discontinuity in the third derivative.     

Spectral and scattering theory for a Schrödinger operator with a class of momentum-dependent long-range potentials

Let be the selfadjoint operator for the static electromagnetic field where W _j for 0, 1, 2, ..., n is a sum of (i) a short-range potential and (ii) a smooth long-range potential decreasing at as |x|^- with in (0, 1]. Then for >1/2, asymptotic completeness holds for the scattering system (H, H ₀).     

Electric field gradient at the Hf site and the phase transition in HfV2

The electric quadrupole interaction at the Hf site in the cubic HfV₂ compound, which shows lattice instability when cooled below 120 K, has been studied using the 482 keV (5/2⁺) state in¹⁸¹Ta. The time differential perturbed angular correlation (TDPAC) studies involving (133+136)-482 keV - cascade at room temperature show that only a fraction of¹⁸¹Ta nuclei see a cubic symmetry while the remaining experience a weak, randomly oriented, electric quadrupole interaction described by a Gaussian frequency distribution with _Q=6.1±1.0 Mrad/sec and the relative width =0.5. There is a dramatic change in TDPAC pattern indicating a phase transition when the compound is cooled down to 77 K. At this temperature the quadrupole interaction frequency is found to be _Q=40.6±3.0 Mrad/sec with =0.35. The experimentally observed EFG both at the Hf and the V sites are compared to the theoretical estimates based on point charge model.     

Temperature Dependence of the Gibbs State in the Random Energy Model

We consider the problem of temperature dependence of the Gibbs states in two spin-glass models: Derrida's Random Energy Model and its analogue, where the random variables in the Hamiltonian are replaced by independent standard Brownian motions. For both of them we compute in the thermodynamic limit the overlap distribution ^N _{i=1 i i} /N[–1,1] of two spin configurations , under the product of two Gibbs measures, which are taken at temperatures T,T respectively. If TT are fixed, then at low temperature phase the results are different for these models: for the first one this distribution is D _{0 0}+D _{1 1}, with random weights D ₀, D ₁, while for the second one it is ₀. We compute consequently the overlap distribution for the second model whenever T–T0 at different speeds as N.     

Analytic study of power spectra of the tent maps near band-splitting transitions

Successive band-splitting transitions occur in the one-dimensional map x_i+1=g(x_i),i=0, 1, 2,... withg(x)=x, (0 x 1/2) –x +, (1/2 <x 1) as the parameter is changed from 2 to 1. The transition point fromN (=2ⁿ) bands to 2Nbands is given by=(2)^1/N (n=0, 1,2,...). The time-correlation function _i=x_ix₀/(x₀)²,x_i x_i–x_i is studied in terms of the eigenvalues and eigenfunctions of the Frobenius-Perron operator of the map. It is shown that, near the transition point=2, _i–[(10–42)/17] _i,0-[(102-8)/51]_i,1 + [(7 + 42)/17](–1)ⁱe^–yi, where2(–2) is the damping constant and vanishes at=2, representing the critical slowing-down. This critical phenomenon is in strong contrast to the topologically invariant quantities, such as the Lyapunov exponent, which do not exhibit any anomaly at=2. The asymptotic expression for _i has been obtained by deriving an analytic form of _i for a sequence of which accumulates to 2 from the above. Near the transition point=(2)^1/N, the damping constant of _i fori N is given by _N=2(^N-2)/N. Numerical calculation is also carried out for arbitrary a and is shown to be consistent with the analytic results.     

A note on nuclear excitation by positron annihilation with K-shell electrons

Experimental data for the nuclear excitation of¹¹⁵In by positron annihilation with K-shell electrons have been examined, taking account of the effective thickness of an In target used, and then the cross section _res for resonance excitation to the 1078 keV energy level by positron annihilation has been reevaluated. The cross section for the process is induced by the measured effective cross section _eff of nuclear excitation of the isomeric level^115mIn. In this work the effective thicknesses of the In target for positrons with kinetic energies 83.9 and 470 keV have been estimated. The effective thickness r has been determined from a relation r=E/ ¦dE/dr¦, where E is the positron energy width at 83.9 or 470 keV, r is the mean distance traversed by them, and ¦dE/dr¦ the stopping power of indium for them. In the present case, _K, the atomic-level width of the K-level of indium, is used as E. Neglecting a contribution from the 1464 keV level being much smaller than that from the 1078 keV level, a reevaluated value of _res has been obtained as 1.7×10^–25 cm².     

Perturbations of flows on Banach spaces and operator algebras

For automorphism groups of operator algebras we show how properties of the difference _t – ' _t are reflected in relations between the generators , . Indeed for a von Neumann algebraM with separable predual we show that if _t – '_t 0.28 for smallt, then = 0(+)°^-1 where is an inner automorphism ofM and is a bounded derivation ofM. If the difference _t – ' _t =O(t) ast ; 0, then = + and if _t – ' _t 0.28 for allt then =. We prove analogous results for unitary groups on a Hilbert space andC ₀,C ₀ ^* groups on a Banach space.This paper subsumes an earlier work of the same title which appeared as a report from Z.I.F. der Universität BielefeldWith partial support of the U.S. National Science Foundation     

Multiple scattering in random media I. Restricted two-body additive approximation

We present a microscopic theory of the problem of finding the properties of a particle interacting with potentials located at random sites. The sites are governed by a general probability distribution. The starting point is the multiple scattering equations for the amplitude k ₁|T |k ₂ in terms of the individual scattering amplitudes k ₁|T |k ₂. We work with quantitiesA defined by k ₁|T |k ₂=k ₁|T |k ₂exp[i(k ₁–k ₂)R ]. The theory is based on a splitting of the fundamental equation forA into equations for the mean A and the fluctuationsAA . Neglect of the fluctuations yields the quasicrystalline approximation. We rearrange the equation forAA to isolate the collective part of the fluctuations. We then make the simplest microscopic truncation which is thatAA is a restricted two-body additive function of the site positions. With the contribution of the collective fluctuations, this yields results forA that are accurate to ordert ⁴.Work supported in part by the National Science Foundation under Contract No. NSF DMRWork supported in part by the National Science Foundation under Contract No. NSF DMR     

Enhancement of activated decay of metastable states by resonant pumping

We consider the effect of a high-frequency pumping cost on the escape rate of a classical underdamped Brownian particle out of a deep potential well. The energy dependence of the oscillation frequency(E) is assumed to be weak on the scale of thermal energy,_E(0)T(0)T/V₀ (0)[_E(0) is the derivative of(E) atE= 0,V ₀ is the barrier height,V ₀ T]. The quadratic-in- contribution to the decay rate is calculated in two different regimes: (1) for the case of resonance of the pumping frequency with the nth harmonic of the internal motion at an energye, when = n(e); (2) for a rollout region of the basic resonance near the bottom of the potential well, when ¦-(0)¦ and is the damping coefficient. In the latter case the absorption spectrum and the enhancement of the decay rate are calculated as functions of two reduced parameters, the anharmonicity of the potential,v _E (0)T/, and the resonance mismatch, [ —(0)]/. It is shown that the effect of the pumping increases with diminishing ¦v¦ and at small v is proportional tov ^–1. In this regime, the dependence on is stepwise: the pumping contribution is large for v > 0 and small for v < 0. In the frame of our theory, the decay rate is invariant against the simultaneous alternation of the signs of andv. The spectrum of the energy absorption has the standard Lorentzian shape in the absence of anharmonicity,v=0, and with increasing of ¦v¦ shifts and widens retaining its bell-shape form.     

Symmetries and half-sided modular inclusions of von Neumann algebras

LetN, be a von Neumann algebras on a Hilbert space , a common cyclic and separating vector. Assume to be cyclic and also separating forN . Denote by , _N , _N the modular operators to (, ), (N, ), resp (N , ). Assume now ^-it N ^it N for allt 0. (Such type of inclusions ((N U, ) , ) are called half-sided modular.) Then the modular groups ^it , _N ^ir , _N ^is ,t, r, s generate a unitary representation of the group S1(2, )/Z ₂ of positive energy.Another result is related to two half-sided modular inclusions (₁ , ) and (₂ , ). Under proper conditions the three modular groups ^it , ₁ ^ir , ₂ ^is ,t, r, s generate the three-dimensional subgroup of O(2, 1) of two commuting translations and the Lorentz transformation.Partly supported by the DFG, SFB 288 Differentialgeometrie und Quantenphysik.     

Nonlinear properties of the laser as an amplifier

Summary The paper presents the results of a theoretical analysis of an optical traveling-wave amplifier, to whose input are applied monochromatic pulses of frequency equal to the transition frequency. At distances zp(c/2) |2N₂–N₁|/ /(N₁–N₂) where p>1, and for <2₀aN₁ all input pulses assume the form of a stationary pulse (see Eq. (33)). The stationary signal intensity, duration and total energy are substantially dependent on the nonresonant loss in the medium, the relaxation time T₂ and the initial inversion population difference N₁. For loss values >2₀aN₁ the input pulses are attenuated.In the amplifier model considered we have assumed the laser medium to have a uniformly broadened line. This enables us to apply the results of the investigation to a ruby laser which satisfies this condition (by excluding low temperatures).The neglect of pumping and the lifetime T₁ is permissible for values of not too close to 2₀aN₁.Izvestiya VUZ. Radiofizika, Vol. 8, No. 5, pp. 899–908, 1965     

Zero value of the Schwarzschild mass for an asymptotically Euclidian system of gravitational waves

A class of the asymptotically Euclidian space-times is shown to exist for which the Schwarzschild mass is equal to zero. The coordinate atlases of these space-times satisfy two additional conditions: _k (-gg ^0k )=0 and _ik ⁰ _0g ^ik - _ik ^k _0g ⁰ⁱ =0. In aT-orthogonal metricgs ²=g ₀₀ dt ² -g dx dx these conditions take a simple form: ₀(detg )=0 and (₀ g )(₀ g )=0.     

A contribution to the theory of the triple crystal diffractometer

The first part of the paper gives a general equation for triple-crystal arrangement with perfect crystals on the assumption that the third crystal is rotated. It is shown that in the case of perfect crystals the shape of the reflection curve is practically independent of the vertical divergence. The case of mosaic crystals is also solved and the possibility of rotation by other than the third crystal is considered. A method is proposed for investigating the imperfection of a crystal which is different from methods used up to now. The paper is supplemented by some experimental results.

---

, . , . , , . , . .

     

Second-harmonic generation at periodic second-order susceptibility gratings with self-phase and cross-phase modulation

Quasi-phase-matched second-harmonic generation at periodic quadratic susceptibility gratings with self-phase and cross-phase modulation owing to Kerr nonlinearities is investigated. A model of interaction of monochromatic plane travelling waves is considered. The solution assumes a nondepleted pump and results for a numerical computation of second-harmonic conversion efficiency for a highly efficient process are presented. It is shown that for high spatial-frequency gratings the efficiency of second-harmonic generation depends on the reduced detuning and the Kerr-nonlinearity coefficient only. In some cases, if the signs of and are opposite, the Kerr nonlinearity can enhance the second-harmonic conversion efficiency due to the compensation of the phase mismatch attained in the below-half-conversion stage by the reversed phase mismatch in the above-half-conversion stage of the process. The computed maximum conversion efficiencies for various values of and are plotted in a contour map on the - plane.     

Two singular integrals containing a partial wave of the two-body coulomb T-matrix

The operatorsT _C,l E+i0)[–G ₀(E+i0)]^1–i andT _C,l(E+i)G ₀[–iG ₀(E+i)]ⁱ acting on spaces of Hölder continuous, differentiable and analytic functions are investigated. The results of their action are expressed in terms of explicit singular factors and terms and Hölder (differentiable, analytic) functions. The most singular part of these operators is shown to be determined by a simple functional.     

A theory of 1/f noise

Letu() be an absolutely integrable function and define the random process where thet _i are Poisson arrivals and thes _i, are identically distributed nonnegative random variables. Under routine independence assumptions, one may then calculate a formula for the spectrum ofn(t), S _n(), in terms of the probability density ofs, p_s(). If any probability density p_s() having the property p_s() I for small is substituted into this formula, the calculated S_n() is such that S_n() 1 for small . However, this is not a spectrum of a well-defined random process; here, it is termed alimit spectrum. If a probability density having the property p_s() for small , where > 0, is substituted into the formula instead, a spectrum is calculated which is indeed the spectrum of a well-defined random process. Also, if the latter p_s is suitably close to the former p_s, then the spectrum in the second case approximates, to an arbitrary, degree of accuracy, the limit spectrum. It is shown how one may thereby have 1/f noise with low-frequency turnover, and also strict 1/f ^1– noise (the latter spectrum being integrable for > 0). Suitable examples are given. Actually, u() may be itself a random process, and the theory is developed on this basis.