Inequalities for the Schattenp-norm. III

We present some inequalities for the Schattenp-norm of operators on a Hilbert space. It is shown, among other things, that ifA is an operator such that ReAa0, then for any operatorX, AX+XA* _p 2aX _p . Also, for any two operatorsA andB, A–B ₂ ² +A*B* ₂ ² 2A–B ₂ ² .     

Random aggregation and random-walking center of mass

Two random aggregation models are used in demonstrating the properties of the random displacementsr _i of the center of mass of aggregating particles. It is found that r _i is a randomly decreasing sequence that scales with the cluster size (steps)s and _i ^=1/s r _i s ^1/D , whereD is the fractal dimension. The center-of-mass random walk is a consistent representation of the dynamics of aggregation.     

Renormalization group equations with several length scales and crossover scaling functions for axial anisotropy

The critical behaviour of axially anisotropicn-vector models is characterized by two distinct length scales, the correlation lengths and for the easy and hard axes. In order to handle the full range of anisotropics from to partial differential renormalization group equations are derived, depending on and . The anisotropicX-Y model is studied in detail near four dimensions. The crossover scaling functions for the susceptibilities are calculated to first order in=4–d. Two distinct crossover regions are found for weak and dominant anisotropy, respectively.     

On the Existence of a Maximizer for the Strichartz Inequality

It is shown that a maximizing function u _*L ² does exist for the Strichartz inequality e ^it _x ² u _L ⁶ _t (L ⁶ _x )Su _L ², with S>0 being the sharp constant.     

Dipolar magnetic field and fermi contact field at the μ+ site in α-Fe at very low temperatures

In a longitudinal ⁺SR experiment on a high-purity-Fe single crystal sphere magnetically saturated in a 111 direction damped oscillations (wiggles) were observed in a temperature range 30 mK to 600 mK and in a certain regime of applied magnetic fieldsB _appl. Meassurements of the wiggle frequency as a function ofB _appl give us directly the Fermi fieldB _Fermi=(–1.13±0.02)T and the dipolar magnetic field ¦B _dip ¦=(0.66±0.03)T.B _dip was used to determine the prefactor in the Arrhenius law obeyed by the ⁺ hopping rate between 100 K and 1000 K. A comparision with the corresponding values for protons and deuterons suggests diffusion via the adiabatic mechanism.     

Superstability estimates for anharmonic systems

We consider an anharmonic crystal described by variablesS _x ,x ^d ,S _x , with one-body interaction ¦S _x ¦ and nearest neighbor (n.n.) two body interaction ¦S _x –S _y ¦. We prove that, for ^d bounded, , where is the correlation function for the free boundary condition Gibbs state in ,>0 and are suitable constants independent of and . This generalizes previous results obtained in the case.Research partially supported by Consiglio Nazionale delle Ricerche.     

Correlation functionals of infinite volume quantum spin systems

The existence and analyticity of the correlation functionals of a quantum lattice in the infinite volume limit is proved. The result is valid at sufficiently high temperatures and for a large class of interactions. Our method estimates the kernelK for a set of Kirkwood-Salzburg equations. While a naive estimate would indicate that K =, we take into account cancellations between different contributions toK in order to show that for sufficiently high temperatures K <1, and this estimate is independent of the volume of the system.Supported in part by the Air Force Office of Scientific Research.     

Multiatom interactions,order and stability in binary transition metal alloys

Interface delocalization or depinning transitions such as wetting or surface induced disorder are considered. At these transitions, the correlation length for transverse correlations parallel to the surface diverges. These correlations are studied in the framework of Landau theory. It is shown the t^–1/2 at all types of transitions for systems with short-range forces wheret measures the distance from bulk coexistence.     

The influence of a surface on Ising and conjugate model spin correlations atT=T c

We calculate rigorously the two-point correlation function in the 2-d Ising model on a quasi-cylindrical lattice at the critical temperatureT _c .The scaling predictions for the surface critical exponents = 5/8, = 1 are confirmed and the change from surface to bulk behavior is studied. By a correspondence relation the results are mapped to the conjugate model, a subcase of the 8-vertex model. We find in this model = 5/4, = 2.     

The Optical Spectra of the Arsenic Triselenide Crystal in the Range from 1 to 10 eV

Full complexes of fundamental optical functions of the As₂Se₃ single crystal in the region from 1 to 10 eV are calculated at 10 K (1–5.5 eV) and 77 K (5.5–10 eV) for three light polarizations (E a, E b, and E c). Their main characteristic features are established. The experimental reflection spectra and the spectra of other optical functions calculated on their basis are compared with the known literature data.     

Real-space renormalization group for directed percolation system

We use the recently proposed real-space renormalization group method to study the critical behavior of directed percolation system in two dimensions. The correlation length exponents and are found to be 1.76 and 1.15. These results are in good agreements with the best known values.     

Instability in degenerate two-photon running wave laser with detuning

The previous stability analysis of the degenerate two-photon running wave laser is extended to the inclusion of detuning between frequencies of cavity and atoms. We derive the analytical equation for the critical pumping and prove analytically that for the special case ofr (/) being unity, there is no Hopf bifurcation instability for the bad cavity. The good cavity case is analysed numerically. The role played by detuning is to raise the critical pumping. In the case ofk (or <K) where there is no Hopf bifurcation instability for the perfect tuning case, the large detuning can give rise to self-pulsing instability.     

Long-range correlations at depinning transitions

Semi-infinite systems are considered with long-range surface fields B _z ^–(1+r) for large distancesz from the surface. The influence of such fields on the global phase diagram and on the critical singularities of depinning transitions is studied within Landau theory. For |B|0, the correlation length diverges as b ^–1/2 withb=|Bln|B^–(1+r). For finiteB, t ^v withv =(2+r)/(2+2r) wheret measures the distance from bulk coexistence. In the latter case, a Ginzburg criterion leads to the upper critical dimensiond ^*=(2+3r)/(2+r).     

Magnetoelastic coupling in a ferrite with magnetic ions in s states

Analytic relations are derived between the macroscopic magnetoelastic coefficients for a crystal of spinel type and the microscopic coefficients inthe Hamiltonian for spin-phonon interaction of S ions. On the basis of Feher's measurements of the components of the tensor C_ij for the one-ion spin-phonon interaction in the octahedral sublattice of a lithium ferrite, it is shown that the correct order of magnitude is obtained from these relations.     

Scaling factors associated withM-furcations of the 1−μ∥x∥z map

A numerical study is made of the scaling behavior associated withM-furcations (M=3, 4, 5) in the mapx _t+1 =1–x _t ^z (z>1). The scaling constants and are calculated as functions ofz, as well as the more general scaling functions andf(a).     

Determination of in-plane lattice parameters of InAs/AlAs strained-layer superlattices by X-ray precession camera

A determination of the lattice parametersa andb of InAs/AlAs short-period strained-layer superlattices grown by Atomic Layer Molecular Beam Epitaxy (ALMBE) on GaAs(001) substrates has been performed by means of an X-ray precession camera using copper radiation. Spots belonging to the superlattice are clearly differentiated from those of the substrate, which confirms that they are partly decoupled from each other. It was also possible to resolve the lattice spots of InAs or In_0.8Ga_0.2 As decoupling buffer layers grown between the substrate and the superlattice. This technique proves to be very useful to characterize, in a very short time and with a reasonable resolution, highly mismatched epitaxial systems in which lattice parameters parallel to the interface play a crucial role in the understanding of the growing behaviour.     

On absence of diffusion near the bottom of the spectrum for a random Schrödinger operator onL 2(ℝ)+

We consider a random Schrödinger operator onL ²(^v) of the form , {C _i} being a covering of ^v with unit cubes around the sites of ^v and {q _i} i.i.d. random variables with values in [0, 1]. We assume that theq _i's are continuously distributed with bounded densityf(q) and that 0<P(q ₀<1/2)=<1. Then we show that an ergodic mean of the quantity dx|x|²|(exp(itH ))(x)|²t ^–1 vanishes provided =g _E(H ), where is well-localized around the origin andg _E is a positiveC -function with support in (0,E),EE*(, |f|). Our estimate ofE*(, |f|) is such that the set {x ^v |V (x) E*(, |f|)} may contain with probability one an infinite cluster of cubes {C _i} which are nearest neighbours. The proof is based on the technique introduced by Fröhlich and Spencer for the analysis of the Anderson model.Work supported in part by C.N.R. (Italy) and NAVF (Norway)On leave of absence from Instituto di Fisica Università di Roma, Italia     

Instability in degenerate two-photon running wave laser

The stability of the homogeneously broadened and degenerate two-photon running wave laser is analysed by using the full set of matter-field equations. The stability depends on the relative size of the relaxation constants. For 2k>1+r(k=/,r=/; is the cavity loss of the field and , are the longitudinal and transversal decay constants, respectively) no stable lasing state exists. Forr<k<(1+r)/2 an instability occurs. With the decrease in pumping the stable lasing state loses its stability due to Hopf-bifurcation.     

The S-matrix for abstract scattering systems

Let S() be the S-matrix at energy for an abstract scattering system. We derive a bound, in terms of the interaction, on integrals of the form h () S()- _HS ² d, where denotes the Hilbert-Schmidt norm.Supported by the Swiss National Science Foundation.     

On the number of invariant measures for higher-dimensional chaotic transformations

LetS be a bounded region inR ^N and letP={S _l } _i=1 ^m be a partition ofS into a finite number of closed subsets having piecewiseC ² boundaries of finite (N–1)-dimensional measure. Let :SS be piecewiseC ² onP and expanding in the sense that there exists 0<<1 such that for anyi=1,2,...,m, DT _i ^–1<, whereDT _i ^–1 is the derivative matrix ofT _i ^–1 and · is the Euclidean matrix norm. We prove that for some classes of such mappings, for example, Jabtonski transformations or convexity-preserving transformations, the number of crossing points constitutes a bound for the number of ergodic absolutely continuous -invariant measures. We give examples showing that in general the simple bound of one-dimensional dynamics cannot be generalized to higher dimensions. In fact, we show that it is possible to construct piecewise expandingC ² transformations on a fixed partition with a finite number of elements but which have an arbitrarily large number of ergodic, absolutely continuous invariant measures.