The spectrum of a Schrödinger operator inL p ℝ v isp-independent

LetH _p =–1/2+V denote a Schrödinger operator, acting inL _p ^v , 1p. We show that (H _p )=(H ₂) for allp[1, ], for rather general potentialsV.     

Massive spin-two particle in a gravitational field

The spin-two particle is described by a symmetric tensorh subject to the subsidiary conditionsh = h =0. Their covariant generalization and the wave equation have been obtained directly from the Eulerian variational equations by algebraic methods only. In addition to the tensor fieldh a symmetric third-rank tensor = as well as a vector fieldA have been added, neither of which enter in the final result. The Lagrangian function is taken as a linear sum of all combinations which can be constructed from these functions, as well as terms involving the curvature tensor and its two possible contractions. Variation with respect toh , andA independently gives the Euler equations. Combining the various trace equations and choice of arbitrary constants yields the subsidiary conditions, while the Euler equations themselves give the connection between the auxiliary functions and the tensorh as well as the generalization of the wave equationD D h + 2R h -R h -R h +g R h +Rh =m ² h Finally, variation with respect tog yields the energy-momentum tensor.     

Upper critical fields of single-crystalline and polycrystalline Ca-Sr-Bi-Cu-O compounds

The ac resistivity of a 110 K phase multiphase polycrystalline Ca-Sr-Bi-Cu-O compound and an 85 K phase single-crystalline Ca_0.9Sr_2.1Cu_2.0O_{8 +} has been measured in various magnetic fields up to 8 T. Values forB _c ^2/ (0) of 71.5 T and forB _c2 (0) of 542 T are found for the 85 K phase sample. A value forB _c2(0) of 57.9 T is estimated for the 110K phase compound.     

Spatial structure in diffusion-limited two-particle reactions

We analyze the limiting behavior of the densities _A(t) and _B(t), and the random spatial structure(r) = ( _A(t)., _B(t)), for the diffusion-controlled chemical reaction A+Binert. For equal initial densities _B(0) = _b(0) there is a change in behavior fromd 4, where _A(t) = _B(t) C/t^d/4, tod 4, where _A(t) = _b(t) C/t ast ; the termC depends on the initial densities and changes withd. There is a corresponding change in the spatial structure. Ind < 4, the particle types separate with only one type present locally, and , after suitable rescaling, tends to a random Gaussian process. Ind >4, both particle types are, after large times, present locally in concentrations not depending on type or location. Ind=4, both particle types are present locally, but with random concentrations, and the process tends to a limit.     

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).     

Far-infrared spectroscopy of the supreconductor YBa2Cu4O8

The frequencies of allB _1u (z) phonon modes predicted by a group-theoretical analysis were measured and found to agree well with recent lattice dynamical calculations for this compound. We report also the determination of two superconducting gap values in YBa₂Cu₄O₈ through phonon self-energy effects in the normal and superconducting conducting state. The gap-to-T _c ratios obtained from an analysis of these effects are 2 ₁/kT _c 2.5 and 5.82 ₂/kT _c 9.2. This coincides with previous results of both phononic and electronic Raman scattering where values of 2.1 and 6.3 were found. We further find anomalous softenings of two phonon modes 40 Kabove T _c , which correlate with an observed deviation from the linear temperature dependence of the average plasma frequency _p (T).     

Mixing properties,differentiability of the free energy and the central limit theorem for a pure phase in the Ising model at low temperature

For the Ising model with nearest neighbour interaction it is shown that the spin correlations _{A B} - _{A B} decrease exponentially asd(A, B) in a pure phase when the temperature is well belowT _c. This is used to prove that the free energyF(,h) is infinitely differentiable in and has one sided derivatives inh of all orders forh=0. The bounds are also used to prove that the central limit theorem holds for several variables such as e.g. the total energy and the total magnetization of the system, the limit distribution being gaussian with variances determined by the second derivatives ofF(,h).     

The electrodynamics and statistical mechanics of linear plasma response functions

The purpose of this paper is to review and to extend, wherever possible, the Kramers-Kronig relations, sum rules, and symmetry properties for the electrodynamic transport tensors of a linear plasma medium. For complete generality, we consider both nonrelativistic and relativistic plasmas with and without external magnetic fields. Our study is carried out first within the framework of classical electrodynamics. We then exploit the statistical-mechanical fluctuation-dissipation theorem to further obtain the Onsager symmetry relations and Kubo sum-rule frequency moments. Of special significance is the emergence of a variety of new Kramers-Kronig formulae andf-sum rules for the inverse dispersion tensor.Nomenclature E(k,) electric field intensity - Ê(k,) electric field in absence of plasma particles, - (k,) electric field due to the plasma particles (=E-Ê) - B(k,) magnetic induction - D(k,) electric induction - H(k,) magnetic field strength - B ₀ constant external magnetic field - A ₀ vector potential corresponding toB ₀ - (k,),j(k, co) charge and current densities due to the plasma particles - (k,),J(k,) charge and current densities of the external agency - (k,,B ₀) dielectric tensor of the plasma medium in the presence of B₀ - (k,,B ₀) diamagnetic tensor - (k, co,B ₀) (k,,B ₀) – 1, electric polarizability tensor - (k,,B ₀) magnetic polarizability tensor - (k,,B ₀) ordinary conductivity tensor - (k,,B ₀) external conductivity tensor - D(k,,B ₀) n²T–(k,,B ₀), dispersion tensor, where T=1-kk is the transverse projection tensor (k being the unit vector in the direction ofk) andn = kc/ the index of refraction - n²T – 1, = vacuum wave operator (value of D in vacuum) - 1/2( + ), Hermitian part of - ^{^} 1/2( – ), Anti-Hermitian part of a - , real and imaginary parts of a - R(r,t) dissipated power per unit volume of plasma - U total energy absorbed by the plasma - R(k,) E^*(k,) · (k,,b ₀) ·E(k,) corresponding spectral energy density - W(r,t) 1/2₀E²(r, 0 + (l/2₀) B²(r,t), field energy density - W(k,) 1/2₀E^*k,) ·E(k,) + (l/2₀)B ^*(k,) · B(k,), energy content in a certain domain of (k,)-space for a single mode - x _i,p _i,v _i coordinate, momentum, and velocity of ith electron - _i [1–(_i ²/c²)]^–1/2 - X _j,P _j,V _j coordinate, momentum, and velocity of jth ion - {A _q}, {E_q} field coordinates and momenta - j_k(t),J _k(t) perturbations in the microscopic electron and ion current densities due to the presence of the small external vector potential agencyâ(r,t) = (1/L³) â_k(t) expi k ·r - Liouville distribution function = ⁰ + - ⁰ macrocanonical distribution function characterizing the equilibrium state of the system in the infinite past - small perturbation due toA - H⁰ Hamiltonian of equilibrium system which includes interaction - H Hamiltonian for the interaction between the system and the small external perturbing agencyA - ⁰ = d^R()⁰ expectation value of any quantity over the equilibrium ensemble (d^R is an element of hypervolume in -phase space) - G(12) two-particle distribution function - F(1) one-particle distribution function - g(¦x₂ – x₁ ¦) [G(12)/F(1)F(2)] – 1, pair correlation function - N total number of electron in volume L³ - n ₀ equilibrium density (of electrons) - ^–1 temperature (in energy units) - ₀ (n₀e²/m₀)^1/2, equilibrium electron plasma frequency - _c ¦e ¦–B ₀/m, electron frequency - ^–1 ( ₀/n₀e²)^1/2, Debye length - ₀ (n₀Ze²/M₀)^1/2, equilibrium ion plasma frequency - _c ZeB₀/M, ion cyclotron frequency     

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     

Logically independent von Neumann lattices

Three definitions of logical independence of two von Neumann latticesP₁,P₂ of two sub-von Neumann algebras ₁, ₂ of a von Neumann algebra are given and the relations of the definitions clarified. It is shown that under weak assumptions the following notion, called logical independence is the strongest:A B 0 for any 0 A P₁, 0 B P₂. Propositions relating logical independence ofP₁,P₂ toC ^*-independence,W ^* independence, and strict locality of ₁, ₂ are presented.     

Cyclotron instabilities of a two-component electron plasma

A study is made of instability at frequencies close to the electron cyclotron frequency _B and its multiples, subject to the presence of two different groups of electrons. It is shown that a mixture of hot and cold electrons ( _ph ² _pc ² ) in the region of frequencies s _B, s2 can be unstable with respect to waves of the flute type (k _z=0) with maximum increment _max ( _ph/_pc). _B, if there exists an interval of transversal velocities in whichF/ >0. When the curvature of the magnetic field is taken into account, even waves with _B can be unstable in such a plasma. The effect of spatial inhomogeneity of the hot component on flute-type instability and on two-beam cyclotron instability is also examined.The author extends his thanks to A. B. Mikhajlovskij for his valuable comments and discussions.     

Schrödinger Operators with Empty Singularly Continuous Spectra

Let H be a semibounded perturbation of the Laplacian H ₀ in L ²( ^d ). For an admissible function sufficient conditions are given for the completeness of the scattering system (H), (H ₀). If is the exponential function and if e^{– H} is an integral operator we denote the kernel of the difference D = e^{– H} – e^{– H} ₀ by D (x, y), > 0. The singularly continuous spectrum of H is empty if^d dx ^d dy |D(x,y)| (1 + |y|²)< for some > 1. This result is applied to potential perturbations and to perturbations by imposing Dirichlet boundary conditions.     

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     

Effect of radiation on hydromagnetic vertical channel flow

The effect of radiation on the combined free and forced convection flow of an electrical conducting viscous fluid through an open-ended vertical channel in slip flow regime and permeated by a constant magnetic field in transverse direction has been considered. The temperature on the walls has been supposed to vary linearly with distance. On the assumption of optically thin limit, the expressions for velocity, induced magnetic field, temperature, flow rate and the heat transfer coefficient due to thermal conduction are obtained and the influence of radiation on these solutions are shown either graphically or in tabulated forms.Notation x, y, z cartesian coordinates - v velocity - 2L distance between the channel walls - B ₀ applied magnetic field - B induced magnetic field - T temperature - p pressure - density - magnetic permeability - v kinematic coefficient of viscosity - conductivity of the fluid - g acceleration due to gravity - coefficient of volume expansion - thermal diffusivity of the fluid - c _p specific heat at constant pressure - q_R radiative heat flux - k absorption coefficient - e _b Plank function - T 0 reference temperature - N vertical temperature gradient - M= B₀L[/(v)]^1/2 Hartmann number - R _a=gNL⁴/(v) Rayleigh number - P_m = magnetic Prandtl number - 2d₁, 2d₂ thickness of the walls - ₁, ₂ conductivities of the walls     

The projection approach to the Fokker-Planck equation. I. Colored Gaussian noise

It is shown that the Fokker-Planck operator can be derived via a projection-perturbation approach, using the repartition of a more detailed operator into a perturbation ₁ and an unperturbed part ₀. The standard Fokker-Planck structure is recovered at the second order in ₁, whereas the perturbation terms of higher order are shown to provoke the breakdown of this structure. To get rid of these higher order terms, a key approximation, local linearization (LL), is made. In general, to evaluate at the second order in ₁ the exact expression of the diffusion coefficient which simulates the influence of a Gaussian noise with a finite correlation time, a resummation up to infinite order in must be carried out, leading to what other authors call the best Fokker-Planck approximation (BFPA). It is shown that, due to the role of terms of higher order in ₁, the BFPA leads to predictions on the equilibrium distributions that are reliable only up to the first order in t. The LL, on the contrary, in addition to making the influence of terms of higher order in ₁ vanish, results in a simple analytical expression for the term of second order that is formally coincident with the complete resummation over all the orders in t provided by the Fox theory. The corresponding diffusion coefficient in turn is shown to lead in the limiting case to exact results for the steady-state distributions. Therefore, over the whole range 0 the LL turns out to be an approximation much more accurate than the global linearization proposed by other authors for the same purpose of making the terms of higher order in ₁ vanish. In the short- region the LL leads to results virtually coincident with those of the BFPA. In the large- region the LL is a more accurate approximation than the BFPA itself. These theoretical arguments are supported by the results of both analog and digital simulation.     

Estimate of foil thickness by stereomicroscopy technique

The determination of foil thickness by stereomicroscopy technique is discussed. The calculation of the foil thickness is performed in a general position of the foil in an electron microscope for a general choice of the marker position. In this case the crystallographic orientation of the foil normal has to be known. But there exists a special case (markers lie in the diffraction plane) where it is not necessary to know the orientation of the foil normal. The relative error in determining the foil thickness is about 10%.List of symbols B direction in the foil parallel with the primary beam in the first micrograph - d vector describing a common straight line of and ₁ ( ₁) - F direction in the foil parallel with the primary beam in the second micrograph - g diffraction vector - M magnification - n foil normal - o vector defined byn ×B - p ₁ first coordinate ofX in the first micrograph - p ₂ first coordinate ofX in the second micrograph - p _1i ,P _2i quantitiesP ₁,P ₂ corresponding to thei-th markers on the top surface - t _B foil thickness measured in theB direction - auxiliary quantity - corresponding to thei-th markers on the top surface - t _d foil thickness measured in thed direction - t _F foil thickness in general - t _n foil thickness measured in then direction - X ₁ point in _B characterizing one of the markers - X ₂ point in _T characterizing one of the markers - X _2i X ₂ corresponding to thei-th markers on the top surface - X _g magnitude ofX _g - X _gi magnitude ofX _gi - X magnitude ofX - X vector connecting pointX ₁ withX ₂ (X X ₂–X ₁ - X _g component ofX perpendicular to the plane ₁ - X _gi X _gi corresponding to thei-th markers on the top surface - X component ofX lying in the plane ₁ - z distance between _{1 T} and the end point ofX divided by cos Greek symbols angle between ₁ and _{1 B} - angle betweeng and _T - ₁, ₂ reflection planes - plane containingn andg - m measure error - p difference betweenp ₂ andp ₁ (p=p ₂–p ₁) - P accuracy of length measurement - (t _B )_max maximum error oft _B - maximum error of - accuracy of angle measurement - angle characterizing the deviation from the foil position when symmetrical tilting is achieved - angle of tilt betweenB andF (between stereomicrographs) - _T plane describing approximately the top surface of the foil - _b plane describing approximately the bottom surface of the foil - screen plane - angle between ₁ andX - plane containingn andB - angle of tilting from starting (after inserting the foil into the microscope) to working position (position during image observation) Special symbol intersection of two planes (common straight line)     

Ground state of a spin-phonon system. II. Adiabatic limit

The phase transition for a spin in a magnetic fieldB coupled to acoustic phonons by a coupling constant is studied. The caseB1 with an upper cutoff of unity for the phonons is studied systematically by using an adiabatic canonical transformation. In leading order the transition line is at =2/B=1. In the normal phase (<1) the ground-state energy is –B/2 plus a function of that is given explicitly as the solution of a pair theory. In the broken symmetry phase (>1) the energy is the classical energy plus the same function of =1/². It is found that the first derivatives of the energy with respect to and with respect toB have finite jumps across the transition line. Quantum fluctuations in both phases are treated. Higher-order terms are a series of powers of 1/B times functions of . The case of a small transverse fieldB is also studied. The sharp transition disappears and is replaced by rapid variation in a region of order (B₁/B)^2/3 about =1.     

Combinations of Observables

This article begins with a review of the framework of fuzzy probability theory.The basic structure is given by the -effect algebra of effects (fuzzy events) E(,A) and the set of probability measures M ⁺ ₁ (, A) on a measurable space (,A). An observable X: B E(, A) is defined, where (, B) is the value spaceof X. It is noted that there exists a one-to-one correspondence between states onE(, A) and elements of M ⁺ (, A) and between observables X: B E(,A) and -morphisms from E(, B) to E(, A). Various combinations ofobservables are discussed. These include compositions, products, direct products,and mixtures. Fuzzy stochastic processes are introduced and an application toquantum dynamics is considered. Quantum effects are characterized from amonga more general class of effects. An alternative definition of a statistical map T:M ⁺ ₁ (, A) M ⁺ ₁ (, B) is given.     

Local energy flux and the refined similarity hypothesis

In this paper we demonstrate the locality of energy transport for incompressible Euler equations both in space and in scale. The key to the proof is the proper definition of a local subscale flux, _t (r), which is supposed to be a measure of energy transfer to length scales <l at the space point r. Kraichnan suggested that for such a quantity the refined similarity hypothesis will hold, which Kolmogorov originally assumed to hold instead for volume-averaged dissipation. We derive a local energy-balance relation for the large-scale motions which yields a natural definition of such a subscale flux. For this definition a precise form of the refined similarity hypothesis is rigorously proved as a big-O bound. The established estimate is _t (r)=O(l _3h-1) in terms of the local Hölder exponenth at the point r, which is also the estimate assumed in the Parisi-Frisch multifractal model. Our method not only establishes locality of energy transfer, but it also clarifies the physical reason that convection effects, which naively violate locality, do not contribute to the subscale flux. In fact, we show that, as a consequence of incompressibility, such effects enter into the local energy balance only as the divergence of a spatial current. Therefore, they describe motion of energy in space and cancel in the integration over volume. We also discuss theorems of Onsager, Eyink, and Constantinet al. on energy conservation for Euler dynamics, particularly to explain their relation with the Parisi-Frisch model. The Constantinet al. proof may be interpreted as giving a bound on the total flux, _t =d ^d r _t (r), of the form _t (r)=O(l _z3h-1), wherez ₃ is the third-order scaling exponent (or Besov index), in agreement with the multifractal model. Finally, we discuss how the local estimates are related to the results of Caffarelli-Kohn-Nirenberg on partial regularity for solutions of Navier-Stokes equations. They provide some heuristic support to a scenario proposed recently by Pumir and Siggia for singularities in the solutions of Navier-Stokes with small enough viscosity.     

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 .