Calculation of bounds of the eigenvalue spectrum of anharmonic phonon collision operator

Extreme bounds on the eigenvalue spectrum of anharmonic phonon collision operator in Ge are calculated using (i) an isotropic continuum model, and (ii) Born-von-Karman model. The maximum occurs for short wave transverse modes. For the isotropic model the minimum occurs for longitudinal modes of the magnitude nearly $\text{3}\text{4}$_qD down to 27 K, and q_D below it. For the dispersive model the minimum occurs for longitudinal modes with $\text{3}\text{4}$_qD for all temperatures.     

On the collision operator for quantum systems with discrete spectrum

The existence of the asymptotic collision operator ψ(0) is established for a class of observables. Sufficient conditions are given for the vanishing of ψ(0) and for the existence of collisional and non-collisional invariants.     

Non-existence of a gap in the eigenvalue spectrum of the linearized collision operator of Peierls' phonon-Boltzmann equation

It is proved that the eigenvalue spectrum of the linearized collision operator of Peierls' phonon-Boltzmann equation has no gap. As a consequence the usual derivation of hydrodynamic equations from a Boltzmann equation is not valid in the case of a phonon system.     

On the spectrum of the linear Boltzmann operator

The continuous spectrum of the linear Boltzmann operator with constant field is derived. It is found that a sufficiency relation for runaway phenomena is consistent with another sufficiency relation for the hydrodynamic regime to exist. There is a further class of systems whose behavior lies in between these two extremes.     

Gyrokinetic Fokker-Planck collision operator

The gyrokinetic linearized exact Fokker-Planck collision operator is obtained in a form suitable for plasma gyrokinetic equations, for arbitrary mass ratio. The linearized Fokker-Planck operator includes both the test-particle and field-particle contributions, and automatically conserves particles, momentum, and energy, while ensuring non-negative entropy production. Finite gyroradius effects in both field-particle and test-particle terms are evaluated. When implemented in gyrokinetic simulations, these effects can be precomputed. The field-particle operator at each time step requires the evaluation of a single two-dimensional integral, and is not only more accurate, but appears to be less expensive to evaluate than conserving model operators.     

The first eigenvalue of the transversal Dirac operator

On a foliated Riemannian manifold with a transverse spin structure, we give a lower bound for the square of the eigenvalues of the transversal Dirac operator. We prove, in the limiting case, that the foliation is a minimal, transversally Einsteinian with constant transversal scalar curvature.     

On the measure of the spectrum for the almost Mathieu operator

Communicated by A. Jaffe     

On the measure of the spectrum for the almost Mathieu operator

We obtain partial results on the conjecture that for the almost Mathieu operator at irrational frequency, , the measure of the spectrum,S(, , )=|4–2|. For ||2 we show that if _n is rational and irrational, then .Dedicated to Res Jost and Arthur WightmanResearch partially supported by U.S. NSF grant number DMS-8801918 and by BSF under grant number 88-00325     

Dissipativity and the spectral property of the collision operator

It has been shown by the Brussels School that there is a physical representation in which the evolution of the density matrix reveals the dissipative structure. In this article, we present a two-dimensional model of the collision operator with a singularity at the origin which characterizes a long-range force. The dissipativity and the approach to equilibrium in the physical representation are studied and it is shown that the second law may be established in spite of the failure of Boltzmann's kinetic equation.     

On the spectrum of the linear transport operator with generalized boundary conditions

The spectrum of the linear transport operator under generalized boundary conditions proposed by Belleni-Morante (1970 a) contains in the right half-plane (|Re>–1) only a finite number of real eigenvalues. Its essential spectrum consists of a fan of continuous spectrum lines with apex at =–1.     

On the essential spectrum of the transfer operator for expanding markov maps

The essential spectrum of the transfer operator for expanding markov maps of the interval is studied in detail. To this end we construct explicityly an infinite set of eigenfunctions which allows us to prove that the essential spectrum inC ^k is a disk whose radius is related to the free energy of the Liapunov exponent.     

Optical phonon spectrum of germanium quantum dots

We have observed additional lines, shifted in both directions relative to the frequency of the bulk phonon of Ge, in the Raman scattering spectra from optical phonons in germanium quantum dots. The observed phonon modes are shown to be due to the straining of the quantum dots as a result of the lattice mismatch of the Ge and Si matrices. The observed frequency shifts, with allowance for optical-phonon localization effects, make it possible to determine the sizes of the regions with different strain states in the quantum dots. Pis’ma Zh. éksp. Teor. Fiz. 70, No. 4, 279–283 (25 August 1999)     

The acoustic phonon spectrum of penrose tilings

The spectral weight function associated with the vibrational modes of a two-dimensional quasicrystal is studied numerically. It is found that plane waves corresponding to special symmetry points in reciprocal space are at least very nearly eigenstates of the system. Near periodicities in lineshape parameters plotted as a function of wavevector are found to occur, and the first Brillouin zone edge is identified with lattice parameters. The contribution of these modes to the specific heat is found to obey a two-dimensional Debye Law.     

A pseudopotential calculation of the phonon spectrum of aluminium

The phonon dispersion curves for Al are calculated from a theory based on Coulomb interactions among the ions and a local pseudopotential interaction between ions and conduction electrons. Conduction electron correlations are taken into account according to a theory of Singwiet al. ^1,2. A dielectric constant is constructed which satisfies the compressibility sum rule. This is especially important for longitudinal phonons. The agreement of the computed frequencies with the experimental dispersion curves of Stedman and Nilsson³ is good. Sites, sizes and widths of Kohn anomalies in the dispersion curves are also calculated. As expected the theoretical curves show larger anomalies than the experimental data. Finally a comparison is made with the recent calculations of Wallace⁴.     

On the phonon self-energy

It is reported that a specific phonon-phonon interaction process described by the sixth-order Hamiltonian can contribute to the real part of the phonon self-energy of an order which involves a quadratic dependence of the resulting frequency shift on temperature, in the high temperature limit. The possible connection between the occurrence of such a process and experimental mental results in α-quartz is discussed.     

Nonadiabatic effects in the phonon spectrum of metal compounds

It is shown that nonadiabatic corrections to the dispersion law of optical phonons in the region of small wave vectors in the case of branches, for which the vibration with a zero wave vector is not accompanied by the appearance of a dipole moment in the ionic lattice, are significant for all possible directions of the wave vector. If a dipole moment arises, nonadiabatic corrections reach a noticeable value only for the wave-vector directions that are almost perpendicular to the direction of the dipole moment.