Energy-level statistics of model quantum systems: Universality and scaling in a lattice-point problem

We investigate the statistics of the numberN(R, S) of lattice pointsnZ ², in an annular domain (R, w)=(R+w)A\RA, whereR, w>0. HereA is a fixed convex set with smooth boundary andw is chosen so that the area of (R, w) isS. The statistics comes fromR being taken as random (with a smooth density) in some interval [c ₁ T,c ₂,T],c ₂>c ₁>0. We find that in the limitT the variance and distribution of N=N(R; S)–S depend strongly on howS grows withT. There is a saturation regimeS/T, asT, in which the fluctuations in N coming from the two boundaries of are independent. Then there is a scaling regime,S/Tz, 0<z<, in which the distribution depends onz in an almost periodic way going to a Gaussian asz0. The variance in this limit approachesz for genericA, but can be larger for degenerate cases. The former behavior is what one would expect from the Poisson limit of a distribution for annuli of finite area.     

Absence of absolutely continuous spectrum of Floquet operators

The spectrum of the Floquet operator associated with time-periodic perturbations of discrete Hamiltonians is considered. If the gap between successive eigenvalues _j of the unperturbed Hamiltonian grows as _j - _j-1 j and the multiplicity of _j grows asj with >0 asj tends to infinity, then the corresponding Floquet operator possesses no absolutely continuous spectrum provided the perturbation is smooth enough.     

Some analytical semigroups occurring in probability theory

One-parameter semigroups occurring in operator-limit distributions are investigated. The topological-algebraic background of the relevant monoids is discussed and Lie semigroup theory is applied to the Urbanik Decomposability Semigroup.     

The matched queueing network PH/M/c→oPH/PH/1

We consider a matched queueing network PH/M/coPH/PH/1, where the output of the first subsystem PH/M/c is regarded as one of the inputs of the second subsystem and matched with another input of PH-renewal process, and they are served by a single server with PH-service-distribution. The stationary state probabilities, the distribution of the virtual waiting times in the two subsystems, and the distribution of the total virtual waiting time are derived.The work was supported by the Max-Planck-Gesellschaft, Germany, and the Chinese Academy ofSciences.     

Sensitivity of sample values to parameter changes

The effect of small changes in parameter values on the sample values is investigated further. The concept of linearity with respect to parameters is introduced, and the linear and the nonlinear cases are treated separately. The relation with the sample-path analysis is discussed.     

Thermodynamics for Coulomb systems: A problem at vanishing particle densities

In this paper I combine techniques recently developed by Charles Fefferman with the well-known methods of Joel Lebowitz and Elliott Lieb to resolve some technical problems left unsettled by Lebowitz and Lieb's fundamental 1972 paper The constitution of matter: Existence of thermodynamics for systems composed of electrons and nuclei.     

Equilibrium theory of two-dimensional simple liquids

Using the Wigner-Kirkwood expansion and bare Lennard-Jones (LJ) (12-6) potential, an effectiveLJ potential is derived, which includes the quantum effects through the expressions of the effective diameter∂(T, λ) and well-depth (T, λ). We use theWCA perturbation theory to calculate the free energy and pressure for theLJ and effectiveLJ potentials. Simple analytic expressions are given for the reference system and the first order correction calculated. The results are quite good at high density. The quantum effects on the free energy and pressure are also discussed.     

Calculation of the proton affinities of polyfluorobenzenes and the activation energies for 1,2-hydrogen shifts in their carbocations

Isolated polyfluorobenzene (PFB) molecules and their protonated forms are investigated by the AM1 method with full geometry optimization. The proton affinities of PFB are estimated for different protonated positions. The proton affinity of PFB averaged over all isomers is shown to decrease monotonically as the number of fluorine atoms in the molecule increases. The relative populations of different isomers of arenonium ions (AI) formed by PFB protonation are determined. From the calculated data, the value of ⁺ for the F atom in theipso-position is estimated as 1.00. The activation energies of the 1,2-hydrogen shifts in AI are calculated. The dependences of the proton affinity and the activation energies of 1,2-hydrogen shifts on the number of halogen atoms are found to have distinct characters for PFB and polychlorobenzenes. The physical reasons for these difference are discussed.Translated fromIzvestiya Akademii Nauk. Seriya Khimicheskaya, No. 11, pp. 1878–1882, November, 1993.     

Evaluation of Multicenter Electronic Attraction, Electric Field and Electric Field Gradient Integrals with Screened and Nonscreened Coulomb Potentials over Integer and Noninteger n Slater Orbitals

By the use of complete orthonormal sets of -exponential-type orbitals, where ( = 1, 0, –1, –2,...) the multicenter electronic attraction (EA), electric field (EF) and electric field gradient (EFG) integrals of nonscreened and Yukawa-like screened Coulomb potentials are expressed through the two-center overlap integrals with the same screening constants and the auxiliary functions introduced in our previous paper (I.I. Guseinov, J. Phys. B, 3 (1970) 1399). The recurrence relations for auxiliary functions are useful for the calculation of multicenter EA, EF and EFG integrals for arbitrary integer and noninteger values of principal quantum numbers, screening constants, and location of slater-type orbitals. The convergence of the series is tested by calculating concrete cases.     

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.