arXiv subjects
Publications and source records attributed to N. E. Frankel.
The series $$S_k(z)=\sum_{m=1}^{\infty}\frac{m^kz^m}{(\{array}{c} 2m m \{array})}$$ is evaluated in non-recursive closed and analytically continued beyond its domain of convergence $0\le |z|<4$ for $k=0,1,2,\...$. From this we provide a firm basis for Lehmer's observation that $π$ emerges from the limiting behavior of $S_k(2)$ as $k\rightarrow\infty$.
A bordering of GUE matrices is considered, in which the bordered row consists of zero mean complex Gaussians N$[0,σ/2] + i {\rm N}[0,σ/2]$ off the diagonal, and the real Gaussian N$[μ,σ/\sqrt{2}]$ on the diagonal. We compute the explicit form of the eigenvalue probability function for such matrices, as well as that for matrices obtained by repeating the bordering. The correlations are in general determinantal, and in the single bordering case the explicit form of the correlation kernel is computed. In the large $N$ limit it is shown that $μ$ and/or $σ$ can be tuned to induce a separation of the largest eigenvalue. This effect is shown to be controlled by a single parameter, universal correlation kernel.
An intensive study for both the weak coupling and strong coupling limits of the ground state properties of this classic system is presented. Detailed results for specific values of finite $N$ are given and from them results for general $N$ are determined. We focus on the density matrix and concomitantly its Fourier transform, the occupation numbers, along with the pair correlation function and concomitantly its Fourier transform, the structure factor. These are the signature quantities of the Bose gas. One specific result is that for weak coupling a rational polynomial structure holds despite the transcendental nature of the Bethe equations. All these new results are predicated on the Bethe ansatz and are built upon the seminal works of the past.
A survey of linearized cosmological fluid equations with a number of different matter components is made. To begin with, the one-component case is reconsidered to illustrate some important mathematical and physical points rarely discussed in the literature. The work of some previous studies of two-component systems are examined and re-analyzed to point out some deficiencies of solutions, and further solutions and physical interpretation are then presented. This leads into a general two-component model with variable velocity dispersion parameters and mass density fractions of each component. The equations, applicable to both hot dark matter (HDM) and cold dark matter (CDM) universes are solved in the long wavelength limit. This region is of interest, because some modes in this range of wavenumbers are Jeans unstable. The mixture Jeans wavenumber of the two-component system is introduced and interpreted, and the solutions are discussed, particularly in comparison to analogous solutions previously derived for plasma modes. This work is applicable to that region in the early Universe ($20 < z < 140$), where large scale structure formation is thought to have occurred.
The equations describing a two-component cosmological fluid with linearized density perturbations are investigated in the small wavelength or large $k$ limit. The equations are formulated to include a baryonic component, as well as either a hot dark matter (HDM) or cold dark matter (CDM) component. Previous work done on such a system in static spacetime is extended to reveal some interesting physical properties, such as the Jeans wavenumber of the mixture, and resonant mode amplitudes. A WKB technique is then developed to study the expanding universe equations in detail, and to see whether such physical properties are also of relevance in this more realistic scenario. The Jeans wavenumber of the mixture is re-interpreted for the case of an expanding background spacetime. The various modes are obtained to leading order, and the amplitudes of the modes are examined in detail to compare to the resonances observed in the static spacetime results. It is found that some conclusions made in the literature about static spacetime results cannot be carried over to an expanding cosmology.
In a recent study we have obtained correction terms to the large N asymptotic expansions of the eigenvalue density for the Gaussian unitary and Laguerre unitary ensembles of random N by N matrices, both in the bulk and at the soft edge of the spectrum. In the present study these results are used to similarly analyze the eigenvalue density for Gaussian and Laguerre random matrix ensembles with orthogonal and symplectic symmetry. As in the case of unitary symmetry, a matching is exhibited between the asymptotic expansion of the bulk density, expanded about the edge, and the asymptotic expansion of the edge density, expanded into the bulk. In addition, aspects of the asymptotic expansion of the smoothed density, which involves delta functions at the endpoints of the support, are interpreted microscopically.
We obtain correction terms to the large N asymptotic expansions of the eigenvalue density for the Gaussian unitary and Laguerre unitary ensembles of random N by N matrices, both in the bulk of the spectrum and near the spectral edge. This is achieved by using the well known orthogonal polynomial expression for the kernel to construct a double contour integral representation for the density, to which we apply the saddle point method. The main correction to the bulk density is oscillatory in N and depends on the distribution function of the limiting density, while the corrections to the Airy kernel at the soft edge are again expressed in terms of the Airy function and its first derivative. We demonstrate numerically that these expansions are very accurate. A matching is exhibited between the asymptotic expansion of the bulk density, expanded about the edge, and the asymptotic expansion of the edge density, expanded into the bulk.
Fisher-Hartwig asymptotics refers to the large $n$ form of a class of Toeplitz determinants with singular generating functions. This class of Toeplitz determinants occurs in the study of the spin-spin correlations for the two-dimensional Ising model, and the ground state density matrix of the impenetrable Bose gas, amongst other problems in mathematical physics. We give a new application of the original Fisher-Hartwig formula to the asymptotic decay of the Ising correlations above $T_c$, while the study of the Bose gas density matrix leads us to generalize the Fisher-Hartwig formula to the asymptotic form of random matrix averages over the classical groups and the Gaussian and Laguerre unitary matrix ensembles. Another viewpoint of our generalizations is that they extend to Hankel determinants the Fisher-Hartwig asymptotic form known for Toeplitz determinants.
The density matrix for the impenetrable Bose gas in Dirichlet and Neumann boundary conditions can be written in terms of $<\prod_{l=1}^n| \cosϕ_1-\cosθ_l| |\cosϕ_2-\cosθ_l|>$, where the average is with respect to the eigenvalue probability density function for random unitary matrices from the classical groups $Sp(n)$ and $O^+(2n)$ respectively. In the large $n$ limit log-gas considerations imply that the average factorizes into the product of averages of the form $<\prod_{l=1}^n|\cosϕ-\cosθ_l>$. By changing variables this average in turn is a special case of the function of $t$ obtained by averaging $\prod_{l=1}^n| t-x_l|^{2q}$ over the Jacobi unitary ensemble from random matrix theory. The latter task is accomplished by a duality formula from the theory of Selberg correlation integrals, and the large $n$ asymptotic form is obtained. The corresponding large $n$ asymptotic form of the density matrix is used, via the exact solution of a particular integral equation, to compute the asymptotic form of the low lying effective single particle states and their occupations, which are proportional to $\sqrt{N}$.
The recent experimental realisation of a one-dimensional Bose gas of ultra cold alkali atoms has renewed attention on the theoretical properties of the impenetrable Bose gas. Of primary concern is the ground state occupation of effective single particle states in the finite system, and thus the tendency for Bose-Einstein condensation. This requires the computation of the density matrix. For the impenetrable Bose gas on a circle we evaluate the density matrix in terms of a particular Painlevé VI transcendent in $σ$-form, and furthermore show that the density matrix satisfies a recurrence relation in the number of particles. For the impenetrable Bose gas in a harmonic trap, and with Dirichlet or Neumann boundary conditions, we give a determinant form for the density matrix, a form as an average over the eigenvalues of an ensemble of random matrices, and in special cases an evaluation in terms of a transcendent related to Painlevé V and VI. We discuss how our results can be used to compute the ground state occupations.
We present here an overview of the history, applications and important properties of a function which we refer to as the Levy integral. For certain values of its characteristic parameter the Levy integral defines the symmetric Levy stable probability density function. As we discuss however the Levy integral has applications to a number of other fields besides probability, including random matrix theory, number theory and asymptotics beyond all orders. We exhibit a direct relationship between the Levy integral and a number theoretic series which we refer to as the generalised Euler-Jacobi series. The complete asymptotic expansions for all natural values of its parameter are presented, and in particular it is pointed out that the intricate exponentially small series become dominant for certain parameter values.
Bosons in the form of ultra cold alkali atoms can be confined to a one dimensional (1d) domain by the use of harmonic traps. This motivates the study of the ground state occupations $λ_i$ of effective single particle states $ϕ_i$, in the theoretical 1d impenetrable Bose gas. Both the system on a circle and the harmonically trapped system are considered. The $λ_i$ and $ϕ_i$ are the eigenvalues and eigenfunctions respectively of the one body density matrix. We present a detailed numerical and analytic study of this problem. Our main results are the explicit scaled forms of the density matrices, from which it is deduced that for fixed $i$ the occupations $λ_i$ are asymptotically proportional to $\sqrt{N}$ in both the circular and harmonically trapped cases.
We investigate the phase-space for trajectories in multi-black hole spacetimes. We find that complete, chaotic geodesics are well described by Lyapunov exponents, and that the attractor basin boundary scales as a fractal in a diffeomorphism invariant manner.
We investigate rotation and rotating structures in (2+1)-dimensional Einstein gravity. We show that rotation generally leads to pathological physical situations.