arXiv subjects
Publications and source records attributed to M. L. Glasser.
It is argued that for certain meromorphic functions $u:\cal{R}\rightarrow\cal{R}$ and analytic function $ A_1$ and for any integrable function $F$, as long as it converges as a Cauchy Principal Value,, $$\int_{-\infty}^{\infty}A_1(x)F[u(x)] dx=\int_{-\infty}^{\infty} A_2(x)F(x) dx,$$ where $A_2$ is also analytic.
An $n$-dimensional generalization of the Onsager Ising partition function integral is reduced to a single integral and applied to evaluate the partition function and residual entropy of an eight vertex model.
This short note investigates a number of index integrals of products of the Lommel functions $s_{μ,ν}(a)$ and uncovers an integral relationship. between this function and the Tchebyshev polynomials $T_{2n}(x)$.
It is proposed that the validity, or not, of the Riemann Hypothesis might be established on the basis of the integral $$\int\frac{ξ(2s)}{ξ(s)}ds$$ where $$ξ(s)=(s-1)π^{-s/2}Γ(1+s/2)ζ(s).$$
In this work we analyze the statistical thermodynamics of Diced lattice carriers employing a Greens function formulation to examine the grand potential, Helmholtz free energy, the grand and ordinary partition functions and entropy. This facilitates the calculation of the specific heat, and all evaluations are carried out for both the degenerate and nondegenerate statistical regimes.
In this note a one-dimensional band model is proposed based on a periodic Dirac comb having an identical mass distribution $m(x)$ . in each unit cell. The mass function is represented as a Hermitian, non-local separable operator. Two specific cases--a constant mass model and a sinusoidal mass model--are examined. The lowest electron and positron bands for the constant mass case are similar to those for the standard relativistic Kronig-Penney model, suggesting that non-locality has little influence. The results for the sinusoidal case are consistent with the expectation that at low wavenumber an electron "feels" it has am average constant mass, but at high wave number, the particle "sees" the periodic mass variation and the band is distorted.
Let $f_0(z) = \exp(z/(1-z))$, $f_1(z) = \exp(1/(1-z))E_1(1/(1-z))$, where $E_1(x) = \int_x^\infty e^{-t}t^{-1}{\,d}t$. Let $a_n = [z^n]f_0(z)$ and $b_n = [z^n]f_1(z)$ be the corresponding Maclaurin series coefficients. We show that $a_n$ and $b_n$ may be expressed in terms of confluent hypergeometric functions. We consider the asymptotic behaviour of the sequences $(a_n)$ and $(b_n)$ as $n \to \infty$, showing that they are closely related, and proving a conjecture of Bruno Salvy regarding $(b_n)$. Let $ρ_n = a_n b_n$, so $\sum ρ_n z^n = (f_0\,\odot f_1)(z)$ is a Hadamard product. We obtain an asymptotic expansion $2n^{3/2}ρ_n \sim -\sum d_k n^{-k}$ as $n \to \infty$, where the $d_k\in\mathbb Q$, $d_0=1$. We conjecture that $2^{6k}d_k \in \mathbb Z$. This has been verified for $k \le 1000$.
This note investigates a number of integrals of and integral equations satisfied by Riemann's $ξ-$function. A different, less restrictive, derivation of one of his key identities is provided. This work centers on the critical strip and it is argued that the line $s=3/2+i t$ , e.g., contains a holographic image of this region.
The Moll-Arias de Reyna integral [1] $$\int_0^{\infty}\frac{dx}{(x^2+1)^{3/2}}\frac{1}{\sqrt{φ(x)+\sqrt{φ(x)}}}$$ $$φ(x)=1+\frac{4}{3}\left(\frac{x}{x^2+1}\right)^2$$ is generalised and several values are given.
We decorate the one-dimensional conic oscillator $\frac{1}{2} \left[-\frac{d^{2} }{dx^{2} } + \left|x \right| \right]$ with a point impurity of either $δ$-type, or local $δ'$-type or even nonlocal $δ'$-type. All the three cases are exactly solvable models, which are explicitly solved and analysed, as a first step towards higher dimensional models of physical relevance. We analyse the behaviour of the change in the energy levels when an interaction of the type $-λ\,δ(x)$ or $-λ\,δ(x-x_0)$ is switched on. In the first case, even energy levels (pertaining to antisymmetric bound states) remain invariant with $λ$ although odd energy levels (pertaining to symmetric bound states) decrease as $λ$ increases. In the second, all energy levels decrease when the form factor $λ$ increases. A similar study has been performed for the so called nonlocal $δ'$ interaction, requiring a coupling constant renormalization, which implies the replacement of the form factor $λ$ by a renormalized form factor $β$. In terms of $β$, even energy levels are unchanged. However, we show the existence of level crossings: after a fixed value of $β$ the energy of each odd level, with the natural exception of the first one, becomes lower than the constant energy of the previous even level. Finally, we consider an interaction of the type $-aδ(x)+bδ'(x)$, and analyse in detail the discrete spectrum of the resulting self-adjoint Hamiltonian.
The exact Green function is constructed for a quantum system, with known Green function, which is decorated by two delta function impurities.It is shown that when two such impurities coincide they behave as a single singular potential with combined amplitude. The results are extended to N impurities.
We determine the exact time-dependent non-idempotent one-particle reduced density matrix and its spectral decomposition for a harmonically confined two-particle correlated one-dimensional system when the interaction terms in the Schrödinger Hamiltonian are changed abruptly. Based on this matrix in coordinate space we derivea precise condition for the equivalence of the purity and the overlap-square of the correlated and non-correlated wave functions as the system evolves in time. This equivalence holds only if the interparticle interactions are affected, while the confinement terms are unaffected within the stability range of the system. Under this condition we also analyze various time-dependent measures of entanglement and demonstrate that, depending on the magnitude of the changes made in the Schrödinger Hamiltonian, periodic, logarithmically incresing or constant value behavior of the von Neumann entropy can occur.
An inversion method is formulated for extracting entanglement-related information on two-particle interactions in a one-dimensional system from measurable one-particle position- and momentum-distribution functions. The method is based on a shell-like expansion of these norm-1 measured quantities in terms of product states taken from a parametric orthonormal complete set. The mathematical constraints deduced from these point-wise expansions are restricted by the underlying physics of our harmonically confined and interacting Heisenberg model. Based on these exact results, we introduce an approximate optimization scheme for different inter-particle interactions and discuss it from the point of view of entropic correlation measures.
We show that the following double integral \[\int_{0}^π{\rm d}x \int_0^x {\rm d}y \frac{1}{\sqrt{1-\smash[b]{p}\cos x}\sqrt{1+\smash[b]{q\cos y}}}\]remains invariant as one trades the parameters $p$ and $q$ for $p'=\sqrt{1-p^2}$ and $q'=\sqrt{1-q^2}$ respectively. This invariance property is suggested from symmetry considerations in the operating characterstics of a semiconductor Hall-effect device.
This paper presents the equality of finite index sums of Bessel func- tions containing arbitrary numbers of terms. These reduce to the familiar three term recursion formulas in simple cases.
We compute and compare the effects due to a uniform perpendicular magnetic field as well as temperature on the static polarization functions for monolayer graphene (MLG), associated with the Dirac point, with that for the two-dimensional electron liquid (2DEL) with the use of comprehensive numerical calculations. Previous results for the 2DEL are discussed and, in particular, we point out a flaw in a reported analytic derivation which was carried out to exhibit the smearing of the Fermi surface for 2DEL. The relevance of our study to the Kohn anomaly in low-dimensional structures and the Friedel oscillations for the screening of the potential for a dilute distribution of impurities is reported. Our results show substantial differences due to screening for the 2DEL and MLG which have not been given adequate attention previously.
We study a one-dimensional singular potential plus three types of regular interactions: constant electric field, harmonic oscillator and infinite square well. We use the Lippman-Schwinger Green function technique in order to search for the bound state energies. In the electric field case the unique bound state coincides with that found in an earlier study as the field is switched off. For non-zero field the ground state is shifted and positive energy "quasibound states" appear. For the harmonic oscillator we find a quantum phase transition of a novel type. This behavior does not occur in the corresponding case of an infinite square well and demonstrates the influence of quantum non-locality.
A Master equation has been previously obtained which allows the analytic integration of a fairly large family of functions provided that they possess simple properties. Here, the properties of this Master equation are explored, by extending its applicability to a general range of an independent parameter. Examples are given for various values of the parameter using Riemann's Zeta function as a template to demonstrate the utility of the equation. The template is then extended to the derivation of various sum rules among the zeros of the Zeta function as an example of how similar rules can be obtained for other functions.