arXiv subjects

A. Horzela

Publications and source records attributed to A. Horzela.

At least 37 records · Page 2Linked to original sources

arXiv2018

Coherence, squeezing and entanglement -- an example of peaceful coexistence

After exhaustive inspection of bosonic coherent states appearing in physical literature two of us, Horzela and Szafraniec, came in 2012 to the reasonably general definition which relies exclusively on reproducing kernels. The basic feature of coherent states, which is the resolution of the identity, is preserved though it now achieves advantageous form of the Segal-Bargmann transform. It turns out that the aforesaid definition is not only extremely economical but also puts under a common umbrella typical coherent states as well as those which are squeezed and entangled. We examine the case here on the groundwork of holomorphic Hermite polynomials in one and two variables. An interesting side of this story is how some limit procedure allows disentangling.

math-ph↗

arXiv2017

The Havriliak-Negami relaxation and its relatives: the response, relaxation and probability density functions

We study functions related to the experimentally observed Havriliak-Negami dielectric relaxation pattern in the frequency domain $\sim[1+(iωτ_{0})^α]^{-β}$ with $τ_{0}$ being some characteristic time. For $α= l/k< 1$ ($l$ and $k$ positive integers) and $β> 0$ we furnish exact and explicit expressions for response and relaxation functions in the time domain and suitable probability densities in their "dual" domain. All these functions are expressed as finite sums of generalized hypergeometric functions, convenient to handle analytically and numerically. Introducing a reparameterization $β= (2-q)/(q-1)$ and $τ_{0} = (q-1)^{1/α}$ $(1 < q < 2)$ we show that for $0 < α< 1$ the response functions $f_{α, β}(t/τ_{0})$ go to the one-sided Lévy stable distributions when $q$ tends to one. Moreover, applying the self-similarity property of the probability densities $g_{α, β}(u)$, we introduce two-variable densities and show that they satisfy the integral form of the evolution equation.

cond-mat.stat-mech↗

arXiv2017

The stretched exponential behavior and its underlying dynamics. The phenomenological approach

We show that the anomalous diffusion equations with a fractional derivative in the Caputo or Riesz sense are strictly related to the special convolution properties of the Lévy stable distributions which stem from the evolution properties of stretched or compressed exponential function. The formal solutions of these fractional differential equations are found by using the evolution operator method where the evolution operator is presented as integral transforms whose kernel is the Green function. Exact and explicit examples of the solutions are reported and studied for various fractional order of derivatives and different initial conditions.

cond-mat.stat-mech↗

arXiv2016

Theory of relativistic heat polynomials and one-sided Lévy distributions

The theory of pseudo-differential operators is a powerful tool to deal with differential equations involving differential operators under the square root sign. These type of equations are pivotal elements to treat problems in anomalous diffusion and in relativistic quantum mechanics. In this paper we report on new and unsuspected links between fractional diffusion, quantum relativistic equations and particular families of polynomials, linked to the Carlitz family, and playing the role of relativistic heat polynomials. We introduce generalizations of these polynomial families and point out their specific use for the solutions of problems of practical importance.

math-ph↗

arXiv2016

Relativistic Heat Equation via Lévy stable distributions: Exact Solutions

We introduce and study an extension of the heat equation relevant to relativistic energy formula involving square root of differential operators. We furnish exact solutions of corresponding Cauchy (initial) problem using the operator formalism invoking one-sided Lévy stable distributions. We note a natural appearance of Bessel polynomials which allow one the obtention of closed form solutions for a number of initial conditions. The resulting relativistic diffusion is slower than the non-relativistic one, although it still can be termed a normal one. Its detailed statistical characterization is presented in terms of exact evaluation of arbitrary moments and is compared with the non-relativistic case.

math-ph↗

arXiv2015

Relativistic Wave Equations: An Operational Approach

The use of operator methods of algebraic nature is shown to be a very powerful tool to deal with different forms of relativistic wave equations. The methods provide either exact or approximate solutions for various forms of differential equations, such as relativistic Schrödinger, Klein-Gordon and Dirac. We discuss the free particle hypotheses and those relevant to particles subject to non-trivial potentials. In the latter case we will show how the proposed method leads to easily implementable numerical algorithms.

math-ph↗

arXiv2014

Squeezing: the ups and downs

We present an operator theoretic side of the story of squeezed states regardless the order of squeezing. For low order, that is for displacement (order 1) and squeeze (order 2) operators, we bring back to consciousness what is know or rather what has to be known by making the exposition as exhaustive as possible. For the order 2 (squeeze) we propose an interesting model of the Segal-Bargmann type. For higher order the impossibility of squeezing in the traditional sense is proved rigorously. Nevertheless what we offer is the state-of-the-art concerning the topic.

math-ph↗

arXiv2014

Photoluminescence decay of silicon nanocrystals and Lévy stable distributions

Recent experiments have shown that photoluminescence decay of silicon nanocrystals can be described by the stretched exponential function. We show here that the associated decay probability rate is the one-sided Levy stable distribution which describes well the experimental data. The relevance of these conclusions to the underlying stochastic processes is discussed in terms of Levy processes.

cond-mat.mes-hall↗

arXiv2013

The Higher-Order Heat-Type Equations via signed Lévy stable and generalized Airy functions

We study the higher-order heat-type equation with first time and M-th spatial partial derivatives, M = 2, 3, ... . We demonstrate that its exact solutions for M even can be constructed with the help of signed Levy stable functions. For M odd the same role is played by a special generalization of Airy Ai function that we introduce and study. This permits one to generate the exact and explicit heat kernels pertaining to these equations. We examine analytically and graphically the spacial and temporary evolution of particular solutions for simple initial conditions.

cond-mat.stat-mech↗

arXiv2013

Squeezed States and Hermite polynomials in a Complex Variable

Following the lines of the recent paper of J.-P. Gazeau and F. H. Szafraniec [J. Phys. A: Math. Theor. 44, 495201 (2011)], we construct here three types of coherent states, related to the Hermite polynomials in a complex variable which are orthogonal with respect to a non-rotationally invariant measure. We investigate relations between these coherent states and obtain the relationship between them and the squeezed states of quantum optics. We also obtain a second realization of the canonical coherent states in the Bargmann space of analytic functions, in terms of a squeezed basis. All this is done in the flavor of the classical approach of V. Bargmann [Commun. Pur. Appl. Math. 14, 187 (1961)].

quant-ph↗

arXiv2012

Graph model of the Heisenberg-Weyl algebra

We consider an algebraic formulation of Quantum Theory and develop a combinatorial model of the Heisenberg-Weyl algebra structure. It is shown that by lifting this structure to the richer algebra of graph operator calculus, we gain a simple interpretation involving, for example, the natural composition of graphs. This provides a deeper insight into the algebraic structure of Quantum Theory and sheds light on the intrinsic combinatorial underpinning of its abstract formalism.

quant-ph↗

arXiv2011

Quasiclassical Asymptotics and Coherent States for Bounded Discrete Spectra

We consider discrete spectra of bound states for non-relativistic motion in attractive potentials V_σ(x) = -|V_{0}| |x|^{-σ}, 0 < σ\leq 2. For these potentials the quasiclassical approximation for n -> \infty predicts quantized energy levels e_σ(n) of a bounded spectrum varying as e_σ(n) ~ -n^{-2σ/(2-σ)}. We construct collective quantum states using the set of wavefunctions of the discrete spectrum taking into account this asymptotic behaviour. We give examples of states that are normalizable and satisfy the resolution of unity, using explicit positive functions. These are coherent states in the sense of Klauder and their completeness is achieved via exact solutions of Hausdorff moment problems, obtained by combining Laplace and Mellin transform methods. For σin the range 0<σ\leq 2/3 we present exact implementations of such states for the parametrization σ= 2(k-l)/(3k-l), with k and l positive integers satisfying k>l.

math-ph↗

arXiv2010

Combinatorial Algebra for second-quantized Quantum Theory

We describe an algebra G of diagrams which faithfully gives a diagrammatic representation of the structures of both the Heisenberg-Weyl algebra H - the associative algebra of the creation and annihilation operators of quantum mechanics - and U(L_H), the enveloping algebra of the Heisenberg Lie algebra L_H. We show explicitly how G may be endowed with the structure of a Hopf algebra, which is also mirrored in the structure of U(L_H). While both H and U(L_H) are images of G, the algebra G has a richer structure and therefore embodies a finer combinatorial realization of the creation-annihilation system, of which it provides a concrete model.

math-ph↗

arXiv2009

Hopf algebras: motivations and examples

This paper provides motivation as well as a method of construction for Hopf algebras, starting from an associative algebra. The dualization technique involved relies heavily on the use of Sweedler's dual.

math-ph↗

arXiv2009

On certain non-unique solutions of the Stieltjes moment problem

We construct explicit solutions of a number of Stieltjes moment problems based on moments of the form $ρ_{1}^{(r)}(n)=(2rn)!$ and $ρ_{2}^{(r)}(n)=[(rn)!]^{2}$, $r=1,2,...$, $n=0,1,2,...$, \textit{i.e.} we find functions $W^{(r)}_{1,2}(x)>0$ satisfying $\int_{0}^{\infty}x^{n}W^{(r)}_{1,2}(x)dx = ρ_{1,2}^{(r)}(n)$. It is shown using criteria for uniqueness and non-uniqueness (Carleman, Krein, Berg, Pakes, Stoyanov) that for $r>1$ both $ρ_{1,2}^{(r)}(n)$ give rise to non-unique solutions. Examples of such solutions are constructed using the technique of the inverse Mellin transform supplemented by a Mellin convolution. We outline a general method of generating non-unique solutions for moment problems generalizing $ρ_{1,2}^{(r)}(n)$, such as the product $ρ_{1}^{(r)}(n)\cdotρ_{2}^{(r)}(n)$ and $[(rn)!]^{p}$, $p=3,4,...$.

math.FA↗

arXiv2009

Heisenberg-Weyl algebra revisited: Combinatorics of words and paths

The Heisenberg-Weyl algebra, which underlies virtually all physical representations of Quantum Theory, is considered from the combinatorial point of view. We provide a concrete model of the algebra in terms of paths on a lattice with some decomposition rules. We also discuss the rook problem on the associated Ferrers board; this is related to the calculus in the normally ordered basis. From this starting point we explore a combinatorial underpinning of the Heisenberg-Weyl algebra, which offers novel perspectives, methods and applications.

quant-ph↗

arXiv2009

Laguerre-type derivatives: Dobinski relations and combinatorial identities

We consider properties of the operators D(r,M)=a^r(a^†a)^M (which we call generalized Laguerre-type derivatives), with r=1,2,..., M=0,1,..., where a and a^†are boson annihilation and creation operators respectively, satisfying [a,a^†]=1. We obtain explicit formulas for the normally ordered form of arbitrary Taylor-expandable functions of D(r,M) with the help of an operator relation which generalizes the Dobinski formula. Coherent state expectation values of certain operator functions of D(r,M) turn out to be generating functions of combinatorial numbers. In many cases the corresponding combinatorial structures can be explicitly identified.

math-ph↗

arXiv2008

Hopf Algebras in General and in Combinatorial Physics: a practical introduction

This tutorial is intended to give an accessible introduction to Hopf algebras. The mathematical context is that of representation theory, and we also illustrate the structures with examples taken from combinatorics and quantum physics, showing that in this latter case the axioms of Hopf algebra arise naturally. The text contains many exercises, some taken from physics, aimed at expanding and exemplifying the concepts introduced.

quant-ph↗