Searcharxiv⌕ Search

arXiv subjects

F. H. Szafraniec

Publications and source records attributed to F. H. Szafraniec.

17 recordsLinked to original sources
arXiv2025

Segal-Bargmann type spaces related to non-rotational measure, and entanglement of bipartite squeezed coherent states

Entanglement of bipartite squeezed states generated by holomorphic Hermite functions of two complex variables is investigated using phase-space approach based on the Wigner distribution function. Orthogonality of the holomorphic Hermite functions implies the relationship between certain real parameter associated with the non-rotational measure in the Bargmann space and the squeezing parameter. The mutual relation between squeezing and entanglement is elucidated with the help of Peres-Horodecki positive partial transpose criterion formulated in the phase-space version for continuous-variable systems. The quantitative characteristics of the entanglement is determined using the log-negativity criterion. The oscillator-like model of a two-particle quantum-mechanical system is developed to illustrate the presented findings.

quant-ph↗
arXiv2018

Holomorphic Hermite polynomials in two variables

Generalizations of the Hermite polynomials to many variables and/or to the complex domain have been located in mathematical and physical literature for some decades. Polynomials traditionally called complex Hermite ones are mostly understood as polynomials in $z$ and $\bar{z}$ which in fact makes them polynomials in two real variables with complex coefficients. The present paper proposes to investigate for the first time holomorphic Hermite polynomials in two variables. Their algebraic and analytic properties are developed here. While the algebraic properties do not differ too much for those considered so far, their analytic features are based on a kind of non-rotational orthogonality invented by van Eijndhoven and Meyers. Inspired by their invention we merely follow the idea of Bargmann's seminal paper (1961) giving explicit construction of reproducing kernel Hilbert spaces based on those polynomials. "Homotopic" behavior of our new formation culminates in comparing it to the very classical Bargmann space of two variables on one edge and the aforementioned Hermite polynomials in $z$ and $\bar{z}$ on the other. Unlike in the case of Bargmann's basis our Hermite polynomials are not product ones but factorize to it when bonded together with the first case of limit properties leading both to the Bargmann basis and suitable form of the reproducing kernel. Also in the second limit we recover standard results obeyed by Hermite polynomials in $z$ and $\bar{z}$.

math.CA↗
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↗
arXiv2016

The Sobolev moment problem and Jordan dilations

Moment problems and orthogonal polynomials, both meant in a single real variable, belong to the oldest problems in Classical Analysis. They have been developing for over a century in two parallel, mostly independent streams. During the last 20 years a rapid advancement of polynomials orthogonal in Sobolev space has been noticed; their moment counterpart seems to be not paid enough attention as it deserves. In this paper we intend to resume the theme and open the door for further, deeper study of the problem.

math.FA↗
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↗
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↗
arXiv2009

Componentwise and Cartesian decompositions of linear relations

Let $A$ be a, not necessarily closed, linear relation in a Hilbert space $\sH$ with a multivalued part $\mul A$. An operator $B$ in $\sH$ with $\ran B\perp\mul A^{**}$ is said to be an operator part of $A$ when $A=B \hplus (\{0\}\times \mul A)$, where the sum is componentwise (i.e. span of the graphs). This decomposition provides a counterpart and an extension for the notion of closability of (unbounded) operators to the setting of linear relations. Existence and uniqueness criteria for the existence of an operator part are established via the so-called canonical decomposition of $A$. In addition, conditions are developed for the decomposition to be orthogonal (components defined in orthogonal subspaces of the underlying space). Such orthogonal decompositions are shown to be valid for several classes of relations. The relation $A$ is said to have a Cartesian decomposition if $A=U+\I V$, where $U$ and $V$ are symmetric relations and the sum is operatorwise. The connection between a Cartesian decomposition of $A$ and the real and imaginary parts of $A$ is investigated.

math.FA↗
arXiv2009

Extending positive definiteness

The main result of the paper gives criteria for extendibility of sesquilinear form-valued mappings defined on symmetric subsets of *-semigroups to positive definite ones. By specifying this we obtain new solutions of: * the truncated complex moment problem, * the truncated multidimensional trigonometric moment problem, * the truncated two-sided complex moment problem, as well as characterizations of unbounded subnormality and criteria for the existence of unitary power dilation.

math.FA↗
arXiv2009

Normals, subnormals and an open question

An acute look at \underbar{basic} facts concerning \underbar{unbounded} subnormal operators is taken here. These operators have the richest structure and are the most exciting among the whole family of beneficiaries of the normal ones. Therefore, the latter must necessarily be taken into account as the reference point for any exposition of subnormality. So as to make the presentation more appealing a kind of comparative survey of the bounded and unbounded case has been set forth. \noindent This piece of writing serves rather as a practical guide to this largely impenetrable territory than an exhausting report.

math.FA↗
arXiv2009

Decomposing numerical ranges along with spectral sets

This note is to indicate the new sphere of applicability of the method developed by Mlak as well as by the author. Restoring those ideas is summoned by current developments concerning $K$-spectral sets on numerical ranges.

math.FA↗
arXiv2007

Operators of the $q$-oscillator

We scrutinize the possibility of extending the result of \cite{ccr} to the case of q-deformed oscillator for $q$ real; for this we exploit the whole range of the deformation parameter as much as possible. We split the case into two depending on whether a solution of the commutation relation is bounded or not. Our {\it leitmotif} is {\it subnormality}. The deformation parameter $q$ is reshaped and this is what makes our approach effective. The newly arrived parameter, the operator $C$, has two remarkable properties: it separates in the commutation relation the annihilation and creation operators from the deformation as well as it $q$-commutes with those two. This is why introducing the operator $C$ seems to be far-reaching.

math.FA↗
arXiv2007

Moments from their very truncations

It is known that positive definiteness is not enough for the multidimensional moment problem to be solved. We would like throw in to the garden of existing in this matter so far results one more, a result which takes into considerations the utmost possible truncations.

math.FA↗
arXiv2007

Adjoints and Formal Adjoints of Matrices of Unbounded Operators

In this paper we {\em discuss} diverse aspects of mutual relationship between adjoints and formal adjoints of unbounded operators bearing a matrix structure. We emphasize on the behaviour of row and column operators as they turn out to be the germs of an arbitrary matrix operator, providing most of the information about the latter {as it is the troublemaker}.

math.FA↗
arXiv2006

A canonical decomposition for linear operators and linear relations

An arbitrary linear relation (multivalued operator) acting from one Hilbert space to another Hilbert space is shown to be the sum of a closable operator and a singular relation whose closure is the Cartesian product of closed subspaces. This decomposition can be seen as an analog of the Lebesgue decomposition of a measure into a regular part and a singular part. The two parts of a relation are characterized metrically and in terms of Stone's characteristic projection onto the closure of the linear relation.

math.FA↗
arXiv2000

On oscillatorlike Hamiltonians and squeezing

Generalizing a recent proposal leading to one-parameter families of Hamiltonians and to new sets of squeezed states, we construct larger classes of physically admissible Hamiltonians permitting new developments in squeezing. Coherence is also discussed.

quant-ph↗