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Fabio Bagarello

Publications and source records attributed to Fabio Bagarello.

At least 109 records · Page 6Linked to original sources
arXiv2013

Pseudo-bosons for the $D_2$ type quantum Calogero model

In the first part of this paper we show how a simple system, a 2-dimensional quantum harmonic oscillator, can be described in terms of pseudo-bosonic variables. This apparently {\em strange} choice is useful when the {\em natural} Hilbert space of the system, $L^2({\bf R}^2)$ in this case, is, for some reason, not the most appropriate. This is exactly what happens for the $D_2$ type quantum Calogero model considered in the second part of the paper, where the Hilbert space $L^2({\bf R}^2)$ appears to be an unappropriate choice, since the eigenvectors of the relevant hamiltonian are not square-integrable. Then we discuss how a certain intertwining operator arising from the model can be used to fix a different Hilbert space more {\em useful}.

math-ph↗
arXiv2012

Quantizations from reproducing kernel spaces

The purpose of this work is to explore the existence and properties of reproducing kernel Hilbert subspaces of $L^2(\C, \, d^2z/π)$ based on subsets of complex Hermite polynomials. The resulting coherent states (CS) form a family depending on a nonnegative parameter $s$. We examine some interesting issues, mainly related to CS quantization, like the existence of the usual harmonic oscillator spectrum despite the absence of canonical commutation rules. The question of mathematical and physical equivalences between the $s$-dependent quantizations is also considered.

math-ph↗
arXiv2012

Damping and Pseudo-fermions

After a short abstract introduction on the time evolution driven by non self-adjoint hamiltonians, we show how the recently introduced concept of {\em pseudo-fermion} can be used in the description of damping in finite dimensional quantum systems, and we compare the results deduced adopting the Schrödinger and the Heisenberg representations.

math-ph↗
arXiv2012

A phenomenological operator description of interactions between populations with applications to migration

We adopt an operatorial method based on the so-called creation, annihilation and number operators in the description of different systems in which two populations interact and move in a two-dimensional region. In particular, we discuss diffusion processes modeled by a quadratic hamiltonian. This general procedure will be adopted, in particular, in the description of migration phenomena. With respect to our previous analogous results, we use here fermionic operators since they automatically implement an upper bound for the population densities.

physics.bio-ph↗
arXiv2012

Few simple rules to fix the dynamics of classical systems using operators

We show how to use operators in the description of {\em exchanging processes} often taking place in (complex) classical systems. In particular, we propose a set of rules giving rise to an {\em hamiltonian} operator for such a system $\Sc$, which can be used to deduce the dynamics of $\Sc$.

physics.soc-ph↗
arXiv2012

Linear Pseudo-fermions

In a recent series of papers we have analyzed a certain deformation of the canonical commutation relations producing an interesting functional structure which has been proved to have some connections with physics, and in particular with quasi-hermitian quantum mechanics. Here we repeat a similar analysis starting with the canonical anticommutation relations. We will show that in this case most of the assumptions needed in the former situation are automatically satisfied, making our construction rather {\em friendly}. We discuss some examples of our construction, again related to quasi-hermitian quantum mechanics, and the bi-coherent states for the system.

math-ph↗
arXiv2012

(Regular) pseudo-bosons versus bosons

We discuss in which sense the so-called {\em regular pseudo-bosons}, recently introduced by Trifonov and analyzed in some details by the author, are related to ordinary bosons. We repeat the same analysis also for {\em pseudo-bosons}, and we analyze the role played by certain intertwining operators, which may be bounded or not.

math-ph↗
arXiv2012

Locally convex quasi $C^*$-normed algebras

If $\ca_0[|\cdot|_0]$ is a $\cs$-normed algebra and $τ$ a locally convex topology on $\ca_0$ making its multiplication separately continuous, then $\widetilde{\ca_0}[τ]$ (completion of $\ca_0[τ]$) is a locally convex quasi *-algebra over $\ca_0$, but it is not necessarily a locally convex quasi *-algebra over the $\cs$-algebra $\widetilde{\ca_0}[|\cdot|_0]$ (completion of $\ca_0[|\cdot|_0]$). In this article, stimulated by physical examples, we introduce the notion of a locally convex quasi $\cs$-normed algebra, aiming at the investigation of $\widetilde{\ca_0}[τ]$; in particular, we study its structure, *-representation theory and functional calculus.

math-ph↗
arXiv2012

Representable linear functionals on partial *-algebras

A GNS - like *-representation of a \pa\ $\A$ defined by certain representable linear functionals on $\A$ is constructed. The study of the interplay with the GNS construction associated with invariant positive sesquilinear forms (ips) leads to the notions of pre-core and of singular form. It is shown that a positive sesquilinear form with pre-core always decomposes into the sum of an ips form and a singular one.

math-ph↗
arXiv2012

Induced and reduced unbounded operator algebras

The induction and reduction precesses of an O*-vector space $\M$ obtained by means of a projection taken, respectively, in $\M$ itself or in its weak bounded commutant $\M'_\w$ are studied. In the case where $\M$ is a partial GW*-algebra, sufficient conditions are given for the induced and the reduced spaces to be partial GW*-algebras again.

math-ph↗
arXiv2011

Weak commutation relations of unbounded operators and applications

Four possible definitions of the commutation relation $[S,T]=\Id$ of two closable unbounded operators $S,T$ are compared. The {\em weak} sense of this commutator is given in terms of the inner product of the Hilbert space $\H$ where the operators act. Some consequences on the existence of eigenvectors of two number-like operators are derived and the partial O*-algebra generated by $S,T$ is studied. Some applications are also considered.

math-ph↗
arXiv2011

Non-isospectral Hamiltonians, intertwining operators and hidden hermiticity

We have recently proposed a strategy to produce, starting from a given hamiltonian $h_1$ and a certain operator $x$ for which $[h_1,xx^\dagger]=0$ and $x^\dagger x$ is invertible, a second hamiltonian $h_2$ with the same eigenvalues as $h_1$ and whose eigenvectors are related to those of $h_1$ by $x^\dagger$. Here we extend this procedure to build up a second hamiltonian, whose eigenvalues are different from those of $h_1$, and whose eigenvectors are still related as before. This new procedure is also extended to crypto-hermitian hamiltonians.

math-ph↗
arXiv2011

Non linear pseudo-bosons versus hidden Hermiticity

The increasingly popular concept of a hidden Hermiticity of operators (i.e., of their Hermiticity with respect to an {\it ad hoc} inner product in Hilbert space) is compared with the recently introduced notion of {\em non-linear pseudo-bosons}. The formal equivalence between these two notions is deduced under very general assumptions. Examples of their applicability in quantum mechanics are discussed.

math-ph↗
arXiv2011

Dissipation evidence for the quantum damped harmonic oscillator via pseudo-bosons

It is known that a self-adjoint, time-independent hamiltonian can be defined for the quantum damped harmonic oscillator. We show here that the two vacua naturally associated to this operator, when expressed in terms of pseudo-bosonic lowering and raising operators, appear to be non square-integrable. This fact is interpreted as the evidence of the dissipation effect of the classical oscillator at a purely quantum level.

math-ph↗
arXiv2011

Pseudo-bosons, so far

In the past years several extensions of the canonical commutation relations have been proposed by different people in different contexts and some interesting physics and mathematics have been deduced. Here, we review some recent results on the so-called {\em pseudo-bosons}. They arise from a special deformation of the canonical commutation relation $[a,a^\dagger]=\1$, which is replaced by $[a,b]=\1$, with $b$ not necessarily equal to $a^\dagger$. We start discussing some of their mathematical properties and then we discuss several examples.

math-ph↗
arXiv2011

Non linear pseudo-bosons

In a series of recent papers the author has introduced the notion of (regular) pseudo-bosons showing, in particular, that two number-like operators, whose spectra are ${\Bbb N}_0:={\Bbb N}\cup\{0\}$, can be naturally introduced. Here we extend this construction to operators with rather more general spectra. Of course, this generalization can be applied to many more physical systems. We discuss several examples of our framework.

math-ph↗
arXiv2011

The dynamical problem for a non self-adjoint Hamiltonian

We review some recent results of the so-called quasi-hermitian quantum mechanics, with particular focus on the quantum dynamics both in the Schrödinger and in the Heisenberg representations. The role of Krein spaces is also discussed.

math.FA↗