arXiv subjects
Publications and source records attributed to F. Bagarello.
We discuss some features of non-self-adjoint Hamiltonians with real discrete simple spectrum under the assumption that the eigenvectors form a Riesz basis of Hilbert space. Among other things, {we give conditions under which these Hamiltonians} can be factorized in terms of generalized lowering and raising operators.
We discuss the role of pseudo-fermions in the analysis of some two-dimensional models, recently introduced in connection with non self-adjoint hamiltonians. Among other aspects, we discuss the appearance of exceptional points in connection with the validity of the extended anti-commutation rules which define the pseudo-fermionic structure.
We show how to construct, out of a certain basis invariant under the action of one or more unitary operators, a second biorthogonal set with similar properties. In particular, we discuss conditions for this new set to be also a basis of the Hilbert space, and we apply the procedure to coherent states. We conclude the paper considering a simple application of our construction to pseudo-hermitian quantum mechanics.
We show how some recent models of PT-quantum mechanics perfectly fit into the settings of $\cal D$ pseudo-bosons, as introduced by one of us. Among the others, we also consider a model of non-commutative quantum mechanics, and we show that this model too can be described in terms of $\cal D$ pseudo-bosons.
The energy levels, generally known as the Landau levels, which characterize the motion of an electron in a constant magnetic field, are those of the one-dimensional harmonic oscillator, with each level being infinitely degenerate. We show in this paper how the associated von Neumann algebra of observables display a modular structure in the sense of the Tomita-Takesaki theory, with the algebra and its commutant referring to the two orientations of the magnetic field. A KMS state can be built which in fact is the Gibbs state for an ensemble of harmonic oscillators. Mathematically, the modular structure is shown to arise as the natural modular structure associated to the Hilbert space of all Hilbert-Schmidt operators.
In this paper we continue our descriptions of stock markets in terms of some non abelian operators which are used to describe the portfolio of the various traders and other {\em observable} quantities. After a first prototype model with only two traders, we discuss a more realistic model of market with an arbitrary number of traders. For both models we find approximated solutions for the time evolution of the portfolio of each trader. In particular, for the more realistic model, we use the {\em stochastic limit} approach and a {\em fixed point like} approximation.
In this paper we continue our systematic analysis of the operatorial approach previously proposed in an economical context and we discuss a {\em mixed} toy model of a simplified stock market, i.e. a model in which the price of the shares is given as an input. We deduce the time evolution of the portfolio of the various traders of the market, as well as of other {\em observable} quantities. As in a previous paper, we solve the equations of motion by means of a {\em fixed point like} approximation.
We apply the orthonormalization procedure previously introduced by two of us and adopted in connection with coherent states to Gabor frames and other examples. For instance, for Gabor frames we show how to construct $g(x)\in L^2(\Bbb{R})$ in such a way the functions $g_{\underline n}(x)=e^{ian_1x}g(x+an_2)$, $\underline n\in\Bbb{Z}^2$ and $a$ some positive real number, are mutually orthogonal. We discuss in some details the role of the lattice naturally associated to the procedure in this analysis.
In a previous paper we have proven that any multi-resolution analysis of $L^2(\R)$ produces, for even values of the inverse filling factor and for a square lattice, a single-electron wave function of the lowest Landau level (LLL) which, together with its (magnetic) translated, gives rise to an orthonormal set in the LLL. We have also discussed the inverse construction. In this paper we simplify the procedure, clarifying the role of the kq-representation. Moreover, we extend our previous results to the more physically relevant case of a triangular lattice and to odd values of the inverse filling factor. We also comment on other possible shapes of the lattice as well as on the extension to other Landau levels. Finally, just as a first application of our technique, we compute (an approximation of) the Coulomb energy for the Haar wavefunction, for a filling $ν={1/3}$.
The unbounded bicommutant $(\mathfrak M_{E'})''$ of the {\em reduction} of an O*-algebra $\MM$ via a given projection $E'$ weakly commuting with $\mathfrak M$ is studied, with the aim of finding conditions under which the reduction of a GW*-algebra is a GW*-algebra itself. The obtained results are applied to the problem of the existence of conditional expectations on O*-algebras.
The notion of completely positive invariant conjugate-bilinear map in a partial *-algebra is introduced and a generalized Stinespring theorem is proven. Applications to the existence of integrable extensions of *-representations of commutative, locally convex quasi*-algebras are also discussed.
Let $(A, A_o)$ be a topological quasi *-algebra, which means in particular that $A_o$ is a topological *-algebra, dense in $A$. Let $π^o$ be a *-representation of $A_o$ in some pre-Hilbert space ${\cal D} \subset {\cal H}$. Then we present several ways of extending $π^o$, by closure, to some larger quasi *-algebra contained in $A$, either by Hilbert space operators, or by sesquilinear forms on ${\cal D}$. Explicit examples are discussed, both abelian and nonabelian, including the CCR algebra.
Morphisms and representations of a class of Banach C*-modules, called CQ*algebras, are considered. Together with a general method for constructing CQ*-algebras, two different ways of extending the GNS-representation are presented.
The completion $\overline{A}[τ]$ of a locally convex *-algebra $A [ τ]$ with not jointly continuous multiplication is a *-vector space with partial multiplication $xy$ defined only for $x$ or $y \in A_{0}$, and it is called a topological quasi *-algebra. In this paper two classes of topological quasi *-algebras called strict CQ$^*$-algebras and HCQ$^*$-algebras are studied. Roughly speaking, a strict CQ$^*$-algebra (resp. HCQ$^*$-algebra) is a Banach (resp. Hilbert) quasi *-algebra containing a C$^*$-algebra endowed with another involution $\sharp$ and C$^*$-norm $\| \|_{\sharp}$. HCQ$^*$-algebras are closely related to left Hilbert algebras. We shall show that a Hilbert space is a HCQ$^*$-algebra if and only if it contains a left Hilbert algebra with unit as a dense subspace. Further, we shall give a necessary and sufficient condition under which a strict CQ$^*$-algebra is embedded in a HCQ$^*$-algebra.
The completion of a (normed) $C^*$-algebra $A_0[\| \cdot \|_0]$ with respect to a locally convex topology $τ$ on $A_0$ that makes the multiplication of $A_0$ separately continuous is, in general, a quasi *-algebra, and not a locally convex *-algebra. In this way, one is led to consideration of locally convex quasi $C^*$-algebras, which generalize $C^*$-algebras in the context of quasi *-algebras. Examples are given and the structure of these relatives of $C^*$-algebras is investigated.
Let $A$ be a partial *-algebra endowed with a topology $τ$ that makes it into a locally convex topological vector space $A[τ]$. Then $A$ is called a topological partial *-algebra if it satisfies a number of conditions, which all amount to require that the topology $τ$ fits with the multiplier structure of $A$ Besides the obvious cases of topological quasi *-algebras and CQ*-algebras, we examine several classes of potential topological partial *-algebras, either function spaces (lattices of $L^p$ spaces on $[0,1]$ or on $\mathbb R$, amalgam spaces), or partial *-algebras of operators (operators on a partial inner product space, O*-algebras).
The main purpose of this paper is to construct *-representations from unbounded C$^*$-seminorms on partial *-algebras and to investigate their *-representations.
We discuss a general strategy which produces an orthonormal set of vectors, stable under the action of a given set of unitary operators $A_j$, $j=1,2,...,n$, starting from a fixed normalized vector in $\Hil$ and from a set of unitary operators. We discuss several examples of this procedure and, in particular, we show how a set of {\em coherent-like} vectors can be produced and in which condition over the lattice spacing this can be done.