arXiv subjects
Publications and source records attributed to F. Bagarello.
In this paper the problem of recovering an algebraic dynamics in a perturbative approach is discussed. The mathematical environment in which the physical problem is considered is that of algebras of unbounded operators endowed with the quasi-uniform topology. After some remarks on the domain of the perturbation, conditions are given for the dynamics to exist as the limit of a net of regularized linear maps.
We describe a method for constructing vector coherent states for quantum supersymmetric partner Hamiltonians. The method is then applied to such partner Hamiltonians arising from a generalization of the fractional quantum Hall effect. Explicit examples are worked out.
In the spirit of some earlier work on the construction of vector coherent states over matrix domains, we compute here such states associated to some physical Hamiltonians. In particular, we construct vector coherent states of the Gazeau-Klauder type. As a related problem, we also suggest a way to handle degeneracies in the Hamiltonian for building coherent states. Specific physical Hamiltonians studied include a single photon mode interacting with a pair of fermions, a Hamiltonian involving a single boson and a single fermion, a charged particle in a three dimensional harmonic force field and the case of a two-dimensional electron placed in a constant magnetic field, orthogonal to the plane which contains the electron. In this last example, an interesting modular structure emerges for two underlying von Neumann algebras, related to opposite directions of the magnetic field. This leads to the existence of coherent states built out of KMS states for the system.
We propose a model of an approximatively two--dimensional electron gas in a uniform electric and magnetic field and interacting with a positive background through the Fröhlich Hamiltonian. We consider the stochastic limit of this model and we find the quantum Langevin equation and the generator of the master equation. This allows us to calculate the explicit form of the conductivity and the resistivity tensors and to deduce a {\it fine tuning condition} (FTC) between the electric and the magnetic fields. This condition shows that the $x$--component of the current is zero unless a certain quotient, involving the physical parameters, takes values in a finite set of physically meaningful rational numbers. We argue that this behaviour is quite similar to that observed in the quantum Hall effect. We also show that, under some conditions on the form factors entering in the definition of the model, also the plateaux and the "almost" linear behaviour of the Hall resistivity can be recovered. Our FTC does not distinguish between fractional and integer values.
We discuss some results concerning fixed point equations in the setting of topological *-algebras of unbounded operators. In particular, an existence result is obtained for what we have called {\em weak $τ$ strict contractions}, and some continuity properties of these maps are discussed. We also discuss possible applications of our procedure to quantum mechanical systems.
In this paper we discuss some physical applications of topological *-algebras of unbounded operators. Our first example is a simple system of free bosons. Then we analyze different models which are related to this one. We also discuss the time evolution of two interacting models of matter and bosons. We show that for all these systems it is possible to build up a common framework where the thermodynamical limit of the algebraic dynamics can be conveniently studied and obtained.
The Banach space $L^p(X,μ)$, for $X$ a compact Hausdorff measure space, is considered as a special kind of quasi *-algebra (called CQ*-algebra) over the C*-algebra $C(X)$ of continuous functions on $X$. It is shown that, for $p \geq 2$, $(L^p(X,μ), C(X))$ is *-semisimple (in a generalized sense). Some consequences of this fact are derived.