arXiv subjects
Publications and source records attributed to Fabio Bagarello.
It is a well known fact that the Black-Scholes equation admits an alternative representation as a Schrödinger equation expressed in terms of a non self-adjoint hamiltonian. We show how {\em pseudo-bosons}, linear or not, naturally arise in this context, and how they can be used in the computation of the pricing kernel.
Motivated by what one observes dealing with PT-symmetric quantum mechanics, we discuss what happens if a physical system is driven by a diagonalizable Hamiltonian with not all real eigenvalues. In particular, we consider the functional structure related to systems living in finite-dimensional Hilbert spaces, and we show that certain intertwining relations can be deduced also in this case if we introduce suitable antilinear operators. We also analyze a simple model, computing the transition probabilities in the broken and in the unbroken regime.
We consider an operatorial model of alliances between three political parties which interact with their electors, with the undecided voters, and with the electors of the other parties. This extends what was done in a previous paper, where this last type of interactions was not considered. Of course, taking them into account makes the system closer to real life. To produce an exactly solvable model, we restrict here to quadratic Hamiltonians, so that the equations of motion turn out to be linear. The dynamics of the so-called {\em decision functions} are deduced, and some explicit situations are considered in details.
We demonstrate that not all generalized Bogoliubov transformations lead to $\cal D$-pseudo-bosons and prove that a correspondence between the two can only be achieved with the imposition of specific constraints on the parameters defining the transformation. For certain values of the parameters we find that the norms of the vectors in sets of eigenvectors of two related apparently non self-adjoint number-like operators possess different types of asymptotic behavior. We use this result to deduce further that they constitute bases for a Hilbert space, albeit neither of them can form a Riesz base. When the constraints are relaxed they cease to be Hilbert space bases, but remain $\cal D$-quasi bases.
The ${\mathcal D}$-pseudo-boson formalism is illustrated with two examples. The first one involves deformed complex Hermite polynomials built using finite-dimensional irreducible representations of the group ${\rm GL}(2,{\mathbb C})$ of invertible $2 \times 2$ matrices with complex entries. It reveals interesting aspects of these representations. The second example is based on a pseudo-bosonic generalization of operator-valued functions of a complex variable which resolves the identity. We show that such a generalization allows one to obtain a quantum pseudo-bosonic version of the complex plane viewed as the canonical phase space and to understand functions of the pseudo-bosonic operators as the quantized versions of functions of a complex variable.
In a recent paper we have introduced several possible inequivalent descriptions of the dynamics and of the transition probabilities of a quantum system when its Hamiltonian is not self-adjoint. Our analysis was carried out in finite dimensional Hilbert spaces. This is useful, but quite restrictive since many physically relevant quantum systems live in infinite dimensional Hilbert spaces. In this paper we consider this situation, and we discuss some applications to well known models, introduced in the literature in recent years: the extended harmonic oscillator, the Swanson model and a generalized version of the Landau levels Hamiltonian. Not surprisingly we will find new interesting features not previously found in finite dimensional Hilbert spaces, useful for a deeper comprehension of this kind of physical systems.
We consider a two-level system such as a two-level atom, interacting with a cavity field mode in the rotating wave approximation, when the atomic transition frequency or the field mode frequency is periodically driven in time. We show that in both cases, for an appropriate choice of the modulation parameters, the state amplitudes in a generic $n${-}excitation subspace obey the same equations of motion that can be obtained from a \emph{static} non-Hermitian Jaynes-Cummings Hamiltonian with ${\mathcal PT}$ symmetry, that is with an imaginary coupling constant. This gives further support to recent results showing the possible physical interest of ${\mathcal PT}$ symmetric non-Hermitian Hamiltonians. We also generalize the well-known diagonalization of the Jaynes-Cummings Hamiltonian to the non-Hermitian case in terms of pseudo-bosons and pseudo-fermions, and discuss relevant mathematical and physical aspects.
This paper consider the possibility of using some quantum tools in decision making strategies. In particular, we consider here a dynamical open quantum system helping two players, $\G_1$ and $\G_2$, to take their decisions in a specific context. We see that, within our approach, the final choices of the players do not depend in general on their initial {\em mental states}, but they are driven essentially by the environment which interacts with them. The model proposed here also considers interactions of different nature between the two players, and it is simple enough to allow for an analytical solution of the equations of motion.
We discuss systematically several possible inequivalent ways to describe the dynamics and the transition probabilities of a quantum system when its hamiltonian is not self-adjoint. In order to simplify the treatment, we mainly restrict our analysis to finite dimensional Hilbert spaces. In particular, we propose some experiments which could discriminate between the various possibilities considered in the paper. An example taken from the literature is discussed in detail.
We introduce the concept of an {\em operator decision making technique} and apply it to a concrete political problem: should a given political party form a coalition or not? We focus on the situation of three political parties, and divide the electorate into four groups: partisan supporters of each party and a group of undecided voters. We consider party-party interactions of two forms: shared or differing alliance attitudes. Our main results consist of time-dependent decision functions for each of the three parties, and their asymptotic values, i.e., their final decisions on whether or not to form a coalition.
We adopt an operatorial method, based on creation, annihilation and number operators, to describe one or two populations mutually interacting and moving in a two--dimensional region. In particular, we discuss how the two populations, contained in a certain two-dimensional region with a non--trivial topology, react when some alarm occurs. We consider the cases of both low and high densities of the populations, and discuss what is changing as the strength of the interaction increases. We also analyze what happens when the region has either a single exit or two ways out.
We demonstrate that a non self-adjoint Hamiltonian of harmonic oscillator type defined on a two-dimensional noncommutative space can be diagonalized exactly by making use of pseudo-bosonic operators. The model admits an antilinear symmetry and is of the type studied in the context of PT-symmetric quantum mechanics. Its eigenvalues are computed to be real for the entire range of the coupling constants and the biorthogonal sets of eigenstates for the Hamiltonian and its adjoint are explicitly constructed. We show that despite the fact that these sets are complete and biorthogonal, they involve an unbounded metric operator and therefore do not constitute (Riesz) bases for the Hilbert space $\Lc^2(\Bbb R^2)$, but instead only D-quasi bases. As recently proved by one of us (FB), this is sufficient to deduce several interesting consequences.
In a series of recent papers we have shown how the dynamical behavior of certain classical systems can be analyzed using operators evolving according to Heisenberg-like equations of motions. In particular, we have shown that raising and lowering operators play a relevant role in this analysis. The technical problem of our approach stands in the difficulty of solving the equations of motion, which are, first of all, {\em operator-valued} and, secondly, quite often nonlinear. In this paper we construct a general procedure which significantly simplifies the treatment for those systems which can be described in terms of fermionic operators. The proposed procedure allows to get an analytic solution, both for quadratic and for more general hamiltonians.
Using as a prototype example the harmonic oscillator we show how losing self-adjointness of the hamiltonian $H$ changes drastically the related functional structure. In particular, we show that even a small deviation from strict self-adjointness of $H$ produces two deep consequences, not well understood in the literature: first of all, the original orthonormal basis of $H$ splits into two families of biorthogonal vectors. These two families are complete but, contrarily to what often claimed for similar systems, none of them is a basis for the Hilbert space $\Hil$. Secondly, the so-called metric operator is unbounded, as well as its inverse. In the second part of the paper, after an extension of some previous results on the so-called $\D$ pseudo-bosons, we discuss some aspects of our extended harmonic oscillator from this different point of view.
We consider some modifications of the two dimensional canonical commutation relations, leading to {\em non commutative bosons} and we show how biorthogonal bases of the Hilbert space of the system can be obtained out of them. Our construction extends those recently introduced by one of us (FB), modifying the canonical anticommutation relations. We also briefly discuss how bicoherent states, producing a resolution of the identity, can be defined.
In some recent papers a loss-gain electronic circuit has been introduced and analyzed within the context of PT-quantum mechanics. In this paper we show that this circuit can be analyzed using the formalism of the so-called pseudo-fermions. In particular we discuss the time behavior of the circuit, and we construct two biorthogonal bases associated to the Liouville matrix $\Lc$ used in the treatment of the dynamics. We relate these bases to $\Lc$ and $\Lc^\dagger$, and we also show that a self-adjoint Liouville-like operator could be introduced in the game. Finally, we describe the time evolution of the circuit in an {\em Heisenberg-like} representation, driven by a non self-adjoint hamiltonian.
We continue our analysis of the consequences of the commutation relation $[S,T]=\Id$, where $S$ and $T$ are two closable unbounded operators. The {\em weak} sense of this commutator is given in terms of the inner product of the Hilbert space $\H$ where the operators act. {We also consider what we call, adopting a physical terminology}, a {\em nonlinear} extension of the above commutation relations.
We propose an alternative definition for pseudo-bosons. This simplifies the mathematical structure, minimizing the required assumptions. Some physical examples are discussed, as well as some mathematical results related to the biorthogonal sets arising out of our framework.