Transition probabilities for non self-adjoint Hamiltonians in infinite dimensional Hilbert spaces

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.Comment: To appear in Annals of Physic

Pseudo-bosons and Riesz bi-coherent states

After a brief review on \D-pseudo-bosons we introduce what we call {\em Riesz bi-coherent states}, which are pairs of states sharing with ordinary coherent states most of their features. In particular, they produce a resolution of the identity and they are eigenstates of two different annihilation operators which obey pseudo-bosonic commutation rules.Comment: This paper is dedicated to the memories of Gerard Emch and of Twareque Ali. Appeared in the Proceedings of XXXIV Workshop on Geometric Methods in Physics, Bialowieza, Polan

Matrix computations for the dynamics of fermionic systems

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.Comment: In press in International Journal of Theoretical Physic

Deformed quons and bi-coherent states

We discuss how a q-mutation relation can be deformed replacing a pair of conjugate operators with two other and unrelated operators, as it is done in the construction of pseudo-fermions, pseudo-bosons and truncated pseudo-bosons. This deformation involves interesting mathematical problems and suggests possible applications to pseudo-hermitian quantum mechanics. We construct bi-coherent states associated to \D-pseudo-quons, and we show that they share many of their properties with ordinary coherent states. In particular, we find conditions for these states to exist, to be eigenstates of suitable annihilation operators and to give rise to a resolution of the identity. Two examples are discussed in details, one connected to an unbounded similarity map, and the other to a bounded map.Comment: in press in Proceedings of the Royal Society

An operator view on alliances in politics

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.Comment: In press in SIAM J. of Applied Mathematic

Some results on the dynamics and transition probabilities for non self-adjoint hamiltonians

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.Comment: in press in Annals of Physic

A quantum-like view to a generalized two players game

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.Comment: in press in International Journal of Theoretical Physic

(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.Comment: arXiv admin note: substantial text overlap with arXiv:1106.011