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

Abstract

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.

Authors:
Publication Date:
OSTI Identifier:
22451166
Resource Type:
Journal Article
Resource Relation:
Journal Name: Annals of Physics; Journal Volume: 356; Other Information: Copyright (c) 2015 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; HAMILTONIANS; HILBERT SPACE; PROBABILITY; QUANTUM SYSTEMS

Citation Formats

Bagarello, F., E-mail: fabio.bagarello@unipa.it. Some results on the dynamics and transition probabilities for non self-adjoint hamiltonians. United States: N. p., 2015. Web. doi:10.1016/J.AOP.2015.02.034.
Bagarello, F., E-mail: fabio.bagarello@unipa.it. Some results on the dynamics and transition probabilities for non self-adjoint hamiltonians. United States. doi:10.1016/J.AOP.2015.02.034.
Bagarello, F., E-mail: fabio.bagarello@unipa.it. Fri . "Some results on the dynamics and transition probabilities for non self-adjoint hamiltonians". United States. doi:10.1016/J.AOP.2015.02.034.
@article{osti_22451166,
title = {Some results on the dynamics and transition probabilities for non self-adjoint hamiltonians},
author = {Bagarello, F., E-mail: fabio.bagarello@unipa.it},
abstractNote = {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.},
doi = {10.1016/J.AOP.2015.02.034},
journal = {Annals of Physics},
number = ,
volume = 356,
place = {United States},
year = {Fri May 15 00:00:00 EDT 2015},
month = {Fri May 15 00:00:00 EDT 2015}
}
  • 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 willmore » find new interesting features not previously found in finite dimensional Hilbert spaces, useful for a deeper comprehension of this kind of physical systems.« less
  • 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.
  • Bagarello, Inoue, and Trapani [J. Math. Phys. 55, 033501 (2014)] investigated some operators defined by the Riesz bases. These operators connect with quasi-Hermitian quantum mechanics, and its relatives. In this paper, we introduce a notion of generalized Riesz bases which is a generalization of Riesz bases and investigate some operators defined by the generalized Riesz bases by changing the frameworks of the operators defined in the work of Bagarello, Inoue, and Trapani.
  • Vibrational transitions in collisions of two polyatomic systems aredescribed in terms of a Hamiltonian bilinear in momentum and positionoperators, for several degrees of freedom. The relative motion is assumed to beclassical and leads to time-dependent coefficients in the Hamiltonian. Thepresent treatment employs a simple procedure that does not require theconstruction of the time-evolution operator and leads to recursion relationsfor transition amplitudes, suitable for numerical applications. As examples, weconsider the special cases of a single degree of freedom, and of the linearlydriven harmonic oscillator.