Heisenberg dynamics for non self-adjoint Hamiltonians: symmetries and derivations

Abstract

In some recent literature the role of non self-adjoint Hamiltonians, $H\ne H^{†}$, is often considered in connection with gain-loss systems. The dynamics for these systems is, most of the times, given in terms of a Schrödinger equation. In this paper we rather focus on the Heisenberg-like picture of quantum mechanics, stressing the (few) similarities and the (many) differences with respected to the standard Heisenberg picture for systems driven by self-adjoint Hamiltonians. In particular, the role of the symmetries, *-derivations and integrals of motion is discussed.

arXiv commentIn press in Mathematical Physics, Analysis and Geometry

References in corpus (7)

• Real Spectra in Non-Hermitian Hamiltonians Having PT Symmetry
• Operator (quasi-)similarity, quasi-Hermitian operators and all that
• Position Dependent Mass Schroedinger Equation and Isospectral Potentials : Intertwining Operator approach
• Mathematical aspects of intertwining operators: the role of Riesz bases
• Some results on the dynamics and transition probabilities for non self-adjoint hamiltonians
• Symmetry and Intertwining Operators for the Nonlocal Gross-Pitaevskii Equation
• A chain of solvable non-Hermitian Hamiltonians constructed by a series of metric operators

Cited by in corpus (2)

• Heisenberg and Heisenberg-Like Representations via Hilbert Space Bundle Geometry in the Non-Hermitian Regime
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