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arXiv:0711.1244 · Journal of Physics A Mathematical and Theoretical

Quasistationary quaternionic Hamiltonians and complex stochastic maps

G Scolarici, L Solombrino

Published Jun 3, 2008• 1 citation
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Abstract

We show that the complex projections of time-dependent η-quasianti-Hermitian quaternionic Hamiltonian dynamics are complex stochastic dynamics in the space of complex quasi-Hermitian density matrices if and only if a quasistationarity condition is fulfilled, i.e. if and only if η is an Hermitian positive time-independent complex operator. An example is also discussed.

DOI
10.1088/1751-8113/41/24/244024

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We show that the complex projections of time-dependent η-quasianti-Hermitian quaternionic Hamiltonian dynamics are complex stochastic dynamics in the space of complex quasi-Hermitian density matrices if and only if a quasistationarity condition is fulfilled, i.e. if and only if η is an Hermitian positive time-independent complex operator. An example is also discussed.
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We propose a scheme to deal with certain time-dependent non-Hermitian Hamiltonian operators H(t)H(t)H(t) that generate a real phase in their time-evolution. This involves the use of invariant operators IPH(t)I_{PH}(t)IPH​(t) that are pseudo-Hermitian with respect to the time-dependent metric operator and which implies that the dynamics is governed by unitary time evolution. Furthermore, H(t)H(t)H(t) is not generally quasi-Hermitian and does not define an observable of the system but IPH(t)I_{PH}(t)IPH​(t) obeys a quasi-hermiticity transformation as in the completely time-independent Hamiltonian systems case. The harmonic oscillator with a time-dependent frequency under the action of a complex time-dependent linear potential is considered as an illustrative example.