Unbounded generators of dynamical semigroups

TL;DR

This paper explores the structure of unbounded generators of dynamical semigroups, providing a detailed description of their standard form, illustrating potential pitfalls, and constructing examples that deviate from this form.

Contribution

It offers a rigorous characterization of unbounded generator forms, clarifies misconceptions, and presents new examples of generators not conforming to the standard form.

Findings

Standard form of unbounded generators is precisely characterized.

Naive interpretation of the standard form can be misleading.

Examples of generators outside the standard form are constructed.

Abstract

Dynamical semigroups have become the key structure for describing open system dynamics in all of physics. Bounded generators are known to be of a standard form, due to Gorini, Kossakowski, Sudarshan and Lindblad. This form is often used also in the unbounded case, but rather little is known about the general form of unbounded generators. In this paper we first give a precise description of the standard form in the unbounded case, emphasizing intuition, and collecting and even proving the basic results around it. We also give a cautionary example showing that the standard form must not be read too naively. Further examples are given of semigroups, which appear to be probability preserving to first order, but are not for finite times. Based on these, we construct examples of generators which are not of standard form.