Let $H$ be a Hilbert space, which we interpret as a space of quantum states.


* If $U(t):H\to H$ is a unitary **norm-continuous** one-parameter group with $U(0)=I$, (essentially) *Cauchy's functional equation* says it has the form $$U(t)=e^{itA}$$ for some **bounded** self-adjoint Hamiltonian $A$.
* If $U(t):H\to H$ is a unitary **strongly-continuous** one-parameter group with $U(0)=I$, *Stone's theorem* says it has the form $$U(t)=e^{itA}$$ for some **unbounded** self-adjoint Hamiltonian $A$.


Now let $\mathcal{S}\_1$ be the set of trace-class operators of trace $1$ on $H$, which we interpret as states in an open quantum system.


* If $\mathcal{T}(t):\mathcal{S}\_1\to\mathcal{S}\_1$ is a trace-preserving completely positive **norm-continuous** one-parameter group with $\mathcal{T}(0)=I$, *Lindblad's theorem* says it has the form $$\mathcal{T}(t)=e^{t\mathcal{L}}$$ where $$\mathcal{L}(\rho)=-i[A,\rho]+\sum\_{k=0}^{\infty}L\_k\rho L\_k^\*+L\_k^\*L\_k\rho$$ is a Lindbladian with **bounded** $A$, $L\_k$ where $A$ is self-adjoint and $\sum\_{k=0}^{\infty}L\_k^\*L\_k$ is strongly summable.


My question is:


**What can we say in this case when $\mathcal{T}(t)$ is only strongly continuous?**



"Open systems" version of Stone's Theorem for one-parameter groups of quantum operations
This has been generalized by Brian Davies to the general case, the article is


Davies, E.B.: Quantum dynamical semigroups and the neutron diffusion equation. Rep. Math.
Phys. 11(2), 169–188 (1977)


A more modern introduction are chapters 5-7 in this book


<https://www.cambridge.org/core/books/quantum-stochastics/0E5897239F5430E41858ACB936DC4386>


