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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?

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1 Answer 1

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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

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