Extended Weak Coupling Limit for Friedrichs Hamiltonians

Details Extended Weak Coupling Limit for Friedrichs Hamiltonians Extended Weak Coupling Limit for Friedrichs Hamiltonians

2006-04-25

arxiv.org/abs/math-ph/0604058v2

Physics

Mathematical Physics

changes in notation and title, minor corrections

Scientific paper

10.1063/1.2405402

We study a class of self-adjoint operators defined on the direct sum of two Hilbert spaces: a finite dimensional one called sometimes a ``small subsystem'' and an infinite dimensional one -- a ``reservoir''. The operator, which we call a ``Friedrichs Hamiltonian'', has a small coupling constant in front of its off-diagonal term. It is well known that under some conditions in the weak coupling limit the appropriately rescaled evolution in the interaction picture converges to a contractive semigroup when restricted to the subsystem. We show that in this model, the properly renormalized and rescaled evolution converges on the whole space to a new unitary evolution, which is a dilation of the above mentioned semigroup. Similar results have been studied before \cite{AFL} in more complicated models and they are usually referred to as "stochastic Limit".

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