Abstract
Using the Trotter-Kato theorem we prove the convergence of the unitary dynamics generated by an increasingly singular Hamiltonian in the case of a single field coupling. The limit dynamics is a quantum stochastic evolution of Hudson-Parthasarathy type, and we establish in the process a graph limit convergence of the pre-limit Hamiltonian operators to the Chebotarev-Gregoratti-von Waldenfels Hamiltonian generating the quantum Itō evolution.
Highlights
1 Introduction In the situation of regular perturbation theory, we typically have a Hamiltonian interaction of the form H = H + Hint with associated strongly continuous one-parameter unitary groups U (t) = e–itH and U(t) = e–itH, we transform to the Dirac interaction picture by means of the unitary family V (t) = U (–t)U(t)
We may have a pair of unitary groups U(·) and U (·) with Stone generators H and H respectively, but where the intersection of the domains of the generators are not dense. This is the situation of a singular perturbation
If we assume at the outset a fixed free dynamics U (·), with Stone generator H, and a strongly continuous unitary left U -cocycle V (·), U(t) = U (t)V (t) will form a strongly continuous one-parameter unitary group with Stone generator H
Summary
In the situation of regular perturbation theory, we typically have a Hamiltonian interaction of the form H = H + Hint with associated strongly continuous one-parameter unitary groups U (t) = e–itH (the free evolution) and U(t) = e–itH (the perturbed evolution), we transform to the Dirac interaction picture by means of the unitary family V (t) = U (–t)U(t). We may have a pair of unitary groups U(·) and U (·) with Stone generators H and H respectively, but where the intersection of the domains of the generators are not dense This is the situation of a singular perturbation. V (·) arises as the Dirac picture evolution for a singular perturbation of a unitary U(·) with some generator H with respect to the time-shift: it was a long standing problem to find an explicit form for H which was resolved by Gregoratti [ ], see [ ]. The strategy is to employ the Trotter-Kato theorem which guarantees strong uniform convergence of the unitaries once graph convergence of the Hamiltonians is established
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