Extended Weak Coupling Limit for Friedrichs Hamiltonians

TL;DR

This paper investigates the weak coupling limit of Friedrichs Hamiltonians, demonstrating that the rescaled evolution converges to a unitary dilation of the subsystem's semigroup, extending previous stochastic limit results.

Contribution

It introduces a new approach showing convergence to a unitary evolution in the weak coupling limit for Friedrichs Hamiltonians, generalizing prior stochastic limit findings.

Findings

Rescaled evolution converges to a unitary dilation of the subsystem semigroup.

The model extends the stochastic limit results to a broader class of Hamiltonians.

Provides a rigorous framework for understanding weak coupling limits in quantum systems.

Abstract

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".