An Example of the Singular Coupling Limit

Abstract

We consider the stochastic differential equation

$$\frac{{dU_{s}^{t}}}{{dt}} = ({{L}_{0}}F(t) + {{L}_{1}}{{F}^{ + }}(t))U_{s}^{t}$$

where

$$F(t) = \int {g(t - s){{a}_{s}}ds}$$

is coloured noise. By letting g → γδwe obtain the quantum stochastic differential equation

$$\frac{{du_{s}^{t}}}{{dt}} = (\gamma {{L}_{0}}{{a}_{t}} + \bar{\gamma }{{L}_{1}}a_{t}^{ + } + \kappa {{L}_{0}}{{L}_{1}})u_{0}^{t},$$

where $\gamma = \smallint g(t)dt$ and $\kappa = \smallint _{0}^{\infty }dt\smallint _{{ - \infty }}^{{ + \infty }}dsg(t - s)\bar{g}( - s).$