  We consider a hybrid diffusion process that is a combination of two
Ornstein-Uhlenbeck processes with different restraining forces. This process
serves as the heavy-traffic approximation to the Markovian many-server queue
with abandonments in the critical Halfin-Whitt regime. We obtain an expression
for the Laplace transform of the time-dependent probability distribution, from
which the spectral gap is explicitly characterized. The spectral gap gives the
exponential rate of convergence to equilibrium. We further give various
asymptotic results for the spectral gap, in the limits of small and large
abandonment effects. It turns out that convergence to equilibrium becomes
extremely slow for overloaded systems with small abandonment effects.
