  We consider a model of surface-mediated diffusion with alternating phases of
pure bulk and surface diffusion. For this process, we compute the mean exit
time from a disk through a hole on the circle. We develop a spectral approach
to this escape problem in which the mean exit time is explicitly expressed
through the eigenvalues of the related self-adjoint operator. This
representation is particularly well suited to investigate the asymptotic
behavior of the mean exit time in the limit of large desorption rate $\lambda$.
For a point-like target, we show that the mean exit time diverges as
$\sqrt{\lambda}$. For extended targets, we establish the asymptotic approach to
a finite limit. In both cases, the mean exit time is shown to asymptotically
increase as $\lambda$ tends to infinity. We also revise the optimality regime
of surface-mediated diffusion. Although the presentation is limited to the unit
disk, the spectral approach can be extended to other domains such as rectangles
or spheres.
