  We give an overview of operator-theoretic tools that have recently proved
useful in the analysis of boundary-value and transmission problems for
second-order partial differential equations, with a view to addressing, in
particular, the asymptotic behaviour of resolvents of physically motivated
parameter-dependent operator families. We demonstrate the links of this rich
area, on the one hand, to functional frameworks developed by S. N. Naboko and
his students, and on the other hand, to concrete applications of current
interest in the physics and engineering communities.
