Motivated by recent results concerning the asymptotic behaviour of differential operators with highly contrasting coefficients, whose effective descriptions have involved generalised resolvents, we construct the functional model for a typical example of the latter. This provides a spectral representation for the generalised resolvent, which can be utilised for further analysis, in particular the construction of the scattering operator in related wave propagation setups. © 2024 Elsevier B.V., All rights reserved.