  The non-localized cluster model provides a new perspective on nuclear cluster
effects and has been applied successfully to study cluster structures in
various bound states and quasi-bound states (i.e., long-lived resonant states).
In this work, we extend the application scope of the non-localized cluster
model further to resonant and scattering states. Following the $R$-matrix
theory, the configuration space is divided into the interior and exterior
regions by a large channel radius such that the nuclear forces and the
antisymmetrization effects become negligible between clusters in the exterior
region. In the interior region, the picture of non-localized clustering is
realized mathematically by adopting the Brink-Tohsaki-Horiuchi-Schuck-R\"opke
(Brink-THSR) wave functions as the bases to construct the interior wave
functions. The Bloch-Schr\"odinger equation is used to match the interior wave
functions continuously with the asymptotic boundary conditions of the resonant
and scattering states at the channel radius, which leads eventually to
solutions of the problem. As a first test of the formalism, the low-lying
resonant states of ${}^{8}$Be and the phase shifts of the $\alpha+\alpha$
elastic scattering are studied. The numerical results agree well with the
experimental data, which shows the validity of the theoretical framework.
