  Domain specific localization of eigenstates has been a persistent observation
for systems with local symmetries. The underlying mechanism for this
localization behaviour has however remained elusive. We provide here an
analysis of locally reflection symmetric tight-binding Hamiltonian which
attempts at identifying the key features that lead to the localized
eigenstates. A weak coupling expansion of closed-form expressions for the
eigenvectors demonstrates that the degeneracy of on-site energies occuring at
the center of the locally symmetric domains represents the nucleus for
eigenstates spreading across the domain. Since the symmetry-related subdomains
constituting a locally symmetric domain are isospectral we encounter pairwise
degenerate eigenvalues that split linearly with an increasing coupling strength
of the subdomains. The coupling to the (non-symmetric) environment in an
extended setup then leads to the survival of a certain system specific fraction
of linearly splitting eigenvalues. The latter go hand in hand with the
eigenstate localization on the locally symmetric domain. We provide a brief
outlook addressing possible generalizations of local symmetry transformations
while maintaining isospectrality.
