  We investigate the spectral properties of a class of hard-wall bounded
systems, described by potentials exhibiting domain-wise different local
symmetries. Tuning the distance of the domains with locally symmetric potential
from the hard wall boundaries leads to extrema of the eigenenergies. The
underlying wavefunction becomes then an eigenstate of the local symmetry
transform in each of the domains of local symmetry. These extrema accumulate
towards eigenenergies which do not depend on the position of the potentials
inside the walls. They correspond to perfect transmission resonances of the
associated scattering setup, obtained by removing the hard walls. We argue that
this property characterizes the duality between scattering and bounded systems
in the presence of local symmetries. Our findings are illustrated at hand of a
numerical example with a potential consisting of two domains of local symmetry,
each one comprised of Dirac ? barriers.
