Duality of bounded and scattering wave systems with local symmetries

We investigate the spectral properties of a class of hard-wall bounded systems, described by potentials exhibiting domainwise 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 wave function 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 through a numerical example with a potential consisting of two domains of local symmetry, each comprised of Dirac delta barriers.