Hearing the shape of a quantum boundary condition

We study the isospectrality problem for a free quantum particle confined in ring with junction, analyzing all self-adjoint realizations of corresponding Hamiltonian terms boundary condition at junction. In particular, by characterizing energy spectrum spectral function, we classify two classes, identifying families isospectral Hamiltonians. These classes turn out to be discerned action parity (i.e. space reflection), which plays central role our discussion.