  In this work we analyze the spectral zeta function for massless scalar fields
propagating in a $D$-dimensional flat space under the influence of a shell
potential. The shell potential is defined in terms of the two-interval
self-adjoint extensions of the Hamiltonian describing the dynamics of the
scalar field. After performing the necessary analytic continuation, we utilize
the spectral zeta function of the system to compute the vacuum energy of the
field.
