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

Vacuum energy of scalar fields on spherical shells with general matching conditions