On $p$-adic spectral zeta functions

The spectral zeta functions have been found many application in several branches of modern physics, including the quantum field theory, the string theory and the cosmology. In this paper, we shall consider the spectral zeta functions and their functional determinants in the $p$-adic field. Our approach for the constructions of $p$-adic spectral zeta functions is to apply the $p$-adic Mellin transforms with respect to locally analytic functions $f$, where $f$ interpolates the spectrum for the Hamiltonian of a quantum model.