Expansion of the symmetrized kinetic exponential

Derive an expansion of the symmetrized lattice kinetic exponential into its eigenfunctions for the scalar-field lattice path integral.

Background

The proposed method is intended to express the vacuum amplitude in a form analogous to a Waring decomposition, with an integration variable replacing the decomposition index and with corresponding eigenfunctions and eigenvalues. The derivation obtains the desired form of the path integral without explicitly constructing the eigenfunction expansion of the symmetrized kinetic factor.

An explicit expansion would provide a direct realization of the decomposition motivating the method and would clarify the interpretation of the functions appearing in the final formula. The paper states that this expansion was not obtained, so it remains an unresolved technical problem within the presented approach.