In this paper, we promote the refinement method for estimating asymptotic expression of the fundamental solutions of a fourth order linear differential equation with discontinuous weight function and transmission conditions. These refinement solutions utilize more accurate asymptotic formulas for the eigenvalues and eigenfunctions for the problem.

This article investigates the spectral theory of the problem of the Regge-type with transmission conditions and discontinuous coefficients. We formulate a new linear operator, by which we can deal with simplicity and boundedness of the eigenpairs of the problem. The aim of this work is to conduct that the problem has an infinite number of simple positive eigenvalues.