Physical phenomena of spectral relationships via quadratic third kind mixed integral equation with discontinuous kernel

Spectral relationships explain many physical phenomena, especially in quantum physics and astrophysics. Therefore, in this paper, we first attempt to derive spectral relationships in position and time for an integral operator with a singular kernel. Second, using these relations to solve a mixed integral equation (MIE) of the second kind in the space $L_2\left[-1,1\right]\times C\left[0,T\right],T<1.$ The way to do this is to derive a general principal theorem of the spectral relations from the term of the Volterra-Fredholm integral equation (V-FIE), with the help of the Chebyshev polynomials (CPs), and then use the results in the general MIE to discuss its solution. More than that, some special and important cases will be devised that help explain many phenomena in the basic sciences in general. Here, the FI term is considered in position, in $L_2\left[-1,1\right],$ and its kernel takes a logarithmic form multiplied by a general continuous function. While the VI term in time, in $C\left[0,T\right],T<1,$ and its kernels are smooth functions. Many numerical results are considered, and the estimated error is also established using Maple 2022.