M-Strum-Liouville problem with modified coulomb potential

In this article, we suggest important spectral data with boundary conditions for under the \(\mathcal{M}\)-derivative the Sturm-Liouville problem having modified Coulomb potential. We present the representation of the solution for the \(\mathcal{M}\)-Sturm-Liouville problem with a modified Coulomb potential, depending on both the boundary and initial conditions. Laplace transform and convolution property of \(\mathcal{M}\)-derivative are available to obtain some important results. In addition, the uniqueness of the \(\mathcal{M}\)-Sturm-Liouville problem solution with modified Coulomb potential is shown. We obtain a more generalized version of the Coulomb potential solution in classical analysis. Therefore, the purpose of this article is to observe the problem by supporting the spectral structure of the modified Coulomb potential \(\mathcal{M}\)-Sturm-Liouville problem with graphs.