The confluent Heun equation with two singularities and the modified Schrödinger equation with two accessory parameters

In this article we establish a connection between the confluent Heun equation (with two singularities 0 and $\infty$) and the Schrödinger equation for the hydrogen atom with two accessory parameters $\mu$ and $\nu$, which “influence” the spectrum of eigenvalues and the Coulomb potential.