Spectral solutions for the Schrödinger equation with a regular singularity

We propose a modification in the Bethe-like ansatz to reproduce the hydrogen atom spectrum and the wave functions. Such a proposal provided a clue to attempt the exact quantization condition (EQC) for the quantum periods associated with potentials V(x) V(x) which are of the form (V(x) = |x| + a/|x| + b/|x|^2 ) (V(x)=|x|+a/|x|+b/|x|2) . We validate the EQC proposal by showing that our computed Voros spectrum in the limit a,b → 0 a,b→0 is matching well with the true spectrum of the familiar |x| |x| potential. Thus we have given a route to obtain the spectral solution for the one dimensional Schrödinger equation involving potentials with regular singularity at the origin.