  In the present contribution, we apply the double exponential Sinc-collocation
method (DESCM) to the one-dimensional time independent Schr\"odinger equation
for a class of rational potentials of the form $V(x) =p(x)/q(x)$. This
algorithm is based on the discretization of the Hamiltonian of the
Schr\"odinger equation using Sinc expansions. This discretization results in a
generalized eigenvalue problem where the eigenvalues correspond to
approximations of the energy values of the corresponding Hamiltonian. A
systematic numerical study is conducted, beginning with test potentials with
known eigenvalues and moving to rational potentials of increasing degree.
