The Sinc Collocation Method for Computing Eigenvalues of the Schrödinger equation
The study of quantum anharmonic oscillators as potentials in the Schrödinger equation has been on
the edge of thrilling and exciting research during the past three decades. The one dimensional
anharmonic oscillator is of great interest to field theoreticians because it models complicated fields in
one- dimensional space-time. Numerous approaches which have been proposed to solve this problem
and while several of these methods yield excellent results for specific cases, it would be favorable to
have one general method that could handle efficiently and accurately any anharmonic potential.
The Sinc collocation methods (SCM) have been used extensively during the last three decades to solve many
problems in numerical analysis. Their applications include numerical integration, linear and nonlinear ordinary differential equations. The double exponential transformation yields optimal accuracy for a given number of function evaluations when using the trapezoidal rule in numerical integration. Recently, combination of the SCM with the double exponential (DE) transformation has sparked great interest.
In this talk, we present a method based on the double exponential Sinc collocation method (DESCM) for
numerically solving the Schrödinger equation with anharmonic oscillator. The eigenvalues are computed
to unprecedented accuracy and efficiency. The DESCM starts by approximating the wave function
as a series of weighted Sinc functions in the eigenvalue problem and evaluating the expression at several
collocation points spaced by a given mesh size h, we obtain a generalized eigen system which can be
transformed into a regular eigenvalue problem. The proposed method is successfully applied to Coulombic
anharmonic oscillator potentials that describe the interaction between charged particles and consistently
arises in physical applications. These applications include interactions in atomic, molecular and particle
physics, and between nuclei in plasma. We will also show how the DESCM can be applied to harmonic
oscillators perturbed by a rational function DESCM leading to an unprecedented accuracy in computing
the energy eigenvalues.