On the numerical solution of the eigenvalue problem in fractional quantum mechanics

• A novel method to solve the eigenvalue problem in fractional quantum mechanics is proposed.
• Homogeneous Dirichlet boundary conditions are introduced for the fractional Laplacian defined on appropriate Sobolev spaces.
• The method is control volume based, thus it applies to irregular domains.
• Eigenfunction approximations is in terms of Radial Basis Functions (RBF) interpolation.
• The method is applied to the potentials: infinite well, quadratic (harmonic oscillator), quartic (anharmonic oscillator).

In this work we propose a Control Volume Function Approximation (CVFA) method to solve equations involving the fractional Laplacian. The function approximation part is carried out with Radial Basis Function (RBF) interpolation. The physical application of interest is the eigenvalue problem for the time independent fractional Schrödinger equation. Fractional derivatives are considered in the Riesz potentials sense.