An efficient finite element method and error analysis for eigenvalue problem of Schrödinger equation with an inverse square potential on spherical domain

Abstract

We provide an effective finite element method to solve the Schrödinger eigenvalue problem with an inverse potential on a spherical domain. To overcome the difficulties caused by the singularities of coefficients, we introduce spherical coordinate transformation and transfer the singularities from the interior of the domain to its boundary. Then by using orthogonal properties of spherical harmonic functions and variable separation technique we transform the original problem into a series of one-dimensional eigenvalue problems. We further introduce some suitable Sobolev spaces and derive the weak form and an efficient discrete scheme. Combining with the spectral theory of Babuška and Osborn for self-adjoint positive definite eigenvalue problems, we obtain error estimates of approximation eigenvalues and eigenvectors. Finally, we provide some numerical examples to show the efficiency and accuracy of the algorithm.

An Efficient Finite Element Method and Error Analysis for Schrödinger Equation with Inverse Square Singular Potential

“…□ Theorem 3. Let u m and u mh be the solutions of problems (18) and (20), respectively. en, the following inequality holds:…”

“…Substituting expression (48) into (20) and taking v mh through all the basis functions in X h , we obtain the following linear system:…”

“…In recent years, more and more attention has been paid to the numerical methods of the schrödinger equations with similar singular potential [1,[16][17][18][19][20][21]. However, many numerical methods are based on low-order finite element methods.…”