An efficient finite element method and error analysis for eigenvalue problem of Schr\xf6dinger equation with an inverse square potential on spherical domain

Yubing Sui,Jun Zhang,Donghao Zhang,Junying Cao

doi:10.1186/s13662-020-03034-9

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.

Highlights

1 Introduction The Schrödinger eigenvalue problem with the inverse-square (IS) or centrifugal potential is widely used in nuclear physics, quantum physics, nonlinear optics, and so on [1–6]
The potential in many electronic equations produces singularity and can describe the attraction or repulsion between objects, which usually leads to strong singularities of the eigenfunctions, and this cannot be regarded as a perturbation term [7–11]
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 the boundary of the domain

Summary

Introduction

The Schrödinger eigenvalue problem with the inverse-square (IS) or centrifugal potential is widely used in nuclear physics, quantum physics, nonlinear optics, and so on [1–6]. The purpose of this work is developing an effective finite element method for the eigenvalue problem of the Schrödinger equation with an IS potential on spherical domain. 2, we obtain a dimensional reduction scheme for the Schrödinger eigenvalue problem with IS potential. For the case a > 0, inserting expression (10) into (5) and (7), we can derive the following one-dimensional eigenvalue problem:. The corresponding numerical scheme of (21) is: Find (λlh, umlh) ∈ R × Sh(l) such that al umlh, vmlh = λlhbl umlh, vmlh , ∀vmlh ∈ Vh. Use the technique of [34], we deduce the following results. Let umli (t) = uml (t), t ∈ Ii. from an error formula of linear interpolating remainder term we derive that umli (t) pli(t). If uml ∈ H01(I) satisfy the condition of Theorem 4, the following inequalities hold: uml – umlh al h, λlh – λl h2.

Implementation of the numerical scheme

Conclusions