Abstract
We provide in this study an effective finite element method of the Schrödinger equation with inverse square singular potential on circular domain. By introducing proper polar condition and weighted Sobolev space, we overcome the difficulty of singularity caused by polar coordinates’ transformation and singular potential, and the weak form and the corresponding discrete scheme based on the dimension reduction scheme are established. Then, using the approximation properties of the interpolation operator, we prove the error estimates of approximation solutions. Finally, we give a large number of numerical examples, and the numerical results show the effectiveness of the algorithm and the correctness of the theoretical results.
Highlights
Schrodinger equation with the inverse square or centrifugal potential plays an important role in quantum mechanics, quantum cosmology, nuclear physics, molecular physics, and so on [1–8]. e potential has the same differential order as the Laplacian operator near the origin, which usually leads to strong singularities and cannot be treated as a lower-order perturbation term [9–14]
By introducing proper polar condition and weighted Sobolev space, we overcome the difficulty of singularity caused by polar coordinates transformation and singular potential and establish the weak form and corresponding discrete scheme based on the dimension reduction format. en, using the approximation properties of interpolation operator, we prove the error estimates of approximation solutions
We present in this paper an efficient finite element method for the Schrodinger equation with the inverse square potential on the circular domain
Summary
Schrodinger equation with the inverse square or centrifugal potential plays an important role in quantum mechanics, quantum cosmology, nuclear physics, molecular physics, and so on [1–8]. e potential has the same differential order as the Laplacian operator near the origin, which usually leads to strong singularities and cannot be treated as a lower-order perturbation term [9–14]. Li et al [15] proposed an efficient finite element method to discuss the numerical solution of time-fractional Schrodinger equations. Us, we need to develop some new numerical methods to solve Schrodinger equation with inverse square singular potential. Many numerical methods are based on low-order finite element methods If we solve these problems directly in two-dimensional domain, it will cost a lot of computing time and memory capacity to obtain high-precision numerical solutions [22–24]. We usually need to solve the Schrodinger equation with inverse square singular potential on circular domain. As far as we know, there are few reports on an effective numerical method for the Schrodinger equation with inverse square potential in circular domain. Us, the purpose of this paper is to propose an effective finite element method of the Schrodinger equation with inverse square singular potential on circular domain.
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