圆域上Schrodinger方程特征值问题有效的谱Galerkin逼近及误差估计

Efficient spectral-Galerkin approximation and error estimates for the eigenvalue problem of Schr?dinger equation in circular domain

提出了圆域上Schrodinger方程特征值问题的一种有效的Legendre-Fourier谱方法。首先利用极坐标变换将笛卡尔直角坐标系下的二阶Schr?dinger方程特征值问题转化为极坐标系下的一种等价形式。其次,我们推导了极条件,克服了极坐标变换引入的极点奇性。再结合特征函数的边界条件和在θ方向的周期性,我们定义了带权的Sobolev空间及其逼近空间,建立了极坐标系下二阶Schrodinger方程特征值问题的一种弱形式和相应的离散格式。基于紧算子的谱理论、非一致带权Sobolev空间中投影算子的逼近性质以及傅里叶基函数的逼近性质,对逼近解的误差估计给出了证明。最后,给出了一些数值实验,数值结果表明我们提出的算法是有效的和高精度的。

We proposed an efficient Legendre-Fourier spectral method for the eigenvalue problem of Schr?dinger equation in a circular domain.Firstly,the eigenvalue problem of the second order Schr?dinger equation in Cartesian coordinate system was transformed into an equivalent form in polar coordinate by using polar coordinate transformation.Then,we derived the polar condition,which eliminated the pole singularity introduced by polar coordinate transformation.Combined with the boundary conditions of the eigenfunction and the periodicity in θ direction,we defined a weighted Sobolev space and its approximation space,and established a weak form and corresponding discrete scheme for the eigenvalue problem of the second-order Schr?dinger equation.Based on the spectral theory of compact operators,the approximation properties of projection operators in nonuniform weighted Sobolev spaces,and the approximation properties of Fourier basis functions,we proved the error estimates of approximate solution.Finally,we present some numerical experiments,and the numerical results show that our algorithm is efficient and of high accuracy.