Efficient Spectral-galerkin Approximation For The Eigenvalue Problem Of Schr(?)dinger Equation In Circular/sphere Domain

In this paper,we propose an efficient spectral-Galerkin approximation for eigenvalue problems of Schr(?)dinger equations in the circular and spherical domains.For the circular domain,an efficient Legendre-Fourier spectral method for the eigenvalue problem of Schr(?)dinger equation is proposed.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 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 high accuracy.For the eigenvalue problem of the second-order Schr(?)dinger equation with variable coefficients on the spherical domain,an effective spectral approximation method is proposed.This method mainly uses spherical coordinate transformation and spherical harmonic function expansion to transform the eigenvalue problem of the second-order Schr(?)dinger equation with variable coefficients in Cartesian coordinates into an equivalent form in spherical coordinates,and then introduces an appropriate Sobolev space to establish its weak form and corresponding discrete format.Secondly,we describe the implementation process of the algorithm in detail,and give some numerical examples.The numerical results show that our algorithm is convergent and spectral accurate.