Efficient Spectral and Spectral Element Methods for Eigenvalue Problems of Schrödinger Equations with an Inverse Square Potential

DOI10.1137/16M1069596zbMath1355.65150arXiv1606.06388OpenAlexW2963080509MaRDI QIDQ2954484

Zhimin Zhang, Hui-yuan Li

Publication date: 13 January 2017

Published in: SIAM Journal on Scientific Computing (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/1606.06388

zbMATH Keywords

eigenvaluesSchrödinger operatorsingularitySchrödinger equationBessel functionseigenfunctionnumerical experimentinverse square potentialexponential orderspectral/spectral element method

Mathematics Subject Classification ID

Spectral, collocation and related methods for boundary value problems involving PDEs (65N35) Estimates of eigenvalues in context of PDEs (35P15) Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs (65N30) Schrödinger operator, Schrödinger equation (35J10) Numerical methods for eigenvalue problems for boundary value problems involving PDEs (65N25)

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Cites Work

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• A posteriori eigenvalue error estimation for a Schrödinger operator with inverse square potential
• A posteriori error estimation of hierarchical type for the Schrödinger operator with inverse square potential
• High order eigenvalues for the Helmholtz equation in complicated non-tensor domains through Richardson extrapolation of second order finite differences
• Isospectral plane domains and surfaces via Riemannian orbifolds
• Approximation properties of the \(h\)-\(p\) version of the finite element method
• Computing ultra-precise eigenvalues of the Laplacian within polygons
• The h, p and h-p versions of the finite element method in 1 dimension. I. The error analysis of the p-version
• The h, p and h-p versions of the finite element method in 1 dimension. II. The error analysis of the h- and h-p versions
• The h, p and h-p versions of the finite element method in 1 dimension. III. The adaptive h-p version
• On Schrödinger operators with multipolar inverse-square potentials
• Solutions to critical elliptic equations with multi-singular inverse square potentials
• Optimal spectral-Galerkin methods using generalized Jacobi polynomials
• Improving the rate of convergence of `high order finite elements' on polygons and domains with cusps
• Optimal error estimates in Jacobi-weighted Sobolev spaces for polynomial approximations on the triangle
• Basics and some applications of the mortar element method
• One cannot hear the shape of a drum
• Finite-Difference Approximations to Singular Sturm-Liouville Eigenvalue Problems
• Eigenmodes of Isospectral Drums
• Efficient Spectral-Galerkin Methods III: Polar and Cylindrical Geometries
• Some planar isospectral domains
• Approximation Theory and Harmonic Analysis on Spheres and Balls
• Spectral Approximation on the Unit Ball
• Elliptic Equations with Multi-Singular Inverse-Square Potentials and Critical Nonlinearity
• Spectral Methods
• Reviving the Method of Particular Solutions
• The Optimal Convergence of theh‐pVersion of the Finite Element Method with Quasi‐Uniform Meshes
• Orthogonal Polynomials of Several Variables
• Can One Hear the Shape of a Drum?
• Approximations and Bounds for Eigenvalues of Elliptic Operators
• Singular Potentials