Error Estimation for Eigenvalues of Unbounded Linear Operators and an Application to Energy Levels in Graphene Quantum Dots

• Published in: Numerical functional analysis and optimization Vol. 38; no. 3; pp. 293 - 305
• Main Authors: Berntsson, Fredrik, Orlof, Anna, Thim, Johan
• Format: Journal Article
• Language: English
• Published: Abingdon Taylor & Francis 04.03.2017; Taylor & Francis Ltd
• Subjects: Eigenvalues; Eigenvectors; Energy levels; Error analysis; error estimation; Estimates; Graphene; Interpolation; linear operator; Linear operators; Mathematical analysis; quantum dot; Quantum dots; spectrum
• ISSN: 0163-0563; 1532-2467; 1532-2467
• DOI: 10.1080/01630563.2017.1279176

The eigenvalue problem for linear differential operators is important since eigenvalues correspond to the possible energy levels of a physical system. It is also important to have good estimates of the error in the computed eigenvalues. In this work, we use spline interpolation to construct approximate eigenfunctions of a linear operator using the corresponding eigenvectors of a discretized approximation of the operator. We show that an error estimate for the approximate eigenvalues can be obtained by evaluating the residual for an approximate eigenpair. The interpolation scheme is selected in such a way that the residual can be evaluated analytically. To demonstrate that the method gives useful error bounds, we apply it to a problem originating from the study of graphene quantum dots where the goal was to investigate the change in the spectrum from incorporating electron-electron interactions in the potential.