Error Estimation for Eigenvalues of Unbounded Linear Operators and an Application to Energy Levels in Graphene Quantum Dots
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 approxim...
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Published in | Numerical functional analysis and optimization Vol. 38; no. 3; pp. 293 - 305 |
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Main Authors | , , |
Format | Journal Article |
Language | English |
Published |
Abingdon
Taylor & Francis
04.03.2017
Taylor & Francis Ltd |
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Abstract | 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. |
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AbstractList | 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. 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 electronâelectron interactions in the potential. |
Author | Berntsson, Fredrik Thim, Johan Orlof, Anna |
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Cites_doi | 10.1088/0953-8984/24/49/495302 10.1103/PhysRevB.84.085405 10.1103/PhysRevB.80.205402 10.1103/PhysRevB.85.165446 10.1103/PhysRevB.88.081406 10.1126/science.1102896 10.1103/PhysRevB.83.235104 10.1088/0953-8984/21/10/102201 10.1103/PhysRevB.50.7757 10.1103/PhysRevB.77.115423 10.1007/978-3-662-07233-2 10.1103/PhysRevB.92.075431 10.1103/PhysRevB.78.195427 |
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SubjectTerms | Eigenvalues Eigenvectors Energy levels Error analysis error estimation Estimates Graphene Interpolation linear operator Linear operators Mathematical analysis quantum dot Quantum dots spectrum |
Title | Error Estimation for Eigenvalues of Unbounded Linear Operators and an Application to Energy Levels in Graphene Quantum Dots |
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