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Computing the Enclosures Eigenvalues Using the Quadratic Method
Department of Mathematics, Faculty of Science, King Abdul-Aziz University, Jeddah, KSA
- 1 Department of Mathematics, Faculty of Science, King Abdul-Aziz University, Jeddah, KSA
Applied Mathematics·Volume 10 (2019)·Pages 212–225·Published 10 April 2019·DOI10.4236/am.2019.104016
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Abstract
In this article, we compute the enclosures eigenvalues (upper and lower bounds) using the quadratic method. The Schrodinger operator (A) (harmonic and anharmonic oscillator model) has used as an example. We study a new technique to get more accurate bounds. We compare our results with Boulton and Strauss method.
KeywordsQuadratic MethodEnclosures EigenvaluesBoulton and Strauss Method
- Kato, T. (1949) On the Upper and Lower Bounds of Eigenvalues. Journal of the Physical Society of Japan, 4, 334-339. https://doi.org/10.1143/JPSJ.4.334
- Strauss, M. (2013) The Second Order Spectrum and Optimal Convergence. Mathematics of Computation, 82, 2305-2325. https://doi.org/10.1090/S0025-5718-2013-02693-2
- Boulton, L. and Strauss, M. (2011) On the Convergence of second order spectra and Multiplicity. Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences, 467, 264-284. https://doi.org/10.1098/rspa.2010.0233
- Boulton, L. and Strauss, M. (2007) Stability of Quadratic Projectione Methods. Operators and Matrices, 17, 217-233. https://doi.org/10.7153/oam-01-15
- Boulton, L. and Hobiny, A. (2013) On the Quality of Complementary Bounds for Eigenvalues. Calcolo, 52, 577-601. https://doi.org/10.1007/s10092-014-0131-y
- Davies, E.B. and Plum, M. (2004) Spectral Pollution. IMA Journal of Numerical Analysis, 24, 417-438. https://doi.org/10.1093/imanum/24.3.417
- Davies, E.B. (1996) Spectral Theory and Differential Operators. Cambridge University Press, Cambridge.
- Levitin, M. and Shargorodsky, E. (2004) Spectral Pollution and second order relative Spectra for Self-Adjoint Operators. IMA Journal of Numerical Analysis, 24, 393-416. https://doi.org/10.1093/imanum/24.3.393
- Boulton, L. and Levitin, M. (2007) On Approximation of the Eigenvalue of Perturbed Periodic Schrodinger Operators. Journal of Physics A Mathematical and Theoretical, 40, 9319. https://doi.org/10.1088/1751-8113/40/31/010
- Boulton, L. and Hobiny, A. (2016) On the Convergence of the Quadratic Method. IMA Journal of Numerical Analysis, 36, 1310-1333. https://doi.org/10.1093/imanum/drv036
- Boulton, L., Garcia del Moral, M.P. and Restuccia, A. (2012) Spectral Properties in Super Symmetric Matrix Models. Nuclear Physics B, 856, 716-747. https://doi.org/10.1016/j.nuclphysb.2011.11.017