Abstract
In this paper, we continue to solve as accurately as possible singular eigenvalues problems attached to the Schrödinger equation. We use the conventional ChC and SiC as well as Chebfun. In order to quantify the accuracy of our outcomes, we use the drift with respect to some parameters, i.e., the order of approximation N, the length of integration interval X, or a small parameter ε, of a set of eigenvalues of interest. The deficiency of orthogonality of eigenvectors, which approximate eigenfunctions, is also an indication of the accuracy of the computations. The drift of eigenvalues provides an error estimation and, from that, one can achieve an error control. In both situations, conventional spectral collocation or Chebfun, the computing codes are simple and very efficient. An example for each such code is displayed so that it can be used. An extension to a 2D problem is also considered.
Authors
Calin-Ioan Gheorghiu
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy
Keywords
Schrödinger eigenproblem; singularities; Chebyshev collocation; Chebfun; error estimation; Weierstrass spectrum; eigenvalue level crossing
Cite this paper as:
C.I. Gheorghiu, Accurate spectral collocation computations of high order eigenvalues for singular Schrödinger equations-revisited, Symmetry, 13 (2021) 5, https://doi.org/10.3390/sym13050761
About this paper
Print ISSN
2227-7390
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1. Gheorghiu, C.-I., Accurate Spectral Collocation Computation of High Order Eigenvalues for Singular Schrödinger Equations. Computation 2021, 9, 2. [CrossRef]
2. Shizgal, B., Spectral Methods in Chemistry and Physics. Application to Kinetic Theory and Quantum Mechanics; Springer: Berlin/Heidelberg, Germany, 2015. [CrossRef]
3. Zettl, A., Sturm–Liouville Theory; A. M. S. Providence: Providence, RI, USA, 2005
4. Frank, W.M. Land, J.D., Spector, R.M. Singular Potentials. Rev. Mod. Phys. 1971, 43, 36–98. [CrossRef]
5. Berry, M.V., Lewis, S.V., On the Weierstrass-Mandelbrot Fractal Function. Proc. R. Soc. Lond. A 1980. [CrossRef]
6. Gheorghiu, C.-I., Spectral Methods for Non-Standard Eigenvalue Problems. Fluid and Structural Mechanics and Beyond; Springer: Berlin/Heidelberg, Germany, 2014.
7. Weideman, J.A.C., Reddy, S.C. A MATLAB Differentiation Matrix Suite. ACM Trans. Math. Softw. 2000, 26, 465–519. [CrossRef]
8. Schonfelder, J.L., Chebyshev Expansions for the Error and Related Functions. Math. Comput. 1978, 32, 1232–1240. [CrossRef]
9. Roy, A.K., The generalized pseudospectral approach to the bound states of the Hulthén and the Yukawa potentials. Pramana J. Phys. 2005, 65, 1–15. [CrossRef]
10. Driscoll, T.A., Bornemann, F., Trefethen, L.N. The CHEBOP System for Automatic Solution of Differential Equations. BIT 2008, 48, 701–723. [CrossRef]
11. Driscoll, T.A., Hale, N., Trefethen, L.N., Chebfun Guide; Pafnuty Publications: Oxford, UK, 2014
12. Trefethen, L.N., Birkisson, A., Driscoll, T.A. Exploring ODEs; SIAM: Philadelphia, PA, USA, 2018
13. Brown, B.M., McCormack, D.K.R., Evans, W.D., Plum, M., On the spectrum of second-order differential operators with complex coefficients. Proc. R. Soc. A 1999, 455, 1235–1257. [CrossRef]
14. Brown, B.M., Langer, M., Marletta, M., Tretter, C., Wagenhofer, M., Eigenvalue bounds for the singular Sturm–Liouville problem with a complex potential. J. Phys. A: Math. Gen. 2003, 36, 3773–3787. [CrossRef]
15. Magherini, C., A corrected spectral method for Sturm–Liouville problems with unbounded potential at one endpoint. J. Comput. Appl. Math. 2020, 364. [CrossRef]
16. Boyd, J.P., A Chebyshev polynomial method for computing analytic solutions to eigenvalue problems with application to the anharmonic oscillator. J. Math. Phys. 1978, 19, 1445–1456. [CrossRef]
17. Pryce, J.D., A Test Package for Sturm–Liouville Solvers. ACM T. Math. Softw. 1999, 25, 21–57. [CrossRef]
18. Hoepffner, J., Implementation of Boundary Conditions. Available online: http://www.lmm.jussieu.fr/hoepffner/boundarycondition. pdf (accessed on 25 August 2012).
19. Boyd, J.P., Traps and Snares in Eigenvalue Calculations with Application to Pseudospectral Computations of Ocean Tides in a Basin Bounded by Meridians. J. Comput. Phys. 1996, 126, 11–20. [CrossRef]
20. Everitt, W.N., Gunson, J., Zettl, A., Some comments on Sturm–Liouville eigenvalue problems with interior singularities. J. Appl.Math. Phys. ZAMP 1987, 38, 813–838. [CrossRef]
21. Volkmer, H.W., Eigenvalue problems for Bessel’s equation and zero-pairs of Bessel functions. Stud. Sci. Math. Hung. 1999, 35, 261–280.
22. Bender, C., Orszag, S., Advanced Mathematical Methods for Scientists and Engineers; McGraw-Hill:, New York, NY, USA, 1978
23. Trefethen, L.N., Analyticity at Eigenvalue Near-Crossings. Available online: https://www.chebfun.org/examples/linalg/ CrossingsAnalyticity.html (accessed on 20 January 2020).
24. Teytel, M., How Rare Are Multiple Eigenvalues? Comm. Pure Appl. Math. 1999, 52, 917–934. [CrossRef]
25. Birkhoff, G., Lynch, R.E., Numerical Solutions of Elliptic Problems; Society for Industrial and Applied Mathematics: University City, PA, USA, 2014; pp. 38–40. [CrossRef]
26. Ixaru, L., New numerical method for the eigenvalue problem of the 2D Schrödinger equation. Comput. Phys. Commun. 2010, 181, 1738–1742.[CrossRef]