×

An improved discretization of Schrödinger-like radial equations. (English) Zbl 1397.81063

Summary: A new discretization of the radial equations that appear in the solution of separable second order partial differential equations with some rotational symmetry (as the Schrödinger equation in a central potential) is presented. It cures a pathology, related to the singular behavior of the radial function at the origin, that suffers in some cases the discretization of the second derivative with respect to the radial coordinate. This pathology causes an enormous slowing down of the convergence to the continuum limit when the two point boundary value problem posed by the radial equation is solved as a discrete matrix eigenvalue problem. The proposed discretization is a simple solution to that problem. Some illustrative examples are discussed.

MSC:

81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
81R05 Finite-dimensional groups and algebras motivated by physics and their representations
34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
65L12 Finite difference and finite volume methods for ordinary differential equations
39A12 Discrete version of topics in analysis

Software:

Matlab; ARPACK
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Landau, L.; Lifchitz, E., Mécanique Quantique, (1974), Moscou: MIR, Moscou
[2] Sommerfeld, A., Partial Differential Equations in Physics, (1949), New York: Academic, New York
[3] Rawitscher, G.; Koltracht, I., An efficient numerical spectral method for solving the schrödinger equation, Comput. Sci. Eng., 7, 58-66, (2005)
[4] Morrison, D.; Riley, J.; Zancanaro, J., Multiple shooting method for two-point boundary value problems, Commun. ACM, 12, 613-614, (1962) · Zbl 0106.31903 · doi:10.1145/355580.369128
[5] Killingbeck, J., Shooting methods for the Schrödinger equation, J. Phys. A: Math. Gen., 20, 1411-1418, (1987) · Zbl 0627.65096 · doi:10.1088/0305-4470/20/6/024
[6] Sakas, D.; Simos, T., Multiderivative methods of eighth algebraic order with minimal phase-lag for the numerical solution of the radial Schrödinger equation, J. Comput. Appl. Math., 175, 161-172, (2005) · Zbl 1063.65067 · doi:10.1016/j.cam.2004.06.013
[7] Ming, Q.; Yang, Y.; Fang, Y., An optimized Runge–Kutta method for the numerical solution of the radial Schrödinger equation, Math. Problems Eng., 2012, (2012) · Zbl 1264.65113 · doi:10.1155/2012/867948
[8] Fang, Y.; You, X.; Ming, Q., A new phase-fitted modified rungekutta pair for the numerical solution of the radial Schrödinger equation, Appl. Math. Comput., 224, 432-441, (2013) · Zbl 1337.65067 · doi:10.1016/j.amc.2013.08.081
[9] Fack, V.; Berghe, G. V., A program for the calculation of energy eigenvalues and eigenstates of a Schrödinger equation, Comput. Phys. Commun., 39, 187-196, (1986) · doi:10.1016/0010-4655(86)90130-X
[10] Simos, T.; Williams, P., On finite difference methods for the solution of the Schrödinger equation, Comput. Chem., 23, 513-554, (1999) · Zbl 0940.65082 · doi:10.1016/S0097-8485(99)00023-6
[11] Vigo-Aguiar, J.; Simos, T., Review of multistep methods for the numerical solution of the radial Schrödinger equation, Int. J. Quantum Chem., 103, 278-290, (2005) · doi:10.1002/qua.20495
[12] Trefethen, L. N., Spectral Methods in Matlab, (2000), Philadelphia, PA: SIAM, Philadelphia, PA
[13] Laliena, V.; Campo, J., Stability of skyrmion textures and the role of thermal fluctuations in cubic helimagnets: a new intermediate phase at low temperature, Phys. Rev. B, 96, (2017) · doi:10.1103/PhysRevB.96.134420
[14] Frank, W.; Land, D.; Spector, R., Singular potentials, Rev. Mod. Phys., 43, 36-98, (1971) · doi:10.1103/RevModPhys.43.36
[15] Toikka, L. A.; Hietarinta, J.; Suominen, K-A, Exact soliton-like solutions of the radial grosspitaevskii equation, J. Phys. A: Math. Theor., 45, 48203, (2012) · Zbl 1282.35359 · doi:10.1088/1751-8113/45/48/485203
[16] Yang, X.; Guo, S.; Chan, F.; Wong, K.; Ching, W., Analytic solution of a two dimensional hydrogen atom. I. Nonrelativistic theory, Phys. Rev. A, 43, 1186-1196, (1991) · doi:10.1103/PhysRevA.43.1186
[17] Kohn, W.; Luttinger, J., Theory of donor states in silicon, Phys. Rev., 98, 915-922, (1955) · Zbl 0064.23802 · doi:10.1103/PhysRev.98.915
[18] Lehoucq, R.; Sorensen, D.; Yang, C., ARPACK Users Guide: Solution of Large-Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods, (1998), Philadelphia, PA: SIAM, Philadelphia, PA · Zbl 0901.65021
[19] ’t Hooft, G., Magnetic monopoles in unified gauge theories, Nucl. Phys. B, 79, 276-284, (1974) · doi:10.1016/0550-3213(74)90486-6
[20] Polyakov, A., Particle spectrum in the quantum field theory, JETP Lett., 20, 194-196, (1974)
[21] Belavin, A.; Polyakov, A., Metastable states of two dimensional isotropic ferromagnets, JETP Lett., 22, 246-247, (1975)
[22] Ivanov, B.; Sheka, D.; Krivonos, V.; Mertens, F., Quantum effects for the 2d soliton in isotropic ferromagnets, Phys. Rev. B, 75, (2007) · doi:10.1103/PhysRevB.75.132401
[23] Skyrme, T. H R., A non-linear field theory, Proc. R. Soc. A, 260, 127-138, (1961) · Zbl 0102.22605 · doi:10.1098/rspa.1961.0018
[24] Bogdanov, A.; Hubert, A., Thermodynamically stable magnetic vortex states in magnetic crystals, J. Magn. Magn. Mater., 138, 255-269, (1994) · doi:10.1016/0304-8853(94)90046-9
[25] Camblong, H.; Epele, L.; Fanchiotti, H.; Canal, C. G., Renormalization of the inverse square potential, Phys. Rev. Lett., 85, 1590, (2000) · doi:10.1103/PhysRevLett.85.1590
[26] Lévy-Leblond, J-M, Electron capture by polar molecules, Phys. Rev., 153, 1-4, (1967) · doi:10.1103/PhysRev.153.1
[27] Desfrançois, C.; Abdoul-Carime, H.; Khelifa, N.; Schermann, P., From 1/r to 1/r2 potentials: electron exchange between rydberg atoms and polar molecules, Phys. Rev. Lett., 73, 2436-2439, (1994) · doi:10.1103/PhysRevLett.73.2436
[28] Denschlag, J.; Umshaus, G.; Schmiedmayer, J., Probing a singular potential with cold atoms: a neutral atom and a charged wire, Phys. Rev. Lett., 81, 737-741, (1998) · doi:10.1103/PhysRevLett.81.737
[29] Gross, E. P., Structure of a quantized vortex in boson systems, Il Nuovo Cimento, 20, 454-457, (1961) · Zbl 0100.42403 · doi:10.1007/BF02731494
[30] Pitaevskii, P., Vortex lines in an imperfect bose gas, Sov. Phys.—JETP, 13, 451-454, (1961)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.