Representation of solutions to the one-dimensional Schrödinger equation in terms of Neumann series of Bessel functions. (English) Zbl 1426.34025
Summary: A new representation of solutions to the equation \(- y^{\prime \prime} + q(x) y = \omega^2 y\) is obtained. For every \(x\) the solution is represented as a Neumann series of Bessel functions depending on the spectral parameter \(\omega \). Due to the fact that the representation is obtained using the corresponding transmutation operator, a partial sum of the series approximates the solution uniformly with respect to \(\omega\) which makes it especially convenient for the approximate solution of spectral problems. The numerical method based on the proposed approach allows one to compute large sets of eigendata with a nondeteriorating accuracy.
MSC:
| 34A25 | Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc. |
| 33C10 | Bessel and Airy functions, cylinder functions, \({}_0F_1\) |
| 34L40 | Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.) |
Keywords:
Sturm-Liouville problem; transmutation operator; one-dimensional Schrödinger equation; Neumann series of Bessel functions; Fourier-Legendre series; numerical solution of spectral problemsReferences:
| [1] | Abramovitz, M.; Stegun, I. A., Handbook of Mathematical Functions (1972), Dover: Dover New York · Zbl 0543.33001 |
| [2] | Baricz, A.; Jankov, D.; Pogány, T. K., Neumann series of Bessel functions, Integral Transform. Spec. Funct., 23, no. 7, 529-538 (2012) · Zbl 1259.40001 |
| [3] | Barnett, A. R., The calculation of spherical Bessel and Coulomb functions, Computational Atomic Physics, Electron and Positron Collisions with Atoms and Ions, 181-202 (1996), Springer-Verlag: Springer-Verlag Berlin |
| [4] | Begehr, H.; Gilbert, R., Transformations, Transmutations and Kernel Functions, 1-2 (1992), Longman Scientific & Technical: Longman Scientific & Technical Harlow · Zbl 0827.35001 |
| [5] | Boumenir, A., Sampling and eigenvalues of non-self-adjoint Sturm-Liouville problems, SIAM J. Sci. Comput., 23, no. 1, 219-229 (2001) · Zbl 1006.34079 |
| [6] | Boumenir, A., The approximation of the transmutation kernel, J. Math. Phys., 47, 013505 (2006) · Zbl 1111.34063 |
| [7] | Camporesi, R.; Scala, A. J.D., A generalization of a theorem of Mammana, Colloq. Math., 122, no. 2, 215-223 (2011) · Zbl 1230.34011 |
| [8] | Campos, H.; Kravchenko, V. V.; Torba, S. M., Transmutations, L-bases and complete families of solutions of the stationary Schrödinger equation in the plane, J. Math. Anal. Appl., 389, no. 2, 1222-1238 (2012) · Zbl 1246.34080 |
| [9] | Carroll, R. W., Transmutation theory and applications, Mathematics Studies, 117 (1985), North-Holland · Zbl 0581.35004 |
| [10] | Castillo, R.; Kravchenko, V. V.; Oviedo, H.; Rabinovich, V. S., Dispersion equation and eigenvalues for quantum wells using spectral parameter power series, J. Math. Phys., 52, 043522 (2011) · Zbl 1316.81023 |
| [11] | Chébli, H.; Fitouhi, A.; Hamza, M. M., Expansion in series of Bessel functions and transmutations for perturbed Bessel operators, J. Math. Anal. Appl., 181, no. 3, 789-802 (1994) · Zbl 0799.34079 |
| [12] | Colton, D. L., Solution of Boundary Value Problems by the Method of Integral Operators (1976), Pitman Publ.: Pitman Publ. London · Zbl 0332.35001 |
| [13] | Davis, P. J.; Rabinowitz, P., Methods of Numerical Integration (2007), Dover Publications: Dover Publications New York · Zbl 1139.65016 |
| [14] | DeVore, R. A.; Lorentz, G. G., Constructive Approximation (1993), Springer-Verlag: Springer-Verlag Berlin · Zbl 0797.41016 |
| [15] | Filippi, S., Angenäherte Tschebyscheff-Approximation einer Stammfunktion-eine Modifikation (German), Numer. Math., 6, 320-328 (1964) · Zbl 0122.12402 |
| [16] | Fitouhi, A.; Hamza, M. M., A uniform expansion for the eigenfunction of a singular second-order differential operator, SIAM J. Math. Anal., 21, no. 6, 1619-1632 (1990) · Zbl 0736.34055 |
| [17] | Gillman, E.; Fiebig, H. R., Accurate recursive generation of spherical Bessel and Neumann functions for a large range of indices, Comput. Phys., 2, 62-72 (1988) |
| [18] | Hryniv, R. O.; Mykytyuk, Y. V., Inverse spectral problems for Sturm-Liouville operators with singular potentials, Inverse Probl., 19, 665-684 (2003) · Zbl 1034.34011 |
| [19] | Hryniv, R. O.; Mykytyuk, Y. V., Transformation operators for Sturm-Liouville operators with singular potentials, Math. Phys. Anal. Geom., 7, no. 2, 119-149 (2004) · Zbl 1070.34041 |
| [20] | Jackson, D., The Theory of Approximation. Reprint of the 1930 Original (1994), American Mathematical Society: American Mathematical Society Providence, RI · JFM 56.0936.01 |
| [21] | Khmelnytskaya, K. V.; Kravchenko, V. V.; Rosu, H. C., Eigenvalue problems, spectral parameter power series, and modern applications, Math. Methods Appl. Sci., 38, 1945-1969 (2015) · Zbl 1347.34128 |
| [22] | Khmelnytskaya, K. V.; Kravchenko, V. V.; Torba, S. M.; Tremblay, S., Wave polynomials and Cauchy’s problem for the Klein-Gordon equation, J. Math. Anal. Appl., 399, 191-212 (2013) · Zbl 1256.35009 |
| [23] | Kravchenko, V. V., A representation for solutions of the Sturm-Liouville equation, Complex Var. Elliptic Equ., 53, 775-789 (2008) · Zbl 1183.30052 |
| [24] | Kravchenko, V. V.; Morelos, S.; Torba, S. M., Liouville transformation, analytic approximation of transmutation operators and solution of spectral problems, Appl. Math. Comput., 273, 321-336 (2016) · Zbl 1410.34071 |
| [25] | Kravchenko, V. V.; Morelos, S.; Tremblay, S., Complete systems of recursive integrals and Taylor series for solutions of Sturm-Liouville equations, Math. Methods Appl. Sci., 35, 704-715 (2012) · Zbl 1243.34011 |
| [26] | Kravchenko, V. V.; Porter, R. M., Spectral parameter power series for Sturm-Liouville problems, Math. Methods Appl. Sci., 33, 459-468 (2010) · Zbl 1202.34060 |
| [27] | Kravchenko, V. V.; Torba, S. M., Spectral problems in inhomogeneous media, spectral parameter power series and transmutation operators, Proceedings of the 2012 International Conference on Mathematical Methods in Electromagnetic Theory (MMET), 18-22 (2012), IEEE Conference Publications |
| [28] | Kravchenko, V. V.; Torba, S. M., Modified spectral parameter power series representations for solutions of Sturm-Liouville equations and their applications, Appl. Math. Comput., 238, 82-105 (2014) · Zbl 1334.34026 |
| [29] | Kravchenko, V. V.; Torba, S. M., Construction of transmutation operators and hyperbolic pseudoanalytic functions, Complex Anal. Oper. Theory, 9, 389-429 (2015) · Zbl 1308.30055 |
| [30] | Kravchenko, V. V.; Torba, S. M., Analytic approximation of transmutation operators and applications to highly accurate solution of spectral problems, J. Comput. Appl. Math., 275, 1-26 (2015) · Zbl 1337.65099 |
| [31] | Kravchenko, V. V.; Torba, S. M., Analytic approximation of transmutation operators and related systems of functions, Bol. Soc. Mat. Mex., 22, 379-429 (2016) · Zbl 1369.34105 |
| [32] | Ledoux, V.; Daele, M. V.; Berghe, G. V., MATSLISE: a MATLAB package for the numerical solution of Sturm-Liouville and Schrödinger equations, ACM Trans. Math. Softw., 31, 532-554 (2005) · Zbl 1136.65327 |
| [33] | Levitan, B. M., Inverse Sturm-Liouville Problems (1987), VSP: VSP Zeist · Zbl 0749.34001 |
| [34] | Marchenko, V. A., Some questions on one-dimensional linear second order differential operators, Trans. Mosc. Math. Soc., 1, 327-420 (1952) · Zbl 0048.32501 |
| [35] | Marchenko, V. A., Sturm-Liouville Operators and Applications: Revised Edition (2011), AMS Chelsea Publishing |
| [36] | Paine, J. W.; de Hoog, F. R.; Anderssen, R. S., On the correction of finite difference eigenvalue approximations for Sturm-Liouville problems, Computing, 26, 123-139 (1981) · Zbl 0436.65063 |
| [37] | Pogány, T. K.; Süli, E., Integral representation for Neumann series of Bessel functions, Proc. Am. Math. Soc., 137, no. 7, 2363-2368 (2009) · Zbl 1171.33003 |
| [38] | Prudnikov, A. P.; Brychkov, Y. A.; Marichev, O. I., Integrals and Series. Vol. 2. Special Functions., 750 (1986), Gordon & Breach Science Publishers: Gordon & Breach Science Publishers New York · Zbl 0733.00004 |
| [39] | Pryce, J. D., Numerical Solution of Sturm-Liouville Problems (1993), Clarendon Press: Clarendon Press Oxford · Zbl 0795.65053 |
| [40] | Pryce, J. D., A test package for Sturm-Liouville solvers, ACM Trans. Math. Softw., 25, 21-57 (1999) · Zbl 0962.65063 |
| [41] | Sitnik, S. M., Transmutations and applications: a survey, (Korobeinik, Y.; G. Kusraev, A., Advances in Modern Analysis and Mathematical Modeling (2008), Vladikavkaz Scientific Center of the Russian Academy of Sciences and Republic of North Ossetia-Alania: Vladikavkaz Scientific Center of the Russian Academy of Sciences and Republic of North Ossetia-Alania Vladikavkaz), 226-293 |
| [42] | Suetin, P. K., Classical Orthogonal Polynomials, 480 (2005), Fizmatlit: Fizmatlit Moscow |
| [43] | Timan, A. F., Theory of Approximation of Functions of A Real Variable (1963), Macmillan: Macmillan New York · Zbl 0117.29001 |
| [44] | Trefethen, L. N., Is Gauss quadrature better than Clenshaw-Curtis?, SIAM Rev., 50, no. 1, 67-87 (2008) · Zbl 1141.65018 |
| [45] | Trimeche, K., Transmutation Operators and Mean-Periodic Functions Associated with Differential Operators (1988), Harwood Academic Publishers: Harwood Academic Publishers London · Zbl 0875.43001 |
| [46] | Wang, H.; Xiang, S., On the convergence rates of Legendre approximation, Math. Comput., 81, 861-877 (2012) · Zbl 1242.41016 |
| [47] | Watson, G. N., A Treatise on the Theory of Bessel Functions (1996), Cambridge University Press: Cambridge University Press Cambridge, UK |
| [48] | Wilkins, J. E., Neumann series of Bessel functions, Trans. Am. Math. Soc., 64, 359-385 (1948) · Zbl 0038.22501 |
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.
