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Applying numerical continuation to the parameter dependence of solutions of the Schrödinger equation. (English) Zbl 1190.65190

Summary: In molecular reactions at the microscopic level, the appearance of resonances has an important influence on the reactivity. It is important to predict when a bound state transitions into a resonance and how these transitions depend on various system parameters such as internuclear distances. The dynamics of such systems are described by the time-independent Schrödinger equation and the resonances are modeled by poles of the S-matrix.

Using numerical continuation methods and bifurcation theory, techniques which find their roots in the study of dynamical systems, we are able to develop efficient and robust methods to study the transitions of bound states into resonances. By applying H. B. Keller’s pseudo-arclength continuation [Application of bifurcation theory, Proc. adv. Semin., Madison/Wis. 1976, 359–384 (1977; Zbl 0581.65043)], we can minimize the numerical complexity of our algorithm. As continuation methods generally assume smooth and well-behaving functions and the S-matrix is neither, special care has been taken to ensure accurate results.

We have successfully applied our approach in a number of model problems involving the radial Schrödinger equation.

65P30Bifurcation problems (numerical analysis)
65H20Global numerical methods for nonlinear algebraic equations, including homotopy approaches
65Y20Complexity and performance of numerical algorithms
37M20Computational methods for bifurcation problems
81U20S-matrix theory, etc. (quantum theory)
[1]Keller, H. B.: Numerical solution of bifurcation and nonlinear eigenvalue problems, Applications of bifurcation theory, 159-384 (1977) · Zbl 0581.65043
[2]Allgower, E. L.; Georg, K.: Continuation and path following, Acta numerica 1, 1-64 (1992) · Zbl 0792.65034
[3]Doedel, E. J.: Lecture notes on numerical analysis of nonlinear equations, , 51-75 (2007)
[4], Numerical continuation methods for dynamical systems (2007)
[5]Pruess, S.; Fulton, C.: Mathematical software for Sturm–Liouville problems, ACM transactions on mathematical software 19, No. 3, 360-376 (1993) · Zbl 0890.65087 · doi:10.1145/155743.155791 · doi:http://www.acm.org/pubs/contents/journals/toms/1993-19/
[6]Ledoux, V.; Vandaele, M.; Berghe, G.: Matslise: A Matlab package for the numerical solution of Sturm–Liouville and Schrödinger equations, ACM transactions on mathematical software 31, 532 (2005) · Zbl 1136.65327 · doi:10.1145/1114268.1114273
[7]Taylor, J. R.: Scattering theory: the quantum theory of nonrelativistic collisions, (2006)
[8]Newton, R. G.: Scattering theory of waves and particles, Texts and monographs in physics (1982)
[9]Burke, P. G.; Joachain, C. J.: Theory of electron-atom collisions, part 1: potential scattering, (1995)
[10]Keller, H. B.; Doedel, E. J.: Sourcebook of parallel computing, , 671-700 (2003)
[11]Deuflhard, P.: Newton methods for nonlinear problems: affine invariance and adaptive algorithms, (2004)
[12]Kelley, C.: Iterative methods for linear and nonlinear equations, (1995)
[13], Bifurcation theory and nonlinear eigenvalue problems (1969) · Zbl 0181.00105
[14]Allgower, E. L.; Georg, K.: Numerical continuation methods–an introduction, Springer series in computational mathematics 13 (1990) · Zbl 0717.65030
[15]Mei, Z.: Numerical bifurcation analysis for reaction–diffusion equations, (2000)
[16]Auto–Software for continuation and bifurcation problems in ordinary differential equations, version AUTO-07p available online, August 2007. URL: http://indy.cs.concordia.ca/auto
[17]Heroux, M. A.; Bartlett, R. A.; Howle, V. E.; Hoekstra, R. J.; Hu, J. J.; Kolda, T. G.; Lehoucq, R. B.; Long, K. R.; Pawlowski, R. P.; Phipps, E. T.; Salinger, A. G.; Thornquist, H. K.; Tuminaro, R. S.; Willenbring, J. M.; Williams, A.; Stanley, K. S.: An overview of the trilinos project, ACM transactions on mathematical software 31, No. 3, 397-423 (2005)
[18]Courant, R.; Hilbert, D.: Methods of mathematical physics, (1966)
[19]Arfken, G.; Weber, H.: Mathematical methods for physicists, (2005)
[20]Messiah, A. M. L.: Quantum mechanics, (1961)
[21]Burke, P.; Berrington, K.: Atomic and molecular processes: an R-matrix approach, (1993)
[22]Alhaidari, A.; Heller, E.; Yamani, H.; Abdelmonem, M.: The J-matrix method, (2008)
[23]Rescigno, T.; Mccurdy, C.: Numerical grid methods for quantum-mechanical scattering problems, Physical review A 62, No. 3, 32706 (2000)
[24]Johnson, B.: New numerical methods applied to solving the one-dimensional eigenvalue problem, The journal of chemical physics 67, 4086-4093 (1977)
[25]Sitenko, A. G.: Scattering theory, (1991)
[26]Amrein, W. O.; Jauch, J. M.; Sinha, K. B.: Scattering theory in quantum mechanics, (1977)
[27]Newton, R.: Connection between the s-matrix and the tensor force, Physical review 100, No. 1, 412-428 (1955) · Zbl 0066.22503
[28]Blatt, J. M.: Practical points concerning the solution of the Schrödinger equation, Journal of computational physics 1, 382-396 (1967) · Zbl 0182.49702 · doi:10.1016/0021-9991(67)90046-0
[29]Nussenzveig, H. M.: The poles of the s-matrix of a rectangular potential well or barrier, Nuclear physics 11, 499-521 (1959)