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Manuscript Title: ODPEVP: A program for computing eigenvalues and eigenfunctions and their first derivatives with respect to the parameter of the parametric self-adjoined Sturm-Liouville problem
Authors: O. Chuluunbaatar, A.A. Gusev, S.I. Vinitsky, A.G. Abrashkevich
Program title: ODPEVP
Catalogue identifier: AEDV_v1_0
Distribution format: tar.gz
Journal reference: Comput. Phys. Commun. 180(2009)1358
Programming language: FORTRAN 77.
Computer: Intel Xeon EM64T, Alpha 21264A, AMD Athlon MP, Pentium IV Xeon, Opteron 248, Intel Pentium IV.
Operating system: OC Linux, Unix AIX 5.3, SunOS 5.8, Solaris, Windows XP.
RAM: depends on
  1. the number and order of finite elements;
  2. the number of points; and
  3. the number of eigenfunctions required.
Test run requires 4 MB
Keywords: eigenvalue problems, Kantorovich method, finite element method, ordinary differential equations, high-order accuracy approximations.
PACS: 02.60.Lj, 31.15.Ja, 32.80.Fb.
Classification: 2.1, 2.4.

External routines: GAULEG [3]

Nature of problem:
The three-dimensional boundary problem for the elliptic partial differential equation with an axial symmetry similar to the Schrödinger equation with the Coulomb and transverse oscillator potentials is reduced to the two-dimensional one. The latter finds wide applications in modeling of photoionization and recombination of oppositively charged particles (positrons, antiprotons) in the magnet-optical trap [4], optical absorption in quantum wells [5], and channeling of likely charged particles in thin doped films [6,7] or neutral atoms and molecules in artificial waveguides or surfaces [8,9]. In the adiabatic approach [10] known in mathematics physics as Kantorovich method [11] the solution of the two-dimensional elliptic partial differential equation is expanded over basis functions with respect to the fast variable (for example, angular variable) and depended on the slow variable (for example, radial coordinate ) as a parameter. An averaging of the problem by such a basis leads to a system of the second-order ordinary differential equations which contain potential matrix elements and the first-derivative coupling terms, (see, e.g., [12,13,14]). The purpose of this paper is to present the finite element method procedure based on the use of high-order accuracy approximations for calculating eigenvalues, eigenfunctions and their first derivatives with respect to the parameter of the parametric self-adjoined Sturm-Liouville problem with the parametric third type boundary conditions on the finite interval. The program developed calculates potential matrix elements - integrals of the eigenfunctions multiplied by their derivatives with respect to the parameter. These matrix elements can be used for solving the bound state and multi-channel scattering problems for a system of the coupled second-order ordinary differential equations with the help of the KANTBP programs [1,2].

Solution method:
The parametric self-adjoined Sturm-Liouville problem with the parametric third type boundary conditions is solved by the finite element method using high-order accuracy approximations [15]. The generalized algebraic eigenvalue problem AF = EBF with respect to a pair of unknown (E,F) arising after the replacement of the differential problem by the finite-element approximation is solved by the subspace iteration method using the SSPACE program [16]. First derivatives of the eigenfunctions with respect to the parameter which contained in potential matrix elements of the coupled system equations are obtained by solving the inhomogeneous algebraic equations.
As a test desk, the program is applied to the calculation of the potential matrix elements for an integrable 2D-model of three identical particles on a line with pair zero-range potentials described in [1,17,18], a 3D-model of a hydrogen atom in a homogeneous magnetic field described in [14,19] and a hydrogen atom on a threedimensional sphere [20].

Restrictions:
The computer memory requirements depend on:
  1. the number and order of finite elements;
  2. the number of points; and
  3. the number of eigenfunctions required.
Restrictions due to dimension sizes may be easily alleviated by altering PARAMETER statements (see Long Write Up and listing for details). The user must also supply DOUBLE PRECISION functions POTCCL and POTCC1 for evaluating potential function U(ρ, z) of Eq. (1) and its first derivative with respect to parameter ρ. The user should supply DOUBLE PRECISION functions F1FUNC and F2FUNC that evaluate functions f1(z) and f2(z)) of Eq. (1). The user must also supply subroutine BOUNCF for evaluating the parametric third type boundary conditions.

Running time:
The running time depends critically upon:
  1. the number and order of finite elements;
  2. the number of points on interval [zmin, zmax]; and
  3. the number of eigenfunctions required.
The test run which accompanies this paper took 2 s with calculation of matrix potentials on the Intel Pentium IV 2.4 GHz.

References:
[1] O. Chuluunbaatar, A.A. Gusev, A.G. Abrashkevich, A. Amaya-Tapia, M.S. Kaschiev, S.Y. Larsen and S.I. Vinitsky, Comput. Phys. Commun. 177 (2007) 649-675
[2] O. Chuluunbaatar, A.A. Gusev, S.I. Vinitsky and A.G. Abrashkevich, Comput. Phys. Commun. 179 (2008) 685-693.
[3] W.H. Press, S.A. Teukolsky, W.T. Vetterling and B.P. Flannery, Numerical Recipes: The Art of Scientific Computing, Cambridge University Press, Cambridge, 1986.
[4] O. Chuluunbaatar, A.A. Gusev, S.I. Vinitsky, V.L. Derbov, L.A. Melnikov and V.V. Serov, Phys. Rev. A 77 (2008) 034702-1-4.
[5] E.M. Kazaryan, A.A. Kostanyan, H.A. Sarkisyan, Physica E 28 (2005) 423-430.
[6] Yu.N. Demkov and J.D. Meyer, Eur. Phys. J. B 42 (2004) 361-365.
[7] P.M. Krassovitskiy and N.Zh. Takibaev, Bulletin of the Russian Academy of Sciences. Physics, 70 (2006) 815-818.
[8] V.S. Melezhik, J.I. Kim and P. Schmelcher, Phys. Rev. A 76 (2007) 053611-1-15.
[9] F.M. Pen kov, Phys. Rev. A 62 (2000) 044701-1-4.
[10] M. Born and X. Huang, Dynamical theory of crystal lattices, The Clarendon Press, Oxford, England, 1954.
[11] L.V. Kantorovich and V.I. Krylov, Approximate Methods of Higher Analysis, Wiley, New York, 1964.
[12] U. Fano, Colloq. Int. C.N.R.S. 273 (1977) 127; A.F. Starace and G.L. Webster, Phys. Rev. A 19 (1979) 1629-1640.
[13] C.V. Clark, K.T. Lu and A.F. Starace, Progress in Atomic Spectroscopy, Part C, eds. H.G. Beyer and H. Kleinpoppen (New-York: Plenum) (1984) 247-320.
[14] O. Chuluunbaatar, A.A. Gusev, V.L. Derbov, M.S. Kaschiev, L.A. Melnikov, V.V. Serov and S.I. Vinitsky, J. Phys. A 40 (2007) 11485-11524.
[15] A.G. Abrashkevich, D.G. Abrashkevich, M.S. Kaschiev and I.V. Puzynin, Comput. Phys. Commun. 85 (1995) 40-64.
[16] K.J. Bathe, Finite Element Procedures in Engineering Analysis, Englewood Cliffs, Prentice Hall, New York, 1982
[17] O. Chuluunbaatar, A.A. Gusev, M.S. Kaschiev, V.A. Kaschieva, A. Amaya-Tapia, S.Y. Larsen and S.I. Vinitsky, J. Phys. B 39 (2006) 243-269.
[18] Yu.A. Kuperin, P.B.Kurasov, Yu.B.Melnikov, S.P.Merkuriev, Ann.Phys. 205 (1991) 330-361.
[19] O. Chuluunbaatar, A.A. Gusev, V.P. Gerdt, V.A. Rostovtsev, S.I. Vinitsky, A.G. Abrashkevich, M.S. Kaschiev and V.V. Serov, Comput. Phys. Commun. 178 (2008) 301-330.
[20] A.G. Abrashkevich, M.S. Kaschiev and S.I. Vinitsky, J. Comp. Phys. 163 (2000) 328 348.