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

A FORTRAN 77 program is presented for calculating with the given accuracy 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 calculates also potential matrix elements – integrals of the eigenfunctions multiplied by their first derivatives with respect to the parameter. Eigenvalues and matrix elements computed by the ODPEVP program 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 [O. Chuluunbaatar, A.A. Gusev, A.G. Abrashkevich, A. Amaya-Tapia, M.S. Kaschiev, S.Y. Larsen, S.I. Vinitsky, Comput. Phys. Commun. 177 (2007) 649–675; O. Chuluunbaatar, A.A. Gusev, S.I. Vinitsky, A.G. Abrashkevich, Comput. Phys. Commun. 179 (2008) 685–693]. 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, a 3D-model of a hydrogen atom in a homogeneous magnetic field and a hydrogen atom on a three-dimensional sphere.Program summaryProgram title: ODPEVPCatalogue identifier: AEDV_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEDV_v1_0.htmlProgram obtainable from: CPC Program Library, Queen's University, Belfast, N. IrelandLicensing provisions: Standard CPC license, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 3001No. of bytes in distributed program, including test data, etc.: 24 195Distribution format: tar.gzProgramming language: FORTRAN 77Computer: Intel Xeon EM64T, Alpha 21264A, AMD Athlon MP, Pentium IV Xeon, Opteron 248, Intel Pentium IVOperating system: OC Linux, Unix AIX 5.3, SunOS 5.8, Solaris, Windows XPRAM: depends on1.the number and order of finite elements;2.the number of points; and3.the number of eigenfunctions required. Test run requires 4 MBClassification: 2.1, 2.4External 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 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=EBFAF=EBF with respect to a pair of unknown (E,FE,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 three-dimensional sphere [20].Restrictions: The computer memory requirements depend on:1.the number and order of finite elements;2.the number of points; and3.the number of eigenfunctions required. Restrictions due to dimension sizes may be easily alleviated by altering PARAMETER statements (see sections below and listing for details). The user must also supply DOUBLE PRECISION functions POTCCL and POTCC1 for evaluating potential function U(ρ,z)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)f1(z) and f2(z)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][zmin,zmax]; and3.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, S.I. Vinitsky, Comput. Phys. Comm. 177 (2007) 649–675[2]O. Chuluunbaatar, A.A. Gusev, S.I. Vinitsky, A.G. Abrashkevich, Comput. Phys. Comm. 179 (2008) 685–693.[3]W.H. Press, S.A. Teukolsky, W.T. Vetterling, 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, 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, J.D. Meyer, Eur. Phys. J. B 42 (2004) 361–365.[7]P.M. Krassovitskiy, N.Zh. Takibaev, Bull. Russian Acad. Sci. Phys. 70 (2006) 815–818.[8]V.S. Melezhik, J.I. Kim, 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, X. Huang, Dynamical Theory of Crystal Lattices, The Clarendon Press, Oxford, England, 1954.[11]L.V. Kantorovich, 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, G.L. Webster, Phys. Rev. A 19 (1979) 1629–1640.[13]C.V. Clark, K.T. Lu, A.F. Starace, in: H.G. Beyer, H. Kleinpoppen (eds.), Progress in Atomic Spectroscopy, Part C, Plenum, New York, 1984, pp. 247–320.[14]O. Chuluunbaatar, A.A. Gusev, V.L. Derbov, M.S. Kaschiev, L.A. Melnikov, V.V. Serov, S.I. Vinitsky, J. Phys. A 40 (2007) 11485–11524.[15]A.G. Abrashkevich, D.G. Abrashkevich, M.S. Kaschiev, I.V. Puzynin, Comput. Phys. Comm. 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, 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, V.V. Serov, Comput. Phys. Comm. 178 (2008) 301–330.[20]A.G. Abrashkevich, M.S. Kaschiev, S.I. Vinitsky, J. Comp. Phys. 163 (2000) 328–348.