MATSLISE: A solver for Schrödinger and Sturm-Liouville equations
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MATSLISE
MATSLISE is a MATLAB application for the numerical study of Schrödinger and Sturm-Liouville problems. MATSLISE allows the fast and accurate computation of the eigenvalues and the visualization of the corresponding eigenfunctions. This is realized by making use of the power of high order piecewise Constant Perturbation Methods (CP methods). For more information on the CP-methods we like to refer to the references mentioned below.
More concrete MATSLISE collects the implementations of the CPM{12,0}, CPM{14,12}, CPM{16,14} and CPM{18,16} algorithms (see [6]).
Also one Piecewise Line Perturbation Method (LP) was added: the LPM[4,2] method. Using the functions defined in the MATSLISE-package, the
user can calculate eigenvalues and eigenfunctions of a Schrödinger or Sturm-Liouville problem. The Matlab environment allows to use the
results in further calculations or in plots. However a Graphical User Interface
(GUI) was built on top of MATSLISE to make the package more user-friendly. This GUI hides all the technical issues from the user. The GUI uses the CPM{16,14}
method in combination with the CPM{18,16} method to obtain accurate eigenvalue/eigenfunctions estimations (see [7]): as shown in [6]
these CPM{16,14} and CPM{18,16} methods are faster than the lower order CPM{12,10}, CPM{14,12} or LPM[4,2] methods.
The four CP methods and the LP method can be called from the command line: see
command line version.
[a,b]
and
a
and b
. Regular boundary conditions are imposed in the endpoints a
and b
,
that is conditions of the formA0*y(a) + B0*y'(a) = 0,
A1*y(b) + B1*y'(b) = 0
,A0
and A1
are real and not both zero; and similarly for
B0
and B1
. a
and b
. In this case the problem is regularized by interval
truncation.( -p(x)y')'+ q(x)y = Ew(x)y,
with p
, q
and w
defined on the closed interval [a,b]
,
continuous and p,w
stricly positive, and
a
and b
. Regular boundary conditions are imposed in the endpoints a
and b
,
that is conditions of the formA0*y(a) + B0*p(a)*y'(a) = 0,
A1*y(b) + B1*p(b)*y'(b) = 0
,A0
and A1
are real and not both zero; and similarly for
B0
and B1
. a
and b
. In this case the problem is regularized by interval
truncation.
y'' = ( l*(l+1)/x^2 + V(x) - E)y, x > 0,
V(x) = S(x)/x + R(x)
,
S(x)
and R(x)
are well-behaved (continuous) functions such that
they have real (finite) limits at 0 and infinity.
References
[1] L. Gr. Ixaru, Numerical Methods for Differential Equations and Applications (Reidel,Dordrecht-Boston-Lancaster, 1984).
[2] L. Gr. Ixaru, H. De Meyer and G. Vanden Berghe, CP methods for the Schrödinger equation revisited, J. Comput. Appl. Math. 88 (1997) 289-314.
[3] L. Gr. Ixaru, CP methods for the Schrödinger equation, J. Comput. Appl. Math. 125 (2000) 347-357.
[4] L. Gr. Ixaru, H. De Meyer and G. Vanden Berghe, SLCPM12 - A program for solving regular Sturm-Liouville problems, Comp. Phys. Comm. 118 (1999) 259-277.
[5] L. Gr. Ixaru, H. De Meyer and G. Vanden Berghe, Highly accurate eigenvalues for the distorted Coulomb potential, Phys. Rev. E 61 (2000) 3151-3159.
[6] V. Ledoux, M. Van Daele and G. Vanden Berghe, CP methods of higher order for Sturm-Liouville and Schrodinger equations, Comp. Phys. Comm. 162 (2004) 151-165.
[7] V. Ledoux, M. Van Daele and G. Vanden Berghe, MATSLISE, A Matlab package for the numerical solution of Sturm-Liouville and Schrodinger equatons, ACM Trans. Math. Software. 31 (2005).
[8] V. Ledoux, M. Rizea, L. Gr. Ixaru, G. Vanden Berghe and M. Van Daele, Solution of the Schrodinger equation by a high order perturbation method based on a linear reference potential. Comp. Phys. Comm. 175 (2006) 424-439.
[9] V. Ledoux, L. Gr. Ixaru, M. Rizea, M. Van Daele and G. Vanden Berghe, Solution of the Schrödinger equation over an infinite integration interval by perturbation methods, revisited, Comp. Phys. Comm. 175 (2006) 612-619.
[10] J. D. Pryce, Numerical Solution of Sturm-Liouville Problems (Oxford Univ. Press, Oxford, 1993).
Requirements
MATSLISE needs the Matlab symbolic toolbox.
Disclaimer
MATSLISE is freely available for non-commercial use.
In no circumstances can the authors be held responsible for any deficiency, fault
or other mishappening with regard to the use or performance of MATSLISE.
Contact
{Veerle.Ledoux, Marnix.VanDaele, Guido.VandenBerghe}@UGent.be
Department of Applied Mathematics and Computer Science
Ghent University
Krijgslaan 281 - S9
B-9000 Gent, Belgium
Regular Schrödinger problem |
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