New method for computation of discrete spectrum of radical Schrödinger operator

Ivan Úlehla; Miloslav Havlíček

Aplikace matematiky (1980)

  • Volume: 25, Issue: 5, page 358-372
  • ISSN: 0862-7940

Abstract

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A new method for computation of eigenvalues of the radial Schrödinger operator - d 2 / d x 2 + v ( x ) , x 0 is presented. The potential v ( x ) is assumed to behave as x - 2 + ϵ if x 0 + and as x - 2 - ϵ if x + , ϵ 0 . The Schrödinger equation is transformed to a non-linear differential equation of the first order for a function z ( x , ) . It is shown that the eigenvalues are the discontinuity points of the function z ( , ) . Moreover, it is shown how to obtain an arbitrarily accurate approximation of eigenvalues. The method seems to be much more economical in comparison with other known methods used in numerical computations on computers.

How to cite

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Úlehla, Ivan, and Havlíček, Miloslav. "New method for computation of discrete spectrum of radical Schrödinger operator." Aplikace matematiky 25.5 (1980): 358-372. <http://eudml.org/doc/15159>.

@article{Úlehla1980,
abstract = {A new method for computation of eigenvalues of the radial Schrödinger operator $-d^2/dx^2+v(x), x\ge 0$ is presented. The potential $v(x)$ is assumed to behave as $x^\{-2+\epsilon \}$ if $x\rightarrow 0_+$ and as $x^\{-2-\epsilon \}$ if $x\rightarrow +\infty , \epsilon \ge 0$. The Schrödinger equation is transformed to a non-linear differential equation of the first order for a function $z(x,\aleph )$. It is shown that the eigenvalues are the discontinuity points of the function $z(\infty , \aleph )$. Moreover, it is shown how to obtain an arbitrarily accurate approximation of eigenvalues. The method seems to be much more economical in comparison with other known methods used in numerical computations on computers.},
author = {Úlehla, Ivan, Havlíček, Miloslav},
journal = {Aplikace matematiky},
keywords = {computation of discrete spectrum; quantum mechanical problem; computation of discrete spectrum; quantum mechanical problem},
language = {eng},
number = {5},
pages = {358-372},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {New method for computation of discrete spectrum of radical Schrödinger operator},
url = {http://eudml.org/doc/15159},
volume = {25},
year = {1980},
}

TY - JOUR
AU - Úlehla, Ivan
AU - Havlíček, Miloslav
TI - New method for computation of discrete spectrum of radical Schrödinger operator
JO - Aplikace matematiky
PY - 1980
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 25
IS - 5
SP - 358
EP - 372
AB - A new method for computation of eigenvalues of the radial Schrödinger operator $-d^2/dx^2+v(x), x\ge 0$ is presented. The potential $v(x)$ is assumed to behave as $x^{-2+\epsilon }$ if $x\rightarrow 0_+$ and as $x^{-2-\epsilon }$ if $x\rightarrow +\infty , \epsilon \ge 0$. The Schrödinger equation is transformed to a non-linear differential equation of the first order for a function $z(x,\aleph )$. It is shown that the eigenvalues are the discontinuity points of the function $z(\infty , \aleph )$. Moreover, it is shown how to obtain an arbitrarily accurate approximation of eigenvalues. The method seems to be much more economical in comparison with other known methods used in numerical computations on computers.
LA - eng
KW - computation of discrete spectrum; quantum mechanical problem; computation of discrete spectrum; quantum mechanical problem
UR - http://eudml.org/doc/15159
ER -

References

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  1. Z. S. Agranovich V. A. Marchenko, Inverse Scattering Theory Problem, N. York 1963. (1963) Zbl0117.06003
  2. F. G. Tricomi, Differential Equations, Blackie and Son, 1961. (1961) Zbl0101.05904MR0138812
  3. M. A. Naymark, Linejnye differencialnye operatory, Moskva, 1969. (1969) 
  4. I. Úlehla, The nucleon-antinucleon bound states, Preprint CEN-Saciay, DPhPE 76-23. 
  5. I. Zborovský, Study of nucleon-antinucleon bound states, Diploma-thesis, Faculty of Matematics and Physics, Prague, 1978 (in Czech). (1978) 
  6. J. Kurzweil, Ordinary Differential Equations, Praha 1978 (in Cezch). (1978) Zbl0401.34001MR0617010
  7. F. Calogero, Variable phase approach to potential scattering, N. York, Ac. Press 1967. (1967) Zbl0193.57501

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