# 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

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topÚlehla, 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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- M. A. Naymark, Linejnye differencialnye operatory, Moskva, 1969. (1969)
- I. Úlehla, The nucleon-antinucleon bound states, Preprint CEN-Saciay, DPhPE 76-23.
- I. Zborovský, Study of nucleon-antinucleon bound states, Diploma-thesis, Faculty of Matematics and Physics, Prague, 1978 (in Czech). (1978)
- J. Kurzweil, Ordinary Differential Equations, Praha 1978 (in Cezch). (1978) Zbl0401.34001MR0617010
- F. Calogero, Variable phase approach to potential scattering, N. York, Ac. Press 1967. (1967) Zbl0193.57501

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