# Article

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Keywords:
computation of discrete spectrum; quantum mechanical problem
Summary:
A new method for computation of eigenvalues of the radial Schrödinger operator $-d^2/dx^2+v(x), x\geq 0$ is presented. The potential $v(x)$ is assumed to behave as $x^{-2+\epsilon}$ if $x\rightarrow 0_+$ and as $x^{-2-\epsilon}$ if $x\rightarrow +\infty, \epsilon \geq 0$. 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.
References:
[1] Z. S. Agranovich V. A. Marchenko: Inverse Scattering Theory Problem. N. York 1963.
[2] F. G. Tricomi: Differential Equations. Blackie and Son, 1961. MR 0138812 | Zbl 0101.05904
[3] M. A. Naymark: Linejnye differencialnye operatory. Moskva, 1969.
[4] 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).
[5] J. Kurzweil: Ordinary Differential Equations. Praha 1978 (in Cezch). MR 0617010 | Zbl 0401.34001
[6] F. Calogero: Variable phase approach to potential scattering. N. York, Ac. Press 1967. Zbl 0193.57501

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