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computation of discrete spectrum; quantum mechanical problem
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.
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