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A new method for computation of eigenvalues of the radial Schrödinger operator $-d^2/d{x}^2+v\left(x\right),x\ge 0$ is presented. The potential $v\left(x\right)$ is assumed to behave as $x^{-2+ϵ}$ if $x\to 0_{+}$ and as $x^{-2-ϵ}$ if $x\to +\infty ,ϵ\ge 0$. 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...