The form of discrete spectrum in the case of high singular potential

Vladimír Lelek; Jan Wiesner

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A new method for computation of eigenvalues of the radial Schrödinger operator $- d^{2} / d x^{2} + v (x), x \geq 0$ is presented. The potential $v (x)$ is assumed to behave as $x^{- 2 + ϵ}$ if $x \to 0_{+}$ and as $x^{- 2 - ϵ}$ if $x \to + \infty, ϵ \geq 0$ . The Schrödinger equation is transformed to a non-linear differential equation of the first order for a function $z (x, ℵ)$ . It is shown that the eigenvalues are the discontinuity points of the function $z (\infty, ℵ)$ . Moreover, it is shown how to obtain an arbitrarily accurate approximation of eigenvalues. The method seems to be much more economical in...