Geodesic
New method for computation of discrete spectrum of radical Schrödinger operator
Applications of Mathematics, Tome 25 (1980) no. 5, pp. 358-372

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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.
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.
DOI : 10.21136/AM.1980.103870
Classification : 34B25, 34L99, 65L15, 81C05, 81Q10
Keywords: computation of discrete spectrum; quantum mechanical problem 
Úlehla, Ivan; Havlíček, Miloslav. New method for computation of discrete spectrum of radical Schrödinger operator. Applications of Mathematics, Tome 25 (1980) no. 5, pp. 358-372. doi: 10.21136/AM.1980.103870
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