Convergent Iteration Method For The Anharmonic Oscillator Schrodinger Eigenvalue Problem

JOURNAL OF THE PHYSICAL SOCIETY OF JAPAN(2012)

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摘要
A new method is presented for solving the Schrodinger eigenvalue problem for anharmonic oscillators, using the Liouville transformation and a systematic iteration procedure. The Liouville transformation changes the potential V in the original eigenvalue equation to Q, which is highly singular. But, the singularity can be softened by rewriting the basic equation. We turn the equation into an integral equation and solve it by iteration. A simple criterion is established for the convergence of the iteration series. Our method is tested on the model of 2 mu x(2) + lambda x(4) potential (mu; lambda > 0) for the purpose of comparing it with the perturbation theory. Most remarkably, our method gives iteration series for the eigenfunctions u(n)(x), converging uniformly in x irrespective of the magnitude of lambda, implying the convergence for the eigenvalues also, while the perturbation theory is known to give divergent series no matter how small lambda is. Our method gives very good results for eigenvalues already at the first iteration, the better for the higher excited states.
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关键词
Schrodinger eigenvalue problem, approximation method, Liouville transformation, convergent iteration, (2 mu x(2) + lambda x(4))-potential
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