Accurate solutions for the spiked oscillators

Abstract

ABSTRACT:The one-dimensional harmonic oscillator potential plus a term of the form /x ␣ is known as the spiked oscillator potential; it constitutes a very interesting system because of its difficulties to accept perturbative and variational solutions for certain regimes of the ␣ parameter. By the use of a numerical method, we obtain accurate energy eigenvalues and eigenfunctions for a wide range of values and a few ␣ values. The accuracy of the present results is by much higher than previously published results.

Confined hydrogen atom: endohedrals H@C36 and H@C60

“…Afterward, the wave functions as well as the eigenvalues are calculated in a sequence by increasing the value of this parameter r 0 until the desired accuracy is achieved. Mathematically, it has been demonstrated that this procedure converges to the exact results of the original system where the spatial confinement is absent (see [41][42][43]). Hence, the artificial spatial confinement gives highly accurate results with respect to the exact values, which can not be guaranteed by other methods [41].…”

“…Hence, the artificial spatial confinement gives highly accurate results with respect to the exact values, which can not be guaranteed by other methods [41]. Some examples to illustrate this point are the free hydrogen atom, an electron in the Yukawa potential, the Hulthen potential, and the singular oscillator (see [41][42][43]).…”

“…It has been recognized as an advantage to study free systems, where bound states exist and the potential vanishes V → 0 at large distances, using the methods applied to the spatially confined systems (see [41][42][43]). In such a procedure the original system with potential V is additionally enclosed inside an impenetrable box of radius r 0 .…”