Substituting Eq. (146) in Eq. (145), we find an equation for the radial function:

Using the method of series expansion, we can solve Eq. (147) exactly. It should be noted that zero and infinity are a regular and an irregular singular point of Eq. (147), respectively. In the region 0 < r, we have a series solution, which has a finite value at r = 0 as where C and D are constants, c0 and d1 are equal to 1, and c1 and d0 are equal to 0, respectively. Noting that cn and dn are equal to 0 for negative n, the other cn can be determined by the recurrence relation

-2wRp + K2R2p]Cn-1 + 2(K2Rp - w)Cn-3 + K]cn-,} /[R2pn(n -1)] (155)

and the dn's obey a similar recurrence relation.

Using the matching conditions at the interface r = r0 and Rp, we can obtain the equation of the eigenenergies e(l) as follows:

0 0

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