Hulthen potential

A standard set of solvable potentials with critical behavior can be found in many text books on quantum mechanics [49,50], like the usual square-well potentials and other piecewise constant potentials. Also there are many potentials that are solvable only at d - 1 or for three-dimensional, v waves like the Hulthen potential, the Eckart potential, and the Posch-Teller potential. These potentials belong to a class of potentials, called shape-invariant potentials, that are exactly solvable using supersymmetric quantum mechanics [51,52], There are also many approaches to make isospectral deformation of these potentials [51,53] therefore it is possible to construct nonsymmetrical potentials with the same critical behavior as that of the original symmetric problem. 

Relationships analogous to Equation (6.16) can also be derived for spherical screened Coulomb potentials like Yukawa and Hulthen potentials. 

Because the Hulthen potential is spherically symmetric, the Green function can be expanded in terms of radial eigenfunctions and the spherical... 

To solve the differential equation with the Hulthen potential, we introduce the new variable x(.r) via ... 

The critical p presented in Fig. 4 is determined using the Hulthen potential rather than the Debye-Htickel potential. The above discussion demonstrates that, in the small curvature limit, the difference between the Debye-Hiickel potential and the Hulthen potential vanishes. From the following considerations, it will... 

Hence, we find the same dependence on k as for the analytical solution exploiting the Hulthen potential [see (25)]. The ratio of (53) and (25) in the limit Ka gives 0.6. Thus, the two results deviate quantitatively by approximately 40%. 

In the weak adsorption limit, we provide an exact analytical solution for polyelectrolyte-sphere adsorption by replacing the Debye-Huckel potential by the Hulthen potential. Other geometries require different approaches. As a generic concept, we propose application of the WKB method of quanmm mechanics, which we adopted to electrostatic polyelectrolyte adsorption problems. We have demonstrate that this description provides valuable analytical solutions and resolves a long-standing puzzle about the scaling properties of critical polyelectrolyte adsorption in curved geometries. 

Again, the first two terms are satisfied by the function (23) and flie remaining part disappears at r = a for = 2. Hence, the critical value p = 2is identical for the Hulthen and the Debye-Hiickel potential in the limit ka I and r — a)/a 1. 

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