T. Dolinszky | Searcharxiv

arXiv2000

Infinitesimally weak coupling, infinitely strong singularity of the scattering potential

In scattering by singular potentials $g^2U(s;r)$, the coupling constant $g^2$ is continuously decreased to zero while the stage $s$ of singularity raised simultaneously beyond all limits by some functional relation $F(g^2;s)=0$. In the extreme situation of this double limit, even the mere existence of a nontrivial physical scattering problem is questionable. By iterating a pair of integral equations, the relevant solution is developed here in terms of wave functions into a pair of convergent series, each of which reduces in the double limit $\{g^2\to 0;s\to\infty\}$ to a single term calculable by quadrature.

math-ph↗

arXiv2000

Supersingular Scattering

In 'supersingular' scattering the potential $g^2U_A(r)$ involves a variable nonlinear parameter $A$ upon the increase of which the potential also increases beyond all limits everywhere off the origin and develops a uniquely high level of singularity in the origin. The problem of singular scattering is shown here to be solvable by iteration in terms of a smooth version of the semiclassical approach to quantum mechanics. Smoothness is achieved by working with a pair of centrifugal strengths within each channel. In both of the exponential and trigonometric regions, integral equations are set up the solutions of which when matched smoothly may recover the exact scattering wave function. The conditions for convergence of the iterations involved are derived for both fixed and increasing parameters. In getting regular scattering solutions, the proposed procedure is, in fact, supplementary to the Born series by widening its scope and extending applicability from nonsingular to singular potentials and from fixed to asymptotically increasing, linear and nonlinear, dynamical parameters.

math-ph↗