We must use the Hankel function of the first type for large .

To solve the problem, we have to match the solutions at the boundary.
First match the wavefunction.

Then match the first derivative.

We can divide the two equations to eliminate the constants to get a condition on the energies.

This is often called matching the

Often, the
term will be sufficient to describe scattering.
For
, the boundary condition is

Dividing and substituting for , we get

This is the

The number of solutions depends on the depth and radius of the well. There can even be no solution.

Jim Branson 2013-04-22