Hi,

I got a question which I'm quite stuck on...

Given the eigenvalue problem,

Show that if

is an eigenvalue, then

must have a specific value and find this value. If

, show that there is exactly one negative eigenvalue.

I believe I've done the first part, as my working is as follows:

Rearrange the HLDE into self adjoint form:

, and taking that

, means the HLDE becomes

. The general solution to this should be of the form

. Using the boundary conditions,

Therefore,

Does this look correct?

Yes.
The next part of the question I get stuck on. The general solution for

is

.

Using the boundary conditions