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I have an equation of the form

[tex](-i \lambda \frac{d}{dr}\sigma_z+\Delta(r)\sigma_x) g =(\epsilon + \frac{\mu \hbar^2}{2mr^2}) g[/tex]

where [itex]\sigma[/itex] refers to the Pauli matrices, g is a two component complex vector and the term on the right hand side of the equation is small compared to the other terms. The authors of the paper where i found this equation (http://www.sciencedirect.com/science/article/pii/0031916364903750) say that this can be handled using perturbation theory.

Ignoring the right hand side gives the possible solutions of g as

[tex]g=const (1,\pm i)^T e^{\pm K}[/tex]

where

[tex]K=\frac{1}{\lambda}\int_0^r \Delta dr[/tex].

Since the known behaviour of [itex]\Delta[/itex] is that it goes to a constant for large r, only the negative sign gives a well behaved function for large r.

Now as I see it both these states are zero energy solutions to the Hamiltonian

[tex]H_0= -i \lambda \frac{d}{dr} \sigma_z +\Delta(r) \sigma_x[/tex]

and the perturbation

[tex]V=\epsilon +\frac{\mu \hbar^2}{2mr^2}[/tex]

being a scalar does not remove the degeneracy between the states. I'm clueless how to proceed here...

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# Degenerate perturbation theory degeneracy not lifted by perturbation

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