A useful similarity transformation

Having discussed the eigenvalues and eigenfunctions of the Dirac-Coulomb equation in some detail, we are now going to scetch a possible method for find- 

We consider the radial Dirac operator (115) and assume that the Coulomb coupling constant is negative, — 1 7 0, which corresponds to an attractive Coulomb potential. 

Mulitplying the eigenvalue equation by the matrix ia2 lets the derivatives appear in the main diagonal  

Written in matrix-form, the new eigenvalue equation has a simple structure, with all r-dependent parts in the off-diagonal of the matrix  

We have obtained a new eigenvalue equation with a simpler matrix. The price we paid is that the new matrix depends on the old eigenvalue e, which has to be determined. The number e now appears as a parameter in the matrix that has to be diagonalized. We have to find those values of the parameter, for which the equation (123) admits square-integrable solutions. 

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