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Step 1

Let me formulate the problem to convey my notation. I have a matrix $A$ which is hermitian - and is diagonalisable by a transformation $$ U_A A\,\,U_A^{-1} = A_{diag}$$

Now the matrix is changed, using the small parameter $\lambda$. Therefore, $$ A \rightarrow A-\lambda B $$ where $B$ can be not Hermitian, in general.

I tried to calculate the first order changes in eigenvalues and eigenvectors the following way -

Step 2

The matrix $A-\lambda B $ has to be rotated by $U^\prime = e^{i\lambda \alpha}U_A$ to be diagonalized, $\alpha$ being a generator of rotation - and it makes sense that the "extra" rotation has to be proportional to $\lambda$.

$$ A^\prime_{diag} = A_{diag} + C_1\lambda + C_2\lambda^2 + \mathcal{O}(\lambda^3) $$

I wrote this down using the BCH formula and set the linear coefficient, $C_1= 0$ and got the constraint $U_AB\;U_A^{-1} = \left[i\alpha,A_{diag}\right]$.

The final form I obtained for $A^\prime_{diag}$ was $$ A^\prime_{diag} = A_{diag} -\frac{1}{2}\left[i\alpha,U_AB\;U_A^{-1}\right]\lambda^2 + \mathcal{O}(\lambda^3)$$


Can anyone tell me if -

  1. I am on the right track - and how to proceed from here to solve for $\alpha$ in terms of the known matrices.

  2. I need to approach this differently.

  3. it would help if $B$ were Hermitian?


This is exactly the problem of doing perturbation theory using the SW transformation. Can anybody cite a reference?

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The Schriffer Wolff transformation was developed to remove the effect of the effect of the perturbation term to the first order by performing a similarity transformation on the hamiltonian. $\tilde{H}=SHS^{-1}$. I am giving you a paper where it was first developed...

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