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[tex]

\mathbb{R}\to\mathbb{C}^{N\times N},\quad t\mapsto A(t)

[/tex]

[tex]

A(t)^{\dagger} = A(t)

[/tex]

[tex]

A(t)v_n(t) = \lambda_n(t)v_n(t)

[/tex]

[tex]

v_n(t)^{\dagger}v_{n'}(t) = \delta_{nn'}

[/tex]

[tex]

\dot{A}(t)v_n(t) + A(t)\dot{v}_n(t) = \dot{\lambda}_n(t)v_n(t) + \lambda_n(t)\dot{v}_n(t)

[/tex]

Suppose that [itex]K>1[/itex] is the dimension of some eigenspace corresponding to some eigenvalue [itex]\lambda_n(0)[/itex]. We can assume the components arranged so that

[tex]

\lambda_n(0)=\lambda_{n+1}(0)=\cdots = \lambda_{n+K}(0)

[/tex]

I have not understood how to solve [itex]\dot{v}_n(0)[/itex].

Isn't the main aim to solve the coefficients [itex]v_{n'}(0)^{\dagger}\dot{v}_n(0)[/itex], like in the non-degenerate case too? If these are known for all [itex]n'[/itex], then the whole vector can be written as

[tex]

\dot{v}_n(0) = \sum_{n'=1}^N \big(v_{n'}(0)^{\dagger}\dot{v}_n(0)\big)v_{n'}(0)

[/tex]

I understand that for those [itex]n'[/itex] such that [itex]\lambda_{n'}(0)\neq\lambda_n(0)[/itex] we have

[tex]

v_{n'}(0)^{\dagger}\dot{v}_n(0) = \frac{v_{n'}(0)^{\dagger}\dot{A}(0)v_n(0)}{\lambda_n(0) - \lambda_{n'}(0)}

[/tex]

I also understand that we can assume

[tex]

v_n(0)^{\dagger}\dot{v}_n(0) = 0

[/tex]

The real part will be zero due to the assumption [itex]\|v_n(t)\|=1[/itex], and the imaginary part can be chosen zero by choosing right complex phase.

What I have not understood is how to obtain the coefficients for such [itex]n'[/itex] that [itex]n'\neq n[/itex] and [itex]\lambda_{n'}(0)=\lambda_n(0)[/itex].

I have Quantum Mechanics Second Edition by Bransden and Joachain. After discussing how to obtain [itex]v_{n'}(0)[/itex] (by some rotation in the eigenspace) they state

Well to me it seems that I cannot obtain all the coefficients in a way "similar" to the non-degenerate case. Once the correct zero-order wave functions [itex]\chi_{nr}^{(0)}[/itex] have been determined, the first-order correction [itex]\psi_{nr}^{(1)}[/itex] to the wave function and the second-order energy correction [itex]E_{nr}^{(2)}[/itex] can be obtained in a way similar to that followed in Section 8.1 for the non-degenerate case.

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# Degenerate perturbation theory, vector derivative

Can you offer guidance or do you also need help?

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