I'm reading Landau's QM book and finding this paragraph confusing. The first order correction solution to wavefunction reads

$$ \psi^{(1)}=\sum_m \frac{V_{mn}}{E_n^{(0)}-E_m^{(0)}}\psi_m^{(0)} $$

which i have no problem to understand. However, Landau then said "the results can be generalized to the case where the operator $H_0$ (the unperturbed Hamiltonian) has a continuous spectrum, but the perturbation is applied to a state of the discrete spectrum as before. Then, for instance, we write

$$ \psi^{(1)}=\sum_m \frac{V_{mn}}{E_n^{(0)}-E_m^{(0)}}\psi_m^{(0)}+\int \frac{V_{\nu n}}{E_n^{(0)}-E_\nu}\psi_{\nu}^{(0)}d\nu $$

What does it mean to say the perturbation is applied to a state in the discrete spectrum while $H_0$ has a continuous spectrum?


In my copy, it says "... can be generalized at once to the case where the operator $H_0$ has also a continuous spectrum ... ." (My emphasis.)

So he is considering a system where the spectrum has both discrete and continuous parts (e.g. the Coulomb system), and he is applying perturbation theory to one of the discrete states.


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