The rank $r_0$ of an hemitian matrix equals to the number of its non-zero eigenvalues. Accordingly, any Hamiltonian \( H_0+ V \) in an $N$ dimensional Hilbert space can be written as \( \sum_{n=1}^N \epsilon_n |n \rangle\langle n| + \sum_{r=1}^{r_0} \lambda_r |v^{(r)}\rangle\langle v^{(r)}| \) .

Consider a rank-1 perturbation. Show that the functional form of the eigenstates in the unperturbed basis can be found explicitly, involving a single normalization constant $C$. Write the equation for the eigen-energies.

Explain why a ring with a delta perturbation at some "0" site is modeled as a rank-1 perturbation.

Consider a more complicated perturbation of the type \( \lambda \left[ |u\rangle\langle v| + |v\rangle\langle u| \right] \) . The representations $u_n$ and $v_n$ are given. Show that the functional form of the eigenstates can be found explicitly in terms of two parameters $C_u$ and $C_v$. Write the reduced eigen-equation for the two parameters, and explain how the eigenvalues are obtained from it.

Explain why a site "0" that is couples to a set of decoupled levels "k" can be modeled as above, and why effctively it is like a rank-1 perturbation.