I would like to solve an eigenvalue problem of a Hamiltonian. I was able to find the lowest eigenvalue by converting the Hamiltonian into a matrix and applying linear algebra eigenvalue techniques. But this method is extremely cumbersome and does not generalize to arbitrary-sized Hamiltonians. I was hoping somebody could point to a more general approach. Here is the definition of the problem:
Let $\vert \psi_N \rangle$ denote the uniform superposition, $$\vert \psi_N \rangle = \frac{1}{\sqrt{N}}\sum^{N-1}_{i=0}\lvert i \rangle.$$ Then $\vert \psi_N \rangle$ is the ground state of the Hamiltonian $H_0 = I - \lvert \psi_N \rangle \langle \psi_N \lvert$ with the lowest eigenvalue $0$. Let $\vert m \rangle = \vert 1 0...0 \rangle$. Then it is the ground state of the Hamiltonian $H_m = I - \vert m \rangle \langle m \vert$.
For $ s \in [0,1]$ define the Hamiltonian $$H(s) = (1-s)H_0 + s H_m.$$
What would be the general approach to solving the following eigenvalue problem for an arbitrary $N$ \begin{align} H(s) \lvert E_k, s \rangle = E_k(s) \lvert E_k, s\rangle \end{align} where $E_k(s)$ is the $k$th eigenvalue at time $s$.
I was able to solve the problem for $N = 4$ by converting the Hamiltonian into a matrix and then using computer algebra I got $$E_0(s) = \displaystyle \frac{1}{2} - \frac{\sqrt{3 s^{2} - 3 s + 1}}{2}.$$ The problem with this approach is that it is not general and requires conversion to matrices and then solving the eigenvalue problem. I suspect that it is possible to get the answers in terms of $N$ and $s$ without fixing the size $N$ and expressing the Hamiltonian as a matrix.
The eigenvalues of $H(s)$ are
$$ 1,1,\dots,\frac{1}{2}\left(1\pm\sqrt{1-Cs+Cs^2} \right) $$
Where
$$ C=\frac{4(N-1)}{N} $$
Aside: Your Hamiltonian has a sort of self-duality: $H_0 \leftrightarrow H_m$, when expressed in terms of the Fourier transformed states
$$ \left|k \right>=\frac{1}{\sqrt{N}}\sum\limits_{n=0}^{N-1}e^{-ink}\left|n \right> $$
With which the transformed Hamiltonian $\tilde{H}$ is
$$ \tilde{H}_0=\mathbb{I}-\left|0 \right> \left< 0\right| \\ \tilde{H}_m=\mathbb{I}-\left|\phi_K \right> \left< \phi_K\right| $$
Here $\left|\phi_K \right>$ denotes the uniform superposition of $k$ states. Consequently
$$ H(s)=\tilde{H}(1-s) $$