The Hilbert-space of this system in the Schrödinger representation is again given by $L_2(-1,1)$. In this Hilbert space, we consider the two functions $g_0(x):=(1+i)\exp(i \pi x)$ and $g_1(x):=\exp(i 2 \pi x)$.
I have shown that $g_0$ and $g_1$ are orthogonal in $L_2([-1,1])$ and I have found that the orthonormal states are $\phi_0=\frac{g_0(x)}{2}$ and $\phi_1=\frac{g_1(x)}{\sqrt{2}}$. Now I have to show that the functions $\phi_0$ and $\phi_1$ are eigenvectors/eigenfunctions for $H=-\frac{\hbar}{2m}\partial^2_x-C$ for $C \in \mathbb{R},$ and I should determine the corresponding eigenvalues. Can anyone help me with an argument. What will the $H$ matrix be? Is C just a constant we can ignore?