If we define the Laplacian as a square matrix with zeroes on the diagonal, and $-1$ on the diagonals exactly above and below the main diagonal, and $0$ everywhere else, how would one go about finding its eigenvalues?
$$\!\!\begin{pmatrix} \ddots \\ & \;\;\;0 & -1 \\ & -1 & \;\;\;0 & -1 \\ & & -1 & \;\;\;0 & -1 \\ & & & -1 & \;\;\;0 & -1 \\ & & & & -1 & \;\;\;0 \\ & & & & & & \ddots \end{pmatrix}$$
Something of this sort.
The reason I wish to find this is because I have a Hamiltonian on an discrete and infinite $L^2$ space, and it takes the form of this matrix. I’d like to diagonalize this and find the spectrum of it.
