Consider the discrete second derivative with Dirichlet boundary conditions on $\mathbb C^n$.

Its eigendecomposition is fully known: https://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors_of_the_second_derivative#Pure_Dirichlet_boundary_conditions_2

It seems like the lowest eigenvalue $\lambda_N$ is the one with the fastest decaying eigenfunction. Is there a way to prove this without using that the eigenfunctions are explicitly known?-Thus, can one show this directly from the matrix?