Given the tridiagonal symmetric infinite matrix of 0 and 1's
$ \left( \begin{matrix} 0&1&0&0&\ldots&0\\ 1&0&1&0&\ldots&0\\ 0&1&0&1&\ldots&0\\ \ldots&\ldots&\ldots&\ldots&\ldots&\ldots\\ 0&0&0&\ldots&0&1\\ 0&0&0&\ldots&1&0\\ \end{matrix}\right) $
How do you go about solving for the largest eigenvalues/eigenvectors? From a physical perspective, this is analogous to coupled harmonic oscillators along a linear chain, thus I expect the eigenvectors to look like the fundmental modes of a string with both ends fixed (i.e. in the continuum limit with scaled coordinates they look something like $\sin(nx)$).
I can solve for any finite matrix, but I'd like to understand how to solve this problem when the number of rows $N \rightarrow \infty$.