I need to find the smallest eigenvalue and the corresponding eigenvector of a sparse matrix $M$ whose dimension is $\approx 10^4$.
Within Matlab enviroment, I use the command
[Evec, Eval]= eigs(M,1,'sa');
The eigenvalue I obtain, $\lambda_0$, is very reasonable, but I suspect that the corresponding eigenvector, $\vec{v}_0$, is wrong. I also suspect that this possibly wrong eigenvector comes from the fact that matrix $M$ is such that the lower part of its spectrum is quasi-degenerate, i.e. $$ \lambda_0 \approx \lambda_1\approx\lambda_2\approx \dots $$ Notice that the spectrum is quasi degenerate and not degenerate, meaning that, in any case, one has that $\lambda_0<\lambda_1<\lambda_2<\dots$.
1) Are my suspects reasonable?
2) Is this a well-known problem in Computational Science?
3) Is there a way to circumvent it, i.e. to obtain the exact eigenvector $\vec{v}_0$ associated to the smallest eigenvale $\lambda_0$ of a matrix $M$ whose lower part of the spectrum is quasi-degenerate?
Additional info: matrix $M$ represents a quantum Hamiltonian. It is symmetric but neither positive definite nor negative definite. Nevertheless, I am interested just in the lower part of its spectrum, i.e to the most negative eigenvalues.