Eigenvalues near the origin

Anyone knows some good package to compute the eigenvalues near the origin?

I have a linear eigen-problem (with Laplacian and some other terms). After spatial discretization (using either Fourier or Hermite functions), it becomes a regular eigenvalue problem. But the hard part is that there are zero eigenvalues, and I want to compute a couple dozen eigenvalues near the origin.

Right now, I am using eigs.m with a small sigma. It doesn't always work: it can only recover the zero eigenvalues upto the accuracy of 0.01.

It seems that the choice of sigma is related to the spatial resolution, which is no surprise b/c the condition number increases quadratically (roughly) w.r.t. the number of Fourier modes. For a linear system, we can solve this issue by using the inverse Laplacian as the pre-conditioner. But with an eigenvalue problem, I don't know how to apply it. Also, I don't explicitly compute the matrix. Any advice will be appreciated. Thanks.