I would like to numerically compute the spectral abscissa of an unbounded linear operator $A$ on a Hilbert space. To give you an idea my operator has the form: $$Af(x,y) = a(y) \partial_x f(x,y) - b(x)\partial_y f(x,y) + c(x,y)\int d(x,y)f(x,y)dy,$$ where $f \in L^2(R^2)$ and $a,b,c,d$ are functions that depend on some parameters. The operator is derived by linearization around a fixed-point and I would like to understand the stability of the fixed point as function of the parameters and also compute the asymptotic growth rate, which is given by the spectral abscissa.
I tried to discretize the operator $A$ and to diagonalize the resulting matrix. However, then I get eigenvalues all over the place, with no apparent convergence for increasing matrix size. I found that many of the eigenvectors of the finite dimensional matrix are not `physical', in the sense that they oscillate fast and thus do not represent a function in the Hilbert space.
How can I find out which of those eigenvectors are actually `physical'? Are there any algorithms that allow to compute the spectral abscissa?
