Question: I have a system of equations, describing a spatially discretized PDE. This system has all negative eigenvalues exept one. This one eigenvalue is zero. I have deduced the eigenvector of this eigenvalue to consist of all ones, as the sum of every row in the matrix is zero. 

Now my question is regarding the physical meaning of this eigenvector. Is there a standard physical interpertation for an eigenvector from a zero eigenvalue? And if not is there a general meaning to an eigenvector containing only ones?

Answer: 
Let's say this is a linear PDE of the form $\dfrac{\partial u}{\partial t} = L u(X,t)$ where $X$ represents the spatial variables and $t$ the time, and your matrix $A$ is the discretized version of the operator $L$ (with appropriate linear homogeneous boundary conditions). The fact that $A {\bf 1} = 0$, where ${\bf 1}$ is a vector of all $1$'s, should correspond to the fact that in the PDE, $L 1 = 0$, i.e. $u(X,t) = constant$ is a solution of the PDE and its boundary conditions, and thus you can add a constant to any solution and still have a solution.
A couple of points:

1) are the matrices symmetric, and or orthogonal and unitary ?

2) are you studying a Quantum Mechanical system, or a classical one ?

The hydrogen atom has negative energy eigenvalues which converge at zero, which is the ionization state for hydrogen, or any other atom.

Is this consistent with your results:

The formula defining the energy levels of a Hydrogen atom are given by the equation: E = -E0/n2, where E0 = 13.6 eV (1 eV = 1.602×10-19 Joules) and n = 1,2,3… and so on. The energy is expressed as a negative number because it takes that much energy to unbind (ionize) the electron from the nucleus.

From **<http://astro.unl.edu/naap/hydrogen/transitions.html>**