Consider the Mobius strip as the unit square with two opposite sides identified (with opposite directions). Consider the eigenvalue equation $\Delta u = \lambda u$ with boundary condition $u=0$. Unlike for orientable manifolds, the least eigenfunction will not be all of one sign; there will be a nodal line. My question generally concerns the behavior of eigenfunctions and eigenvalues in the non-orientable case, but to ask some specific questions: (1) is the eigenspace of the first eigenvalue still one-dimensional? (2) does there have to be just ONE nodal line? (3) does any nodal line have to meet the boundary in two points?
Sorry, I couldn't get an umlaut to appear in Mobius; neither TeX nor HTML worked to accomplish that.