Initial-value problems and signature change

We make a rigorous study of classical field equations on a two-dimensional signature-changing spacetime using the techniques of operator theory. Boundary conditions at the surface of signature change are determined by forming self-adjoint extensions of the Schrödinger Hamiltonian. We show that the initial-value problem for the Klein - Gordon equation on this spacetime is ill-posed, in the sense that its solutions are unstable. Furthermore, if the initial data are smooth and compactly supported away from the surface of signature change, the solution has a divergent -norm after finite time.