Waves in the Witten Bubble of Nothing and the Hawking Wormhole
We investigate the propagation of the scalar waves in the Witten space-time called "bubble of nothing" and in its remarkable sub-manifold, the Lorentzian Hawking wormhole. Due to the global hyperbolicity, the global Cauchy problem is well-posed in the functional framework associated with the energy. We perform a complete spectral analysis that allows to get an explicit form of the solutions in terms of special functions. If the effective mass is non zero, the profile of the waves is asymptotically almost periodic in time. In contrast, the massless case is dispersive. We develop the scattering theory, classical as well as quantum. The quantized scattering operator leaves invariant the Fock vacuum: there is no creation of particles. The resonances can be defined in the massless case and they are purely imaginary.