On the Absolutely Continuous Spectrum of Self-Adjoint Extensions

Let Ω be any bounded domain in Rd, d > 1, and J a gap of the minimal Laplacian on Ω. We show that within J each kind of absolutely ontinuous spectrum can be generated by a self-adjoint realization of the Laplacian on Ω and in addition give results on mixed types of spectra, i.e., absolutely continuous, singular continuous and point spectrum. Thus for bounded domains Ω in Rd with smooth boundary we give self-adjoint realizations of the Laplacian on Ω with spectral properties very different from the properties of the self-adjoint realizations studied before. Both in order to have very simple and clear concepts and in order to enlarge the possible range of applications we shall work within the much more general framework of self-adjoint extensions of so called "significantly deficient" operators.