On inverse spectral theory for self{adjoint extensions: mixed types of spectra

Let H be a symmetric operator in a separable Hilbert space H. Suppose that H has some gap J . We shall investigate the question about what spectral properties the self{adjoint extensions of H can have inside the gap J and provide methods how to construct self{adjoint extensions of H with prescribed spectral properties inside J . Under some weak assumptions about the operator H which are satised, e. g., provided the de ciency indices of H are in nite and the operator (H ) 1 is compact for one regular point of H, we shall show that for every (auxiliary) self{adjoint operator M 0 in the Hilbert space H and every open subset J0 of the gap J of H there exists a self{adjoint extension ~ H of H such that inside J the self{adjoint extension ~ H of H has the same absolutely continuous and the same point spectrum as the given operator M 0 and the singular continuous spectrum of ~ H in J equals the closure of J0 in J . Moreover we shall present a method how to construct such a self{adjoint extension ~ H. Via our methods it is possible to construct new kinds of self{ adjoint realizations of the Laplacian on a bounded domain in Rd , d > 1, with spectral properties very di erent from the spectral properties of the self{adjoint realizations known before. Mathematics Subject Classi cation (1991). 47A10, 47A60, 47B25, 47E05, 47F05