On Inverse Spectral Theory for Self-Adjoint Extensions: Mixed Types of Spectra

LetHbe a symmetric operator in a separable Hilbert space H. Suppose thatHhas some gapJ. We shall investigate the question about what spectral properties the self-adjoint extensions ofHcan have inside the gapJand provide methods on how to construct self-adjoint extensions ofHwith prescribed spectral properties insideJ. Under some weak assumptions about the operatorHwhich are satisfied, e.g., provided the deficiency indices ofHare infinite and the operator (H−λ)−1is compact for one regular pointλofH, we shall show that for every (auxiliary) self-adjoint operatorM′ in the Hilbert space H and every open subsetJ0of the gapJofHthere exists a self-adjoint extensionHofHsuch that insideJthe self-adjoint extensionHofHhas the same absolutely continuous and the same point spectrum as the given operatorM′ and the singular continuous spectrum ofHinJequals the closure ofJ0inJ. Moreover we shall present a method of how to construct such a self-adjoint extensionH. Via our methods it is possible to construct new kinds of self-adjoint realizations of the Laplacian on a bounded domainΩin Rd,d>1, with spectral properties very different from the spectral properties of the self-adjoint realizations known before. Mathematics Subject classification (1991): 47A10; 47A60; 47B25; 47E05; 47F05.