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 sp ectral properties the self-adjoint extensions of H can have inside the gap J and provide methods on how to construct self-adjoint extensions of H with prescribed spectral properties inside J. Under some weak as sumptions about the operator H which are satisfied, e.g., provided the deficiency indices of H are infinite and the operator (H-lambda)(-1) is compact for one regular point lambda of H, we shall show that for e very (auxiliary) self-adjoint operator M' in the Hilbert space H and e very open subset J(0) of the gap J of H there exists a self-adjoint ex tension (H) over tilde of H such that inside J the self-adjoint extens ion (H) over tilde of H has the same absolutely continuous and the sam e point spectrum as the given operator M' and the singular continuous spectrum of (H) over tilde in J equals the closure of J(0) in J. Moreo ver we shall present a method of how to construct such a self-adjoint extension (H) over tilde. Via our methods it is possible to construct new kinds of self-adjoint realizations of the Laplacian on a bounded d omain Omega in R-d, 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. (C) 1998 Academic Press.