On Inverse Spectral Theory for Self-Adjoint Extensions: Mixed Types of Spectra
✍ Scribed by Sergio Albeverio; Johannes Brasche; Hagen Neidhardt
- Publisher
- Elsevier Science
- Year
- 1998
- Tongue
- English
- Weight
- 458 KB
- Volume
- 154
- Category
- Article
- ISSN
- 0022-1236
No coin nor oath required. For personal study only.
✦ Synopsis
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 on how to construct self-adjoint extensions of H with prescribed spectral properties inside J. Under some weak assumptions about the operator H which are satisfied, e.g., provided the deficiency indices of H are infinite and the operator (H&*) &1 is compact for one regular point * of H, we shall show that for every (auxiliary) selfadjoint operator M$ in the Hilbert space H and every open subset J 0 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$ and the singular continuous spectrum of H in J equals the closure of J 0 in J. Moreover we shall present a method of 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 0 in R d , d>1, with spectral properties very different from the spectral properties of the self-adjoint realizations known before.