On the unitary equivalence of absolutely continuous parts of self-adjoint extensions
- Malamud, Mark M.
- Neidhardt, Hagen
2010 Mathematics Subject Classification
- 47A57 47B25, 47A55
- Symmetric operators, self-adjoint extensions, boundary triplets, Weyl functions, spectral multiplicity, unitary equivalence, direct sums of symmetric operators, Sturm-Liouville operators with operator potentials
The classical Weyl-von Neumann theorem states that for any self-adjoint operator $A$ in a separable Hilbert space $gotH$ there exists a (non-unique) Hilbert-Schmidt operator $C = C^*$ such that the perturbed operator $A+C$ has purely point spectrum. We are interesting whether this result remains valid for non-additive perturbations by considering self-adjoint extensions of a given densely defined symmetric operator $A$ in $mathfrak H$ and fixing an extension $A_0 = A_0^*$. We show that for a wide class of symmetric operators the absolutely continuous parts of extensions $widetilde A = widetilde A^*$ and $A_0$ are unitarily equivalent provided that their resolvent difference is a compact operator. Namely, we show that this is true whenever the Weyl function $M(cdot)$ of a pair $A,A_0$ admits bounded limits $M(t) := wlim_yto+0M(t+iy)$ for a.e. $t in mathbbR$. This result is applied to direct sums of symmetric operators and Sturm-Liouville operators with operator potentials.
- J. Funct. Anal., 360 (2011) pp. 613--638.