WIAS Preprint No. 1427, (2009)

On the unitary equivalence of absolutely continuous parts of self-adjoint extensions



Authors

  • Malamud, Mark M.
  • Neidhardt, Hagen

2010 Mathematics Subject Classification

  • 47A57 47B25, 47A55

Keywords

  • Symmetric operators, self-adjoint extensions, boundary triplets, Weyl functions, spectral multiplicity, unitary equivalence, direct sums of symmetric operators, Sturm-Liouville operators with operator potentials

Abstract

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.

Appeared in

  • J. Funct. Anal., 360 (2011) pp. 613--638.

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