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

2009/07/03 by Mark M. Malamud, Hagen Neidhardt, Malamud, Mark M. +1
Mathematics · Physics and Astronomy · #47A55 #47A57 #47B25 #FOS: Mathematics #FOS: Physical sciences #Functional Analysis (math.FA) #Mathematical Physics (math-ph) #math-ph #math.FA #math.MP #msc:47A55 #msc:47A57 #msc:47B25

paper · pdf · doi:10.48550/arxiv.0907.0650

arxiv created 2009/07/03 · arxiv updated 2009/12/01

Abstract

The classical Weyl-von Neumann theorem states that for any self-adjoint operator A in a separable Hilbert space \mathfrak H 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 A₀ = A₀^*. We show that for a wide class of symmetric operators the absolutely continuous parts of extensions \widetilde A = \widetilde A^* and A₀ are unitarily equivalent provided that their resolvent difference is a compact operator. Namely, we show that this is true whenever the Weyl function M(⋅) of a pair \A,A₀\ admits bounded limits M(t) := \wlim_y→+0M(t+iy) for a.e. t ∈ ℝ. This result is applied to direct sums of symmetric operators and Sturm-Liouville operators with operator potentials.

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