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On the unitary equivalence of absolutely continuous parts of self-adjoint extensions : dedicated to the memory of M. S. Birman
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.
Sturm-Liouville boundary value problems with operator potentials and unitary equivalence
Consider the minimal Sturm-Liouville operator A = A_rm min generated by the differential expression A := -fracd^2dt^2 + T in the Hilbert space L^2(R_+,cH) where T = T^*ge 0 in cH. We investigate the absolutely continuous parts of different self-adjoint realizations of cA. In particular, we show that Dirichlet and Neumann realizations, A^D and A^N, are absolutely continuous and unitary equivalent to each other and to the absolutely continuous part of the Krein realization. Moreover, if infsigma_ess(T) = infgs(T) ge 0, then the part wt A^acE_wt A(gs(A^D)) of any self-adjoint realization wt A of cA is unitarily equivalent to A^D. In addition, we prove that the absolutely continuous part wt A^ac of any realization wt A is unitarily equivalent to A^D provided that the resolvent difference (wt A