On the Kato-Rosenblum and Weyl-von Neumann theorems (Q610562)

The classical Weyl-von~Neumann theorem states that, for any selfadjoint operator \(A_0\) on a separable Hilbert space, there exists a (non-unique) Hilbert-Schmidt operator \(C=C^*\) such that the perturbed operator \(A_0+C\) has pure point spectrum. Instead of the Hilbert-Schmidt class \(\mathfrak S_2\), it is possible to take the Schatten-von Neumann class \(\mathfrak S_p\), \(p>1\). The authors consider a similar problem for non-additive perturbations. Namely, they consider the set \(\text{Ext}_A\) of selfadjoint extensions of a given densely defined symmetric operator \(A\). Let \(A_0,\tilde{A}\) be extensions from \(\text{Ext}_A\). It is shown that the absolutely continuous parts of \(A_0\) and \(\tilde{A}\) are unitarily equivalent provided that the resolvent difference \(K=(\tilde{A}-i)^{-1}-(A_0-i)^{-1}\) is compact and the Weyl function \(M(\cdot )\) of the pair \({A,A_0}\) has weak boundary limits \(M(t)=w-\lim_{y\to +0}M(t+iy)\) for almost all \(t\in \mathbb R\). This result generalizes the classical Kato-Rosenblum theorem; in a well-known extension of the latter to non-additive perturbations [\textit{M. Sh. Birman} and \textit{M. G. Kreĭn}, Sov. Math., Dokl. 3, 740--744 (1962); translation from Dokl. Akad. Nauk SSSR 144, 475--478 (1962; Zbl 0196.45004)], the requirement regarding the Weyl function was absent, but it was assumed that \(K\) is of trace class. The above result demonstrates that for such pairs \({A,A_0}\), the Weyl-von Neumann theorem is in general not true in the class \(\text{Ext}_A\). The authors apply the general result to several specific classes of symmetric operators, in particular, to direct sums of sequences of symmetric operators, the Sturm-Liouville operators with operator potentials, and boundary value problems for the Schrödinger operator on a half-space. For a detailed exposition of these results, see also [\textit{M. M. Malamud} and \textit{H. Neidhardt}, J. Funct. Anal. 260, No. 3, 613--638 (2011; Zbl 1241.47011)].