On semibounded restrictions of self-adjoint operators
Consider a symmetric operator \(H\) in a separable Hilbert space with domain \(D(H)\), which is not semibounded from below. Then it was pointed out by \textit{J. von Neumann} in his Satz 47 [Math. Ann. 102, 49-131 (1929; JFM 55.0824.02); ``Collected works'' (ed. by \textit{A. H. Taub}) Pergamon Press, Oxford, Vol. II, 3-85 (1961; Zbl 0100.00202)] and re-noted by the authors in their previous paper [Rev. Math. Phys. 10, No.6, 829-850 (1998; Zbl 0920.47003)] that if \(H\) isnot semibounded from above, for every \(b\in\mathbb{R}\) there is a dense subspace \({\mathcal D}_b\) of \(D(H)\) such that the restriction \(H|_{{\mathcal D}_b}\) is bounded from below by \(b\): \((Hf,f)\geq b(f,f)\), \(f\in{\mathcal D}_b\). Such a domain \({\mathcal D}_b\) is called a \(b\)-stability domain. In the paper mentioned above, they studied this issue for a specific Schrödinger operator in \(L^2(\mathbb{R})\) with a singular potential, which is not semibounded from below. In the paper under review, they in turn apply the theory to the momentum operator \(D= -i{d\over dx}\) in \(L^2(\mathbb{R})\), which is needless to say not semibounded from below, either, to describe corresponding stability domains. Further, the paper studies generalizations to the case of real-valued functions of the operator \(D\) and discusses spectral properties of selfadjoint extensions of the selfadjoint restriction to a maximal stability domain.
scientific article; zbMATH DE number 1204340