The von Neumann problem for nonnegative symmetric operators
\textit{J. von Neumann} [Math. Ann. 102, 49--131 (1929; JFM 55.0824.02)] proposed a problem about the existence and the description of all self-adjoint extensions preserving the lower bound of a given densely defined symmetric operator which is bounded from below. It is the aim of the paper under review to present a new operator-valued parametrization of the domains and associated closed forms of all nonnegative self-adjoint extensions (among them the minimal Kreĭn--von Neumann extension) of a densely defined, closed nonnegative symmetric operator with, in general, zero lower bound. The approach considered by the present authors, based on the Friedrichs extension, is entirely new and provides a complete solution to von Neumann's problem in an explicit form. In this context, \textit{M.~S.~Birman}'s description of all nonnegative self-adjoint extensions of a positive definite symmetric operator [Mat. Sb. 38(80), 431--450 (1956; Zbl 0070.12102)] as well as Kreĭn's resolvent formula about canonical and generalized resolvents of a symmetric operator are obtained as consequences. An application to the description of all nonnegative quantum Hamiltonians in the point-interaction model with a finite number of point-interactions in \({\mathbb R}^3\) concludes the paper.
Friedrichs and Kreĭn--von Neumann extension
Weyl--Titchmarsh operator-valued function
scientific article; zbMATH DE number 2162375