On the Heisenberg commutation relation. II
The paper investigates couples of self-adjoint operators which satisfy the Heisenberg commutation relation on a common core and for which the Weyl relation is not satisfied in general. Let P and Q be self-adjoint operators on a Hilbert space H and let \({\mathcal D}\) be a dense linear submanifold of \({\mathcal H}\). We say (P,Q;\({\mathcal D})\) belongs to the class \({\mathcal C}\) if the following five conditions are fulfilled: (1) P\({\mathcal D}\subseteq {\mathcal D}\), Q\({\mathcal D}\subseteq {\mathcal D}\), (2) \(PQ\phi -QP\phi =-i\phi\) for \(\phi\in {\mathcal D}\), (3) \(P\upharpoonright D\) is essentially self-adjoint, (4) \({\mathcal D}_ a(Q):=\{\phi \in {\mathcal D}:\) \(\phi\) is an analytic vector for \(Q\}\) is dense in \({\mathcal H}\), (5) The Op*-algebra generated by \(P\upharpoonright {\mathcal D}\), \(Q\upharpoonright {\mathcal D}\) and \(I\upharpoonright {\mathcal D}\) is closed on \({\mathcal D}.\) Let \(P_ 0\) be the differential operator -id/dx on the Hilbert space \({\mathcal H}=\sum_{n}\oplus L_ 2(a_ n,b_ n),\) \(Q=x\) and \({\mathcal D}_ 0=\cap_{j,k=0}{\mathcal D}(P_ 0^ jQ^ k).\) Here \(a_ n\in R\cup \{- \infty \}\) and \(b_ n\in R\cup \{+\infty \}\). The first purpose of the paper is to classify all triples (P,Q;\({\mathcal D})\in {\mathcal C}\) for which \(P_ 0\subseteq P\) and \({\mathcal D}_ 0\subseteq {\mathcal D}\). They are described in terms of certain unitary operators (defining self-adjoint extensions of \(P_ 0\) via boundary conditions) and certain vector spaces of boundary values of functions and their derivatives. Under some additional assumptions, the classification of an arbitrary triple (P,Q;\({\mathcal D})\in {\mathcal C}\) can be reduced to this case. The second purpose of the paper is to develop some methods for the construction of triples (P,Q;\({\mathcal D})\in {\mathcal C}\) for which the Weyl relation is not satisfied. [For part I see J. Funct. Anal. 70, 8-49 (1983; Zbl 0507.47010).]
Heisenberg commutation relation on a common core
scientific article; zbMATH DE number 3888380
scientific article