On self-adjoint operators in Krein spaces constructed by Clifford algebra \(\mathcal Cl_{2}\) (Q2901953)

Let \(S\) be a closed densely defined symmetric operator with equal deficiency indices in a Hilbert space \((\mathcal H, (\cdot,\cdot))\). It is always assumed that \(S\) commutes with the elements of a given complex Clifford algebra \(Cl_2(J,R):=\) span\(\,\{I,J,R,iJR\}\) with generating elements \(J\) and \(R\) which are bounded self-adjoint operators in \(\mathcal H\) satisfying \(J^2=I\), \(R^2=I\), and \(JR=-RJ\). We mention that self-adjoint operators \(J\) (and \(R\), respectively) with \(J^2=I\) (resp.\ \(R^2=I\)) are called fundamental symmetries. Fundamental symmetries like \(J\) give rise to an indefinite inner product \([\cdot,\cdot]\) via \([x,y]:= (Jx,y)\) and, hence, they are closely connected to the theory of Krein spaces.NEWLINENEWLINEIn this paper, non-self-adjoint extensions of \(S\) are investigated. Special attention is paid to those extensions \(A\) of \(S\) which satisfy \(AJ_{\vec{\alpha}}= J_{\vec{\alpha}}A^*\), where \(J_{\vec{\alpha}}\) equals a linear combination of \(J,R\) and \(iJR\), namely, \(J_{\vec{\alpha}}=\alpha_1 J + \alpha_2 R + \alpha_3 iJR\) such that the vector \(\vec{\alpha}:= (\alpha_1 \;\; \alpha_2 \;\; \alpha_3)^\top\) is a unit vector in \(\mathbb R^3\). In this situation, \(J_{\vec{\alpha}}\) is a fundamental symmetry. The set of all such extensions is denoted by \(\Sigma_{J_{\vec{\alpha}}}\) and an element from \(\Sigma_{J_{\vec{\alpha}}}\) is a \(J_{\vec{\alpha}}\)-self-adjoint operator, i.e., an operator which is self-adjoint with respect to the Krein space inner product given by \(J_{\vec{\alpha}}\), that is, \([x,y]:= (J_{\vec{\alpha}}x,y)\).NEWLINENEWLINEUsually, boundary triplets are used to describe all extensions of a given symmetric operator. Here, the relation between \(J_{\vec{\alpha}}\) and a boundary triplet and its associated Weyl function is investigated. In particular, the set \(\Sigma_{J_{\vec{\alpha}}}\) allows a description in terms of a boundary triplet and all unitary operators in the auxiliary space of the boundary triplet.NEWLINENEWLINEAs an application, the one-dimensional Schrödinger equation with a non-integrable singularity at zero (in the limit circle case) is considered.