On stable \(\mathcal C\)-symmetries for a class of \(\mathcal P\mathcal T\)-symmetric operators. (English) Zbl 1289.47047

The author considers \(\mathcal P\mathcal T\)-symmetric extensions of a symmetric operator with the deficiency index (2,2). This is motivated by quantum mechanical models with \(\mathcal P\mathcal T\)-symmetric Hamiltonians; see also S. O. Kuzhel’ and O. M. Patsyuk [Ukr. Math. J. 64, No. 1, 35–55 (2012); translation from Ukr. Mat. Zh. 64, No. 1, 32–49 (2012; Zbl 1275.47085)]. In the paper under review, necessary and sufficient conditions are found for the existence of a stable \(\mathcal C\)-symmetry for a class of \(\mathcal P\mathcal T\)-symmetric extensions. As an example, a one-dimensional Schrödinger operator with a distribution potential (involving \(\delta\)- and \(\delta'\)-terms) is considered.


47B25 Linear symmetric and selfadjoint operators (unbounded)
81Q12 Nonselfadjoint operator theory in quantum theory including creation and destruction operators


Zbl 1275.47085