On the theory of \(\mathcal{PT}\)-symmetric operators. (English. Russian original) Zbl 1275.47085

Ukr. Math. J. 64, No. 1, 35-55 (2012); translation from Ukr. Mat. Zh. 64, No. 1, 32-49 (2012).
It is known that the operator \(H=-\frac{d^2}{dx^2}+x^2(ix)^{\varepsilon}\), \(0\leq \varepsilon <2\), is non-selfadjoint in \(L_2(\mathbb{R})\) and its spectrum is real and positive. C. M. Bender and S. Boettcher [Phys. Rev. Lett. 80, No. 24, 5243–5246 (1998; Zbl 0947.81018)] explained such a property of the non-selfadjoint operator by its \(\mathcal{PT}\)-symmetry. This paper gives an abstract definition of \(\mathcal{PT}\)-symmetric operators in an arbitrary Hilbert space \(\mathfrak{H}\) and analyzes the status of \(\mathcal{PT}\)-symmetric operators as selfadjoint operators in Krein spaces. The authors note that \(\mathcal{PT}\)-symmetric operators are not always selfadjoint relative to the definite metric generated in the space \(\mathfrak{H} \) by the operator \(\mathcal{P}\). However, it is shown that \(\mathcal{PT}\)-symmetric quasi-selfadjoint extensions of symmetric operators with deficiency indices \(\langle 2,2 \rangle\) can play a role as selfadjoint operators in Krein spaces. Lastly, the results are illustrated with \(\mathcal{PT}\)-symmetric Schrödinger operators with singular potentials.


47B50 Linear operators on spaces with an indefinite metric
47B25 Linear symmetric and selfadjoint operators (unbounded)
81Q12 Nonselfadjoint operator theory in quantum theory including creation and destruction operators
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis


Zbl 0947.81018
Full Text: DOI


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