Operator-based approach to $${\mathcal {P}}{\mathcal {T}}$$-symmetric problems on a wedge-shaped contour

Abstract

We consider a second-order differential equation $$\begin{aligned} -y''(z)-(iz)^{N+2}y(z)=\lambda y(z), \quad z\in \varGamma \end{aligned}$$ with an eigenvalue parameter $$\lambda \in {\mathbb {C}}$$ . In $${\mathcal {P}}{\mathcal {T}}$$ quantum mechanics z runs through a complex contour $$\varGamma \subset {\mathbb {C}}$$ , which is in general not the real line nor a real half-line. Via a parametrization we map the problem back to the real line and obtain two differential equations on $$[0,\infty )$$ and on $$(-\infty ,0].$$ They are coupled in zero by boundary conditions and their potentials are not real-valued. The main result is a classification of this problem along the well-known limit-point/limit-circle scheme for complex potentials introduced by Sims 60 years ago. Moreover, we associate operators to the two half-line problems and to the full axis problem and study their spectra.