Operator based approach to PT-symmetric problems on a wedge-shaped contour

Abstract

We consider a second-order differential equation $$-y^{\prime \prime }(z)-(iz{)}^{N+2}y(z)=\lambda y(z),z\in \Gamma$$ with an eigenvalue parameter $\lambda \in \mathbb{C}$. In $\mathcal{P}\mathcal{T}$ quantum mechanics $z$ runs through a complex contour $\Gamma \subset \mathbb{C}$, 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 A.R.\ Sims 60 years ago. Moreover, we associate operators to the two half-line problems and to the full axis problem and study their spectra.

References in corpus (8)

• Real Spectra in Non-Hermitian Hamiltonians Having PT Symmetry
• Complex Extension of Quantum Mechanics
• Pseudo-Hermitian Representation of Quantum Mechanics
• Supersymmetry and the spontaneous breakdown of PT symmetry
• On the reality of the eigenvalues for a class of PT-symmetric oscillators
• Equivalence of a Complex $\cP\cT$-Symmetric Quartic Hamiltonian and a Hermitian Quartic Hamiltonian with an Anomaly
• Krein-Space Formulation of PT-Symmetry, CPT-Inner Products, and Pseudo-Hermiticity
• PT-Symmetric Quantum Theory Defined in a Krein Space