Chapter 2: $\mathcal{P}\mathcal{T}$-Symmetric Eigenvalue Problems

Perhaps the most remarkable feature of a $\mathcal{P}\mathcal{T}$-symmetric quantummechanical Hamiltonian is that its eigenvalues can be all real even if the Hamiltonian is not Hermitian. The purpose of this chapter is to explain the mathematics (complex asymptotic methods, WKB approximation, Stokes sectors) needed to define and solve the eigenvalue problems associated with $\mathcal{P}\mathcal{T}$-symmetric Hamiltonians. We apply these techniques to a wide variety of examples that illustrate the parametric deformation (analytic continuation) of differential-equation eigenvalue problems…