Chapter 6: Proof of Reality for Some Simple Examples

Abstract:

The purpose of this chapter is to study the $\mathcal{P}\mathcal{T}$-symmetric eigenvalue problem for the Hamiltonian $H=p^2+x^2{\left(ix\right)}^{\epsilon }$ at a rigorous level. We start with the $\mathcal{P}\mathcal{T}$-symmetric cubic oscillator $\left(\epsilon =1\right)$ that, as explained in the preface of this book, brought the issue of spectral reality to the attention of Bender and Boettcher [Bender and Boettcher (1998b)] and initiated the current upsurge of interest in the subject of $\mathcal{P}\mathcal{T}$ symmetry. The Hamiltonian for this oscillator is

and the corresponding time-independent Schrödinger equation is To set up a well-defined eigenvalue problem, this equation requires a boundary condition for $\psi \left(x\right)$. For now we take this to be that $\psi \left(x\right)$ be squareintegrable on the real line, which is equivalent to the statement that the possibly-complex number E belongs to the spectrum of H if and only if (6.2) has a solution $\psi \left(x\right)$ that decays both as $x\to -\infty$ and as $x\to +\infty$…