Interface between Hermitian and non-Hermitian Hamiltonians in a model calculation

We consider the interaction between the Hermitian world, represented by a real delta-function potential $-\alpha\delta(x)$, and the non-Hermitian world, represented by a PT-symmetric pair of delta functions with imaginary coefficients $i\beta(\delta(x-L)-\delta(x+L))$. In the context of standard quantum mechanics, the effect of the introduction of the imaginary delta functions on the bound-state energy of the real delta function and its associated wave-function is small for L large. However, scattering from the combined potentials does not conserve probability as conventionally defined. Both these problems can be studied instead in the context of quasi-Hermiticity, whereby quantum mechanics is endowed with a new metric $\eta$, and consequently a new wave-function $\Psi(x)$, defined in terms of the original wave-function $\psi(x)$ by means of $\eta$. In this picture, working perturbatively in $\beta$, the bound-state wave-function is actually unchanged from its unperturbed form for $|x|<< L$. However, the scattering wave-function, for $|x|>>L$, is changed in a significant manner. In particular there are incoming and outgoing waves on both sides of the potential. One can then no longer talk in terms of reflection and transmission coefficients, but the total right-moving flux is now conserved.