Method for generating a discrete state in the continuum part of the spectrum

摘要

We present a systematic method for the construction of a discrete state embedded in the continuum part of the spectrum of the differential equation -y″+f(x)y=λy. Starting from an arbitrary preselected eigenvalue λ0, we generate a family of functions yielding identical eigenspectrum as f(x). The nature of the corresponding eigenfunctions remains unaltered, except at λ0, for which we obtain a discrete eigenfunction. The procedure is exemplified using the simplest case of f(x) = 0.