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The free particle in quantum mechanics in real space is endowed withsupersymmetry, which enables a natural extension to complex spectra with abuilt-in parity (P) and time reversal (T) symmetry. It also explains the originof unbroken and broken phases of the PT-symmetry and their relationship withthe real and complex eigenvalues respectively, the latter further displayingzero-width resonances. This is possible as the extension of the eigenvalueproblem to the complex plane enables the incorporation of bound and decayingstates in the enlarged Hilbert space. The inherent freedom of modification ofthe potential without changing the spectra in supersymmetry naturally explainsthe connection of complex breather solutions of KdV with PT-symmetry and thefree particle on the complex plane. Further, non-trivial zero-width resonancesin the broken PT phase mandate a generalization that is directly connected tothe sl(2, R) potential algebra.

Free Particle to Complex KdV breathers through Isospectral Deformation