Reflectionless potentials and symmetry

Large families of Hamiltonians that are non-Hermitian in the conventional sense have been found to have all eigenvalues real, a fact attributed to an unbroken symmetry. The corresponding quantum theories possess an unconventional scalar product. The eigenvalues are determined by differential equations with boundary conditions imposed in wedges in the complex plane. For a special class of such systems, it is possible to impose the -symmetric boundary conditions on the real axis, which lies on the edges of the wedges. The -symmetric spectrum can then be obtained by imposing the more transparent requirement that the potential be reflectionless.