A new PT symmetric complex Hamiltonian with a real spectra

We construct an isospectral system in terms of a real and a complex potential to show that the underlying PT symmetric complex Hamiltonian possesses a real spectra which is shared by its real partner. ∗bbagchi @ cucs.ernet.in †raj @ www.isical.ac.in Complex potentials have found a wide usage [1] in the literature especially in connection with scattering problems. Recently it has been emphasized [2] that by enforcing a PT symmetry, one can obtain new classes of complex Hamiltonian which exhibit a real spectra of energy eigenvalues. The purpose of this letter is to bring to light a new complex Hamiltonian which is PT symmetric and possess a real energy spectra. Consider potentials of the form V (1),(2) = U2±U ′ where U is complex function of x and a dash denotes a derivative with respect to x. Let us express U explicitly as a(x) + ib(x), where a(x) and b(x) are certain real, continuously differentiable functions in R. We have , V (1),(2) = (a − b ± a′) + i(2ab± b′) (1) In the following we investigate the possibility when one of the potentials defined by (1) is real but the other is complex. To this end we choose, for the sake of concreteness, V (2) to be real thus resticting the function a to be given by