D-Wave Resonances in Positronium Hydride

The problem of calculating the position and width of the lowest-lying (L = 0, 1) resonances in the Ps + H scattering system has recently been re-examined by the authors [1]. In the model used, resonances axe assumed generated by bound Rydberg states in the closed re-arranged channel [e(+) + H(-)]. This infinite series of Coulomb bound states are shifted somewhat by the coupling with the open scattering channel [Ps + H]. Estimates using this approach, first carried out by one of the authors [2] using a simple form of scattering wavefunction, indicated an anticipated small shift of resonance away from the Rydberg energy in the S-wave (L = 0). More recent results by Ho and Yan [3] using the complex-rotation method showed unexpected significant shifting in the P (L = 1) and D (L = 2)-wave resonance energies from their respective unperturbed values. Our re-calculation of the resonances [I I for L = 0 and I indicates that, in each case, reasonably consistent resonance energies are obtained by coupling the two lowest lying states (IS and 2S for L = 0 and 2P and 3P for L = 1). For each L it is the higher state (2S and 3P, respectively) that produces the resonance. The lower state (1S and 2P, respectively) is effective only in increasing the non-resonant phase shift. In this work we extend the model to L = 2 (D-waves). Our results show that, unlike the S and P-waves, the 313-state alone produces a resonant energy that is very close to the result of Yan and Ho [3]. Inclusion of the 4D-state produces a second resonance in the scattering system shifted up slightly from its unperturbed Rydberg energy.