DIMENSIONAL SCALING FOR QUASI-STATIONARY STATES

Complex energy eigenvalues which specify the location and width of quasibound or resonant states are computed to good approximation by a simple dimensional scaling method. As applied to bound states, the method involves minimizing an effective potential function in appropriately scaled coordinates to obtain exact energies in the D --> infinity limit, then computing approximate results for D = 3 by a perturbation expansion in I/D about this limit. For resonant states, the same procedure is used, with the radial coordinate now allowed to be complex. Five examples are treated: the repulsive exponential potential (e(-r)); a squelched harmonic oscillator (r2e(-r)); the inverted Kratzer potential (r-1 repulsion plus r-2 attraction); the Lennard-Jones potential (r-12 repulsion, r-6 attraction); and quasibound states for the rotational spectrum of the hydrogen molecule (X 1SIGMA(g)+, v = 0, J = 0 to 50). Comparisons with numerical integrations and other methods show that the much simpler dimensional scaling method, carried to second-order (terms in 1/D2), yields good results over an extremely wide range of the ratio of level widths to spacings. Other methods have not yet evaluated the very broad H-2 rotational resonances reported here (J> 39), which lie far above the centrifugal barrier.