Determination of the quantization function for deep potentials with attractive tails

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Potentials decaying faster than −1/r2, as typically occur in atomic and molecular systems, support at most a finite number of bound states. Properties of bound states below and scattering states above threshold are correlated via a quantization function F(E), whose near-threshold behaviour is dominantly determined by the far tail of the potential. Knowledge of F(E) makes it possible to derive important properties of near-threshold states by simple, model-independent methods which are competitive with and sometimes superior to popular ab initio methods, mainly because they do not require any knowledge of the short-ranged behaviour of the potential.For an energy range extending over several near-threshold bound states, the behaviour of F(E) is well understood qualitatively, but quantitative applications have so far been restricted to homogeneous potential tails proportional to −1/r6. The aim of the this project is to determine F(E) quantitatively for other powers and also for significantly nonhomogeneous tails. This will enable a wide range of applications in atomic and molecular systems.