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Mathematical methods are presented that give pairs of linearly independent solutions for an attractive 1/r 6 potential. These solutions represent a different class of special functions that are important to the understanding of molecular vibration spectra near the dissociation threshold and slow atomic collisions.

We present a multichannel quantum-defect theory for slow atomic collisions that takes advantages of the analytic solutions for the long-range potential, and both the energy and the angular-momentum insensitivities of the short-range parameters. The theory provides an accurate and complete account of scattering processes, including shape and Feshbach resonances, in terms of a few parameters such as the singlet and the triplet scattering lengths. As an example, results for $^{23}$Na-$^{23}$Na scattering are presented and compared close-coupling calculations.Comment: 8 pages, 3 figure

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We present some of the universal properties in ion-atom interaction derived from a newly formulated quantum-defect theory for −1/r 4 type of long-range interactions. For bound states, we present the universal bound spectrum, namely the equivalent of the Rydberg formula, for ion-atom systems. For scattering, we introduce the concept of universal resonance spectrum to give a systematic understanding of many resonances present in ion-atom scattering. The theory further provides a method for an accurate spectroscopic determination of the atomic polarizability. It also suggests the existence of atom-like molecules, in which multiple atoms orbit around a heavy ion.PACS numbers: 34.10.+x,34.50.Cx,03.65.Nk As experimental techniques for preparing and manipulating cold atomic ions improve [1], there is a rapidly growing interest in ion-atom interactions at ultracold temperatures [2][3][4][5]. While it is intuitively clear that an ion can exert a much stronger influence on its environment than either an atom or a molecule, the precise nature of this influence, especially at the quantum level, is far from being understood. For example, an ionatom system is one of the few types of two-body systems for which their loosely-bound states, namely the highly excited rovibrational states right below the dissociation limit, are yet to be observed experimentally or fully characterized theoretically [6].This work presents part of a newly formulated quantum-defect theory (QDT) for ion-atom interactions. It gives precise explanation and characterization of the meaning and the consequence of the "strong influence" of an ion on a neighboring atom. On a more technical level, it is a version of the QDT for −1/r 4 type of longrange potential [7][8][9] that brings our understanding of ion-atom interactions to the same level as, and in certain areas exceeding, our current understanding of ultracold atom-atom interactions [10]. The theory is a result of combining conceptual developments, built further upon Refs. [11][12][13], with improved mathematical understanding of the modified Mathieu functions [14-16], especially for negative energies. The latter development, which allows for the efficient determination of all QDT functions for −1/r 4 potential, is achieved by solving the modified Mathieu equation using techniques we have previously developed for the analytic solutions of 1/r 6 [17] and 1/r 3 [18] potentials.We focus here on the universal spectrum for ion-atom interactions. For bound states, it is the equivalent of the Rydberg formula for −1/r 4 type of long-range potential, formulated in a way to take advantage of the angular momentum insensitivity of the short-range parameter that is a characteristic of atom-atom and ion-atom interactions [11,13,19]. It allows for the determination of the entire rovibrational spectrum in the threshold region from a single short-range parameter, such as the quantum defect. More importantly, the concept of universal spectrum is generalized here to positive energies to include scattering resonance posi...

From a rigorous multichannel quantum-defect formulation of bimolecular processes, we derive a fully quantal and analytic model for the total rate of exoergic bimolecular reactions or inelastic processes that is applicable over a wide range of temperatures including the ultracold regime. The theory establishes a connection between the ultracold chemistry and the regular chemistry by showing that the same theory that gives the quantum threshold behavior agrees with the classical Gorin model at higher temperatures. In between, it predicts that the rates for identical bosonic molecules and distinguishable molecules would first decrease with temperature outside of the Wigner threshold region, before rising after a minimum is reached.

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