Quantum integrability and differential equations.

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Over the last decade, progress in the study of ordinary differential equations defined in the complex plane and that of quantum integrable models has advanced with the help of a surprising correspondence between these previously-separate fields. The link is now called the ODE/IM correspondence. Functional relations lie at the heart of the correspondence, forming a bridge between the two subjects and allowing techniques from one island to be applied to its neighbour, and vice versa. This has led to significant applications, for example in PT-symmetric quantum mechanics and boundary integrable quantum field theory. Each of the functional relations has an infinite set of solutions, which are known to fall into families due to the integrable model relation with conformal field theory. For all cases except the su(2) case, only the highest-weight state in each family has been explored and matched with either an ordinary differential equation or a pseudo-differential equation. The first task is to map out the full set of differential equations which correspond to the excited states of the integrable model. We will begin with a simple case and aim to deduce the general picture. The second aim of the proposed research is to shed light on the hidden role the Lie algebra symmetry has to play in the differential equation side of the picture. From the integrable model side we expect each node of the associated Dynkin diagram to correspond to a different differential equation, up to the symmetry of the diagram. We shall address the issue of the missing differential equations, enlarging the known set of equations beyond the first node of most of the Dynkin diagrams. The Bethe ansatz and related techniques play a central part in all areas of integrable models and are important in many related fields. The research described here will expand the current toolbox of nonlinear integral equations used for solving Bethe ansatz equations.