Method of difference-differential equations for some Bethe-ansatz-solvable models

In studies of one-dimensional Bethe ansatz solvable models, a Fredholm integral equation the second kind with difference kernel on finite interval often appears. This does not generally admit closed-form solution and hence its analysis is quite complicated. Here we study family such equations, concentrating their moments. We find exact relations between moments in form difference-differential equations. The latter results significantly advance analysis, enabling one to practically determine all from explicit knowledge lowest one. As applications, several examples are considered. First, quasimomentum distribution Lieb-Liniger model analytical results. basic quantities, e.g., $N$-body local correlation functions. prove equivalence different expressions found literature for three-body functions an result four-body function terms distributions. eventually three- asymptotic series regimes weak strong interactions. Next, low-energy spectrum magnon (a polaron) excitation two-component Bose gas described by Yang-Gaudin model. form, which depends distributions Then, address seemingly unrelated problem capacitance circular capacitor express parametric form. most interesting case short plate separations, has single logarithmic term. should be contrasted that complicated structure logarithms.