Bethe Ansatz and the Spectral Theory of affine Lie algebra--valued connections II. The non simply--laced case

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We assess the ODE/IM correspondence for the quantum g-KdV model, for a non-simply laced Lie algebra g. This is done by studying a meromorphic connection with values in the Langlands dual algebra of the affine Lie algebra g⁽¹⁾, and constructing the relevant Ψ-system among subdominant solutions. We then use the Ψ-system to prove that the generalized spectral determinants satisfy the Bethe Ansatz equations of the quantum g-KdV model. We also consider generalized Airy functions for twisted Kac--Moody algebras and we construct new explicit solutions to the Bethe Ansatz equations. The paper is a continuation of our previous work on the ODE/IM correspondence for simply-laced Lie algebras.