  The first part of this work consists of a study of the ODE/IM correspondence
for simply-laced affine Toda field theories. It is a first step towards a full
generalisation of the results of Lukyanov and Zamolodchikov on $\hat{\mathfrak
a}_1$ to a general affine Lie-Ka\v{c}-Moody algebra $\hat{\mathfrak g}$. In
order to achieve our goal, we investigate the structure of evaluation
representations of $\hat{\mathfrak g}$ and show how their tensor products are
related by what we call projected isomorphisms. These isomorphisms are used to
construct a set of quadratic functional relations, called $\psi$-system, for
the solutions to complex differential equations associated to $\hat{\mathfrak
g}$. Finally, from the $\psi$-system for the algebras $\mathfrak a$ and
$\mathfrak d$, we derive a set of Bethe Ansatz equations satisfied by the
eigenvalues of some particular boundary problem for the above mentioned
differential equations.
  The second part of this work deals with the study of one-point functions in
sine- and sinh-Gordon models. The approach to the computation of these
quantities follows a powerful method, which we call fermionic basis, developed
by Boos et al. for the XXZ, quantum Liouville and quantum sine-Gordon models.
We show how the determinant formula for one-point functions obtained there can
be generalised to the sinh-Gordon model. In doing so we give an interpretation
of the fermionic basis in terms of certain symmetries of the system. This new
perspective will also allow us to solve trivially the reflection relations
introduced by Fateev et al. We then provide analytical and numerical results
supporting our finding.
