Some results on the inverse spectral theory for the Sturm-Liouville operator on the line

We discuss some results concerning the inverse spectral theory of the Sturm-Liouville operator
$$L:=\dfrac{1}{y(x)}\left(-\dfrac{d}{dx}\left(p(x)\dfrac{d}{dx}\right)+q\right),$$
where the functions $p(x),q(x),y(x)$ are continuous and bounded, and the weight function $y(x)$ is strictly positive. In particular, we focus our attention on two main problems related to the inverse spectral theory for $L$:

• the scattering theory on the whole line, by developing a Gel'fand-Levitan-Marchenko theory for $L$;
• the algebro-geometric theory, by obtaining trace formulas for $L$, and studying the properties of $p(x),q(x)$ and $y(x)$ in a suitable algebraic surface.

The Weyl $m$-functions $m_\pm$ will play a crucial role, both in defining and in solving the inverse problems.
Applications to the study of solutions of some hierarchies of nonlinear evolution equations will be considered, including the well-known Korteweg-de Vries and Camassa-Holm ones.