For a bounded and sufficiently smooth domain $\mathrm{\Omega }$ in ${\mathbb{R}}^N$, $N\ge 2$, let ${\left({\lambda }_k\right)}_{k=1}^{\mathrm{\infty }}$ and ${\left({\phi }_k\right)}_{k=1}^{\mathrm{\infty }}$ be respectively the eigenvalues and the corresponding eigenfunctions of the problem (with Neumann boundary conditions) $$-\text{div}\left(a\left(x\right)\nabla {\phi }_k\right)+q\left(x\right){\phi }_k={\lambda }_k\varrho \left(x\right){\phi }_k\text{ in }\mathrm{\Omega },a\frac{\partial }{\partial \mathbf{n}}{\phi }_k=0\text{ su }\partial \mathrm{\Omega }.$$ We prove that knowledge of the Dirichlet boundary spectral data ${\left({\lambda }_k\right)}_{k=1}^{\mathrm{\infty }}$, ${\left({\phi }_{k|\partial \mathrm{\Omega }}\right)}_{k=1}^{\mathrm{\infty }}$ determines uniquely the Neumann-to-Dirichlet (or the Steklov- Poincaré) map $\gamma$ for a related elliptic problem. Under suitable hypothesis on the coefficients $a,q,\varrho$ their identifiability is then proved. We prove also analogous results for Dirichlet...

Dubrovin type equations for the N -gap solution of a completely integrable system associated with a polynomial pencil is constructed and then integrated to a system of functional equations. The approach used to derive those results is a generalization of the familiar process of finding the 1-soliton (1-gap) solution by integrating the ODE obtained from the soliton equation via the substitution u = u(x + λt).