Some questions in the theory of inverse problems for differential operators

V. A. Strakhov

Matematičeskie zametki, Tome 21 (1977) no. 2, pp. 151-160.

Résumé

For the two operators $$ Ly=y^{(n)}+\sum_{k=0}^{n-2}p_k(x)y^{(k)}\quad\text{and}\quad Ry=y^{(n)}+\sum_{k=0}^{n-2}P_k(x)y^{(k)} $$ with a common set of boundary conditions we establish a connection between $p_k(x)$ and $\overline p_k(x)$ in the case where the weight numbers coincide and a finite number of the eigenvalues do not coincide, in terms of the eigenfunctions of these operators corresponding to the noncoincident eigenvalues and the derivatives of these functions. This enables us to recover the operator $L$ from the operator $R$ by solving a system of nonlinear ordinary differential equations. For Sturm–Liouville operators an analogous relation is proved for the case where infinitely many eigenvalues do not coincide.