Inverse problem for Sturm–Liouville operators in the complex plane

The inverse problem for the standard Sturm – Liouville equation with a spectral parameter $\rho$ and a potential function, piecewise-entire on a rectifiable curve $\gamma \subset \mathbf{C}$, on which only the starting point is given, is studied for the first time. A function $Q$ that is bounded on a curve $\gamma$ is piecewise-entire on it if $\gamma$ can be splitted by a finite number of points into parts on which $Q$ coincides with entire functions, different in neighboring parts. The split points, the initial and final points of the curve are called critical points. The problem is to find all the critical points of the curve $\gamma $ and the potential on it by the column or row of the transfer matrix $\hat P$ along $\gamma$. On the basis of the obtained asymptotics of matrix $\hat P$ for $| \rho| \to \infty$, it is proved that if at least one of its elements is bounded for $\forall \rho \in \mathbf {C}$, then the curve $\gamma$ degenerates to a point after removing all ,,invisible loops". An ,,invisible loop" is a loop of the curve $\gamma$ (with a given piecewise-entire function) whose knot coincides with two successive critical points. The uniqueness of the solution of the inverse problem for curves without ,,invisible loops" is proved. On the example of the inverse problem for the equation $\frac {d}{dx} \left (\frac {1}{r (x)} \frac {dy}{dx} \right) + \left (q(x)-r(x) \lambda^2 \right) y(x) = 0$ with a piecewise-entire function $q(x)$ and a piecewise constant function $r(x)\ne 0$ on the segment of the real axis, the usefulness of the results obtained in the article is shown for the study of inverse problems for generalized Sturm–Liouville equations, which can be reduced to the type studied in the article.