Inverse problem for the Sturm–Liouville equation with piecewise entire potential and piecewise constant weight on a curve

A Sturm–Liouville equation with a piecewise entire potential and a non-zero piecewise constant weight function on a curve of an arbitrary shape lying on the complex plane is considered. For such equation, the inverse spectral problem is posed with respect to the ratio of elements of one column or one row of the transfer matrix along the curve. The uniqueness of the solution to the problem is proved with the help of the method of unit transfer matrix using the study of asymptotic solutions of the Sturm–Liouville equation for large values of the absolute value of the spectral parameter. The obtained results allowed to consider inverse problem for a previously unexplored class of Sturm–Liouville equations with three unknown coefficients on a segment of the real axis.