Determination of the boundary condition of the basis of a finite set of eigenvalues (Q2703944)

Consider the boundary value problem \(L\) NEWLINE\[NEWLINE y''+p_1(x,\lambda)y'+p_2(x,\lambda)y=0, 0<x<1,\quad a(\lambda)y'(0)+b(\lambda)y(0)=c(\lambda)y'(1)+d(\lambda)y(1)=0, NEWLINE\]NEWLINE where \(p_1, p_2, a, b\) and \(c\) are smooth functions, and \(d(\lambda)= d_0+d_1\lambda +\ldots +d_{m-1}\lambda^{m-1}\) is a polynomial. Fix \(k\), \(k\leq m\). The author proves a uniqueness theorem for the solution to the inverse problem of recovering \(k\) of the \(m\) coefficients \(d_0, \ldots, d_{m-1}\) from the given \(k\) eigenvalues of \(L,\) provided that the remaining \(m-k\) coefficients \(d_j\) are known. The functions \(p_1, p_2, a, b\) and \(c\) are also assumed to be known.