Sturm-Liouville eigenvalue problems with finitely many singularities (Q1353480)

The authors consider the inverse problem associated with the classical two-point boundary-value problem posed with the equations \[ y''(x)- p(x)y(x)= -\lambda y(x),\;x\in(0,\pi),\quad y(0)= y(\pi)=0, \] which aims to recover the coefficient \(p(x)\) from some knowledge of the eigenvalues. Their main aim is to recover the jumps in the derivatives rather than to recover \(p(x)\) exactly. Therefor they study first the effect of the singularities of \(p(x)\) on the distribution of the eigenvalues. They show that much of the data describing the singularities of \(p(x)\) can be recovered from the eigenvalues by Fourier analysis of the function \(y(\pi,\lambda)\), which can be constructed from the eigenvalues sequence. They examine also the stability of their method when the eigenvalues are subjected to random perturbations.