An inverse problem for a Sturm-Liouville operator
Consider an identification problem for the coefficient \(q(x)\) in the differential equation (1) \(-u''(x) + q(x) u(x) = \psi (x)\), \(0<x<1\), \(u(0) = 0\), \(u(1) = 0\). It is assumed throughout that 0 is not a Dirichlet eigenvalue of \(-d^ 2/dx^ 2 + q(x)\) on \((0,1)\), so the forward solution to (1) for a given \(\psi (x)\) is unique. The authors prove the following result: Let \(u_ j (x) = u_ j (x;q)\) denote the solution of (1) corresponding to the function \(\psi_ j (x)\) in the basis \(\{\psi_ j (x)\}^ \infty_{j=1}\) of \(L^ 2(0,1)\). Then \(q(x) \in L^ 2 (0,1)\) is uniquely determined by the measurements \(\{u_ j'(0)\}^ \infty_{j=1}\). In Section 2 the special case when \(\psi_ j(x) = \sin \pi jx\) is investigated. In Section 3, a Newton algorithm is used to construct approximations of \(q(x) \approx 0\) from the finite data \(\{u_ j'(0)\}^ N_{j=1}\).
scientific article; zbMATH DE number 567032