A reconstruction method for a two-dimensional inverse eigenvalue problem
Given \(\{\lambda_k : k = 1, \ldots, m\}\), the authors consider the problem of recovering a function \(q\) defined on \(R = ] 0, \pi/a [\times ]0, \pi[\) such that \(q (\pi/a - x,y) = q(x,y) = q(x, \pi - y)\) and such that \(\{\lambda_k : k = 1, \ldots, m\}\) is the set of eigenvalues of \(- \Delta u + qu = \lambda u\) in \(R\), \(u = 0\) on \(\partial R\) and, using the Rayleigh-Ritz approach of computing eigenvalues, the problem is reduced to an inverse eigenvalue problem for a matrix, as the entries of the matrix are linear combinations of Fourier coefficients of \(q\). The problem is solved if \(\lambda_k (k = 1, \ldots, m)\) are sufficiently close to the smallest \(m\) eigenvalues of the problem with \(q = 0\) and such \(m\) eigenvalues are simple or, if multiple, satisfy a convenient hypothesis.
Rayleigh-Ritz approach of computing eigenvalues
scientific article; zbMATH DE number 696960
scientific article