On Ambarzumyan-type theorems (Q1940694)

Let \(\ell\) be a self-adjoint lower semi-bounded linear ordinary differential operator with only discrete spectrum, say \(\lambda_0\leq\lambda_1\leq\dots\). Set \(\tilde{\ell}y+\hat{q}y=\ell y\) for \(\hat{q}\in L_1\) and \(y\) in the domain of \(\ell\). Denoting the spectrum of \(\tilde{\ell}\) by \(\tilde{\lambda}_0\leq\tilde{\lambda}_1\leq\dots\). If \(\tilde{y}_0\) is a normalized eigenfunction to the eigenvalue \(\tilde{\lambda}_0\) of \(\tilde{\ell}\) and \(\lambda_0=\tilde{\lambda}_0+(\hat{q}\tilde{y}_0,\tilde{y}_0)\), then, in this paper, it is shown that \(\hat{q}=\lambda_0-\tilde{\lambda}_0\) a.e. Applications are given.