Inverse problems on the least eigenvalue
The paper deals with generalizations of the classical Ambarzumyan theorem, which, as is well-known, can be formulated in the following way: if the first (i.e., smallest) eigenvalue \(\lambda_0\) of the Sturm-Liouville operator \[ Ay:=-y''+q(x)y, \quad 0<x<\pi, \quad y'(0)=y'(\pi)=0 \] with a real-valued potential \(q(x)\in L(0,\pi)\) equals to the mean value of the potential, i.e. \[ \lambda_0=\frac1\pi\int_0^\pi q(x)\,dx, \] then \(q(x)=\lambda_0\) a.e. on \((0,\pi).\) Using the variational principle the authors obtain an analogous result for bounded from below operators \(A\) in an abstract separable Hilbert space. As a corollary they obtain the corresponding facts for selfadjoint second-order differential operators with operator-valued potentials, higher-order partial differential operators, etc.
scientific article; zbMATH DE number 6305234