Half inverse problems for quadratic pencils of Sturm-Liouville operators (Q1758314)

The paper deals with boundary value problems for the Sturm-Liouville-type quadratic pencil \[ -y''+[q(x)+2\lambda p(x)]y = \lambda^2 y \] on \((0,\pi)\). First, the authors obtain the asymptotic formulae for the eigenvalues of problems with separated boundary conditions. Second, a uniqueness theorem for a ``half-inverse'' problem of the Hochstadt-Lieberman type is proved. The theorem states that a single spectrum (with fixed boundary conditions at the right end) uniquely determines both \(q\) and \(p\) provided they are given on the right half-interval \([\pi/2,\pi]\). The basic tool is the integral representation of the solution of the initial value problem for the quadratic pencil with kernels independent of the spectral parameter \(\lambda\).