SOME NEW INVERSE EIGENVALUE PROBLEMS

WILLIAM RUNDELL

rundell@math.tamu.edu

Department of Mathematics
Texas A&M University
College Station, TX 77843-3368, USA

We will look at two quite different problems to recover the potential in a second order differential operator from information on its spectrum. The first concerns the classical equation $-u'' + q u = \lambda u$ with (say) the boundary condition $u(0)=0$. At the other end point, $x=1$ $u'(1)=\sqrt{\lambda}u(1)$. This problem has complex eigenvalues and we will show that a single such spectrum suffices to determine $q$.

In the second problem we have the equation $-u'' + \ell(\ell+1)/x^2 + q u = \lambda u$ with fixed conditions at $x=1$ and boundedness at $x=0$. We examine the conjecture that two complete spectra $\{\lambda_{n,\ell}\}$ for $n = 1,2, \ldots$ and for $ell$ equals two distinct values $\ell_1$, $\ell_2$ is sufficient to determine $q$.