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$.


EPSRC Gregynog Workshop, 21-26 July 2002

Gregynog Abstracts