UNIQUENESS THEOREMS IN INVERSE SPECTRAL THEORY

CHRISTER BENNEWITZ

christer.bennewitz@math.lu.se

Department of Mathematics
Lund University
Box 118
SE-22100 Lund, Sweden

The celebrated Borg-Marchenko uniqueness theorem says that given the spectral measure of a selfadjoint operator generated by a Sturm-Liouville operator $-u''+qu=\lambda u$, then the potential $q$ is determined. I will review this result and then discuss uniqueness theorems of the same nature, but for more general situations. I will discuss matrix-valued equations, higher order equations, the general Sturm-Liouville equation $-(pu')'+qu=\lambda wu$ and so called left-definite Sturm-Liouville equations. For some of these results method similar to those originally employed by Borg may be used, but some results depend on a new approach, using a generalized version of the classical Paley-Wiener theorem for the Fourier transform.