F. Gesztesy, B. Simon, and G. Teschl
Spectral Deformations of One-Dimensional Schr\"odinger Operators
(167K, amstex)
ABSTRACT. We provide a complete spectral characterization of a new method of
constructing isospectral (in fact, unitary) deformations of general
Schr\"odinger operators H on the real line. Our technique is connected
to Dirichlet data, that is, the spectrum of an associated Schr\"odinger
operator with an additional Dirichlet boundary condition at a reference
point x_0. The transformation moves a single Dirichlet eigenvalue and
perhaps flips which side of x_0 the eigenvalue lives. On the remainder
of the spectrum, the transformation is realized by a unitary operator.
For cases where the potential V(x) tends to infinity asymptotically,
such that V is uniquely determined by the spectrum of H and the
Dirichlet data, our result implies that the specific Dirichlet data
allowed are determined only by their asymptotics as E tends to infinity.