Spectral Deformations of One-dimensional Schrödinger Operators

We provide a complete spectral characterization of a new method of constructing isospectral (in fact, unitary) deformations of general Schrödinger operators H = − d2 dx2 + V in L2(R). Our technique is connected to Dirichlet data, that is, the spectrum of the operator HD on L2((−∞, x0))⊕L2((x0 ,∞)) with a Dirichlet boundary condition at x0. The transformation moves a single eigenvalue of HD and perhaps flips which side of x0 the eigenvalue lives. On the remainder of the spectrum, the transformation is realized by a unitary operator. For cases such as V (x) → ∞ as |x| → ∞, where 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 the asymptotics as E → ∞. §