Spectral deformations of one-dimensional Schrödinger operators

Abstract

We provide a complete spectral characterization of a new method of constructing isospectral (in fact, unitary) deformations of general Schrödinger operators H =− d^2/dx^2 + V in H =− d^2/dx^2 + VinL^2(ℝ). Our technique is connected to Dirichlet data, that is, the spectrum of the operator H^D on L^2((−∞,x_0)) ⊕ L^2((x_0, ∞)) with a Dirichlet boundary condition at x_0. The transformation moves a single eigenvalue of H^D 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 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 → ∞.