Spectral Deformations of One-dimensional Schrodinger Operators W Introduction W Preliminaries on the Dirichlet Deformation Method Proof since and the Associated Dirichlet Deformation

We provide a complete spectral characterization of a new method of constructing isospectral (in fact, unitary) deformations of general Schrrdinger operators H =-d2/d.x 2 + V in L2(/l~). Our technique is connected to Dirichlet data, that is, the spectrum of the operator H D on L2((-~,x0)) @ L2((x0, oo)) with a Difichlet 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 Ixl-" ~, 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 ~ oo. Spectral deformations of Schrrdinger operators in L2(/R), isospectral and certain classes of non-isospectral ones, have attracted a lot of interest over the past three decades due to their prominent role in connection with a variety of topics , including the Korteweg-de Vries (KdV) hierarchy, inverse spectral problems, supersymmetric quantum mechanical models, level comparison theorems, etc. In fact, the construction of N-soliton solutions of the KdV hierarchy (and more generally , the construction of solitons relative to reflectionless backgrounds) is a typical example of a non-isospectral deformation of H =-d2/dx 2 in L2(R) since the resulting deformation [1 =-d2/dx 2 + re" acquires an additional point spectrum (cr(.) abbreviating the spectrum). On the other hand, the solution of the inverse periodic problem and the corresponding solution of the algebro-geometric quasi-periodic finite-gap inverse problem for the KdV hierarchy (and certain almost-periodic limiting situations thereof) are intimately connected with isospectral (in fact, unitary) deformations of a given base (background) operator H =-d 2/dx 2 + V. Although not a complete bibliography on applications of spectral deformations in [47], and the references cited therein. Our main motivation in writing this paper descends from our interest in inverse spectral problems. As pointed out later (see Remarks 4.5, 4.7, and 4.8), spectral deformation methods can provide crucial insights into the isospectral class of a given base potential V, and in some cases can even determine the whole class Iso(V) = {15" E LI~(/~) [ o(-d2/dx 2 q-(1) _~ o-(_d2/dx2q_ V)} of such potentials. A particularly "annoying" open problem in inverse spectral theory concerns the characterization of the isospectral class of potentials V with …