96-326 F. Gesztesy, B. Simon, and G. Teschl

Spectral Deformations of One-Dimensional Schr\"odinger Operators (167K, amstex) Jul 4, 96

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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.

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