Article

**J. d'Analyse Math. 70, 267-324 (1996)**[DOI: 10.1007/BF02820446]

## Spectral Deformations of One-Dimensional Schrödinger

### Fritz Gesztesy, Barry Simon, and Gerald Teschl

We provide a complete spectral characterization of a new method of constructing
isospectral (in fact, unitary) deformations of general Schrödinger operators

*H=-d*in^{2}/dx^{2}+V*L*. Our technique is connected to Dirichlet data, that is, the spectrum of the operator^{2}(ℝ)*H*on^{D}*L*with a Dirichlet boundary condition at^{2}((-∞, x_{0}))⊕ L^{2}((x_{0}, ∞))*x*. The transformation moves a single eigenvalue of_{0}*H*and perhaps flips which side of^{D}*x*the eigenvalue lives. On the remainder of the spectrum, the transformation is realized by a unitary operator. For cases such as_{0}*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→∞*.
** MSC91:** Primary 34B24, 34L05; Secondary 34B20, 47A10

**Keywords:** *Spectral deformations, Sturm-Liouville operators, isospectral*

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