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=-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→∞.

MSC91: Primary 34B24, 34L05; Secondary 34B20, 47A10
Keywords: Spectral deformations, Sturm-Liouville operators, isospectral

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