Journal d’Analyse Mathématique

, Volume 70, Issue 1, pp 267–324

Spectral deformations of one-dimensional Schrödinger operators


  • F. Gesztesy
    • Department of MathematicsUniversity of Missouri
  • B. Simon
    • Division of Physics, Mathematics and AstronomyCalifornia Institute of Technology
  • G. Teschl
    • Institut für Reine und Angewandte MathematikRWTH Aachen

DOI: 10.1007/BF02820446

Cite this article as:
Gesztesy, F., Simon, B. & Teschl, G. J. Anal. Math. (1996) 70: 267. doi:10.1007/BF02820446


We provide a complete spectral characterization of a new method of constructing isospectral (in fact, unitary) deformations of general Schrödinger operatorsH=−d2/dx2+V in\(H = - d^2 /dx^2 + V in \mathcal{L}^2 (\mathbb{R})\). Our technique is connected to Dirichlet data, that is, the spectrum of the operatorHD onL2((−∞,x0)) ⊕L2((x0, ∞)) with a Dirichlet boundary condition atx0. The transformation moves a single eigenvalue ofHD and perhaps flips which side ofx0 the eigenvalue lives. On the remainder of the spectrum, the transformation is realized by a unitary operator. For cases such asV(x)→∞ as |x|→∞, whereV is uniquely determined by the spectrum ofH and the Dirichlet data, our result implies that the specific Dirichlet data allowed are determined only by the asymptotics asE→∞.

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© Hebrew University of Jerusalem 1996