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Spectral deformations of one-dimensional Schrödinger operators

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Abstract

We provide a complete spectral characterization of a new method of constructing isospectral (in fact, unitary) deformations of general Schrödinger operatorsH=−d 2/dx 2+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 operatorH D onL 2((−∞,x 0)) ⊕L 2((x 0, ∞)) with a Dirichlet boundary condition atx 0. The transformation moves a single eigenvalue ofH D and perhaps flips which side ofx 0 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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Authors and Affiliations

  1. Department of Mathematics, University of Missouri, 65211, Columbia, MO, USA

    F. Gesztesy

  2. Division of Physics, Mathematics and Astronomy, California Institute of Technology, 91125, Pasadena, CA, USA

    B. Simon

  3. Institut für Reine und Angewandte Mathematik, RWTH Aachen, 52056, Aachen, Germany

    G. Teschl

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  1. F. Gesztesy
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  2. B. Simon
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  3. G. Teschl
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This material is based upon work supported by the National Science Foundation under Grant No. DMS-9401491. The Government has certain rights in this material.

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Gesztesy, F., Simon, B. & Teschl, G. Spectral deformations of one-dimensional Schrödinger operators. J. Anal. Math. 70, 267–324 (1996). https://doi.org/10.1007/BF02820446

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