Communications in Mathematical Physics

, Volume 359, Issue 2, pp 499–514 | Cite as

Reflectionless Discrete Schrödinger Operators are Spectrally Atypical

  • Tom VandenBoom


We prove that, if an isospectral torus contains a discrete Schrödinger operator with nonconstant potential, the shift dynamics on that torus cannot be minimal. Consequently, we specify a generic sense in which finite unions of nondegenerate closed intervals having capacity one are not the spectrum of any reflectionless discrete Schrödinger operator. We also show that the only reflectionless discrete Schrödinger operators having zero, one, or two spectral gaps are periodic.


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© Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Department of MathematicsRice UniversityHoustonUSA

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