The m-functions of discrete Schrödinger operators are sparse compared to those for Jacobi operators

Hur, Injo

doi:10.1016/j.jde.2017.09.009

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“…We also would be remiss not to mention the recent work [13], wherein the methods of canonical systems are used to prove a result on the sparsity of m-functions of DSOs amongst those for Jacobi operators.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Reflectionless Discrete Schrödinger Operators are Spectrally Atypical

VandenBoom

2017

Commun. Math. Phys.

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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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Section: Introduction and Main Resultsmentioning

confidence: 99%