Reflectionless discrete Schr\"odinger operators are spectrally atypical

Abstract

We prove that, if an isospectral torus contains a discrete Schr\"odinger 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 Schr\"odinger operator. We also show that the only reflectionless discrete Schr\"odinger operators having zero, one, or two spectral gaps are periodic.