@inbook{8767ffeea26b471dba003b6ce92e5bd8,

title = "Non-periodic one-gap potentials in quantum mechanics",

abstract = "We construct a broad class of bounded potentials of the one-dimensional Schr{\"o}dinger operator that have the same spectral structure as periodic finite-gap potentials, but that are neither periodic nor quasi-periodic. Such potentials, which we call primitive, are non-uniquely parametrized by a pair of positive H{\"o}lder continuous functions defined on the allowed bands. Primitive potentials are constructed as solutions of a system of singular integral equations, which can be efficiently solved numerically. Simulations show that these potentials can have a disordered structure. Primitive potentials generate a broad class of bounded non-vanishing solutions of the KdV hierarchy, and we interpret them as an example of integrable turbulence in the framework of the KdV equation.",

keywords = "Integrable turbulence, KdV equation, Riemann–Hilbert problem, Schr{\"o}dinger operator",

author = "Dmitry Zakharov and Vladimir Zakharov",

note = "Funding Information: The authors would like to thank Sergey Dyachenko for the numerical experiments. The first author is thankful for the support of NSF grant DMS-1716822. The second author is grateful for the support of NSF grant DMS-1715323 and RSF grant No. 14-22-00174. Publisher Copyright: {\textcopyright} 2018, Springer International Publishing AG.",

year = "2018",

doi = "10.1007/978-3-319-63594-1_22",

language = "English (US)",

series = "Trends in Mathematics",

publisher = "Springer International Publishing",

number = "9783319635934",

pages = "221--233",

booktitle = "Trends in Mathematics",

edition = "9783319635934",

}