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<p>We consider the Novikov problem, namely, the problem of describingthe global geometry of level lines of quasiperiodic functions on a plane,for a special class of two-dimensional potentials. Potentials of thisclass play an important role in the physics of two-dimensional systemsand are defined by superpositions of two periodic potentials withthe same rotational symmetry. For different orientations of the periodsof the original potentials, the resulting potential can have 4 quasiperiodesor be periodic. The main result of the paper is a proof that quasiperiodicpotentials of this class can have open level lines at only one energy level.This property brings these potentials closer to random potentials on a plane,as well as to potentials with 3 quasiperiodes possessing “chaotic” level lines.The paper also presents an estimate for the energy interval containing openlevel lines of periodic potentials arising at “magic” angles of rotationof the original potentials relative to each other.</p>

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On the Novikov Problem for Superposition of Periodic Potentials

  • 期刊论文

  • 发表日期：

    2026年2月27日

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  • Andrey Ya. Maltsev

摘要

We consider the Novikov problem, namely, the problem of describingthe global geometry of level lines of quasiperiodic functions on a plane,for a special class of two-dimensional potentials. Potentials of thisclass play an important role in the physics of two-dimensional systemsand are defined by superpositions of two periodic potentials withthe same rotational symmetry. For different orientations of the periodsof the original potentials, the resulting potential can have 4 quasiperiodesor be periodic. The main result of the paper is a proof that quasiperiodicpotentials of this class can have open level lines at only one energy level.This property brings these potentials closer to random potentials on a plane,as well as to potentials with 3 quasiperiodes possessing “chaotic” level lines.The paper also presents an estimate for the energy interval containing openlevel lines of periodic potentials arising at “magic” angles of rotationof the original potentials relative to each other.