In this paper, we consider the spectral problem for the magnetic Schrödinger operator on the 2-D plane (x₁, x₂) with the constant magnetic field normal to this plane and with the potential V having the form of a harmonic oscillator in the direction x₁ and periodic with respect to variable x₂. Such a potential can be used for modeling a long molecule. We assume that the magnetic field is quite large, this allows us to make the averaging and to reduce the original problem to a spectral problem for a 1-D Schrödinger operator with effective periodic potential. Then we use semiclassical analysis to construct the band spectrum of this reduced operator, as well as that of the original 2-D magnetic Schroödinger operator.

中文翻译：

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具有一维和单向周期势的二维电磁薛定ding算子的平均和谱带

在本文中，我们考虑了二维Schrödinger算子在二维平面（x ₁，x ₂）上的频谱问题，该平面具有垂直于该平面的恒定磁场，并且电位V呈谐波振荡器形式。方向x ₁和关于变量x _2的周期。这样的潜力可用于建模长分子。我们假设磁场很大，这使我们可以进行平均，并将具有有效周期性电势的一维Schrödinger算子的原始问题简化为频谱问题。然后，我们使用半经典分析来构造该简化算子以及原始2D磁性Schroödinger算子的能谱。