We study the Bernstein–Landau paradox in the collisionless motion of an electrostatic plasma in the presence of a constant external magnetic field. The Bernstein–Landau paradox consists in that in the presence of the magnetic field, the electric field and the charge density fluctuation have an oscillatory behavior in time. This is radically different from Landau damping, in the case without magnetic field, where the electric field tends to zero for large times. We consider this problem from a new point of view. Instead of analyzing the linear magnetized Vlasov–Poisson system, as it is usually done, we study the linear magnetized Vlasov–Ampère system. We formulate the magnetized Vlasov–Ampère system as a Schrödinger equation with a selfadjoint magnetized Vlasov–Ampère operator in the Hilbert space of states with finite energy. The magnetized Vlasov–Ampère operator has a complete set of orthonormal eigenfunctions, that include the Bernstein modes. The expansion of the solution of the magnetized Vlasov–Ampère system in the eigenfunctions shows the oscillatory behavior in time. We prove the convergence of the expansion under optimal conditions, assuming only that the initial state has finite energy. This solves a problem that was recently posed in the literature. The Bernstein modes are not complete. To have a complete system it is necessary to add eigenfunctions that are associated with eigenvalues at all the integer multiples of the cyclotron frequency. These special plasma oscillations actually exist on their own, without the excitation of the other modes. In the limit when the magnetic fields goes to zero the spectrum of the magnetized Vlasov–Ampère operator changes drastically from pure point to absolutely continuous in the orthogonal complement to its kernel, due to a sharp change on its domain. This explains the Bernstein–Landau paradox. Furthermore, we present numerical simulations that illustrate the Bernstein–Landau paradox. In Appendix 2 we provide exact formulas for a family of time-independent solutions.

我们在存在恒定外部磁场的情况下研究了静电等离子体的无碰撞运动中的伯恩斯坦-朗道悖论。Bernstein-Landau悖论在于，在存在磁场的情况下，电场和电荷密度波动随时间具有振荡行为。在没有磁场的情况下，这与Landau阻尼有根本的不同，在这种情况下，电场会长时间趋于零。我们从新的角度考虑这个问题。与其像通常那样分析线性磁化的Vlasov-Poisson系统，不如研究线性磁化的Vlasov-Ampère系统。我们用有限能量的希尔伯特状态空间中的自伴磁化Vlasov-Ampère算子将磁化Vlasov-Ampère系统公式化为Schrödinger方程。磁化的Vlasov–Ampère算子具有一整套正交函数本征函数，其中包括Bernstein模式。磁化的Vlasov-Ampère系统解在本征函数中的展开表明了时间上的振荡行为。我们仅在初始状态具有有限能量的情况下证明了在最佳条件下扩张的收敛性。这解决了文献中最近提出的问题。伯恩斯坦模式不完整。为了拥有一个完整的系统，有必要在回旋加速器频率的所有整数倍处添加与本征值关联的本征函数。这些特殊的等离子体振荡实际上是独立存在的，没有其他模式的激发。在极限处，当磁场变为零时，由于磁畴的急剧变化，被磁化的Vlasov–Ampère算子的频谱从纯点急剧地变化到与其核正交的绝对连续。这解释了伯恩斯坦-朗道悖论。此外，我们提供了数值模拟，说明了伯恩斯坦-朗道悖论。在附录2中，我们提供了一系列与时间无关的解决方案的精确公式。