  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-Amp\`ere system. We formulate the
magnetized Vlasov-Amp\`ere system as a Schr\"odinger equation with a
selfadjoint magnetized Vlasov-Amp\`ere operator in the Hilbert space of states
with finite energy. The magnetized Vlasov-Amp\`ere operator has a complete set
of orthonormal eigenfunctions, that include the Bernstein modes. The expansion
of the solution of the magnetized Vlasov-Amp\`ere 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-Amp\`ere 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.
