  The physics of the pulsar inner magnetosphere remains poorly constrained by
observations. Although about 2000 pulsars have been discovered to date, little
is known about their emission mechanism. Large vacuum gaps probably exist and a
non-neutral plasma made of electrons in some regions and of positrons in some
other regions fills space to form an electrosphere. The purpose of this work is
to study the stability properties of the differentially rotating equatorial
disk in the pulsar's electrosphere for which the magnetic field is assumed to
be dipolar. In contrast to previous studies, the magnetic field is not
restricted to be uniform. A pseudo-spectral Galerkin method using Tchebyshev
polynomials expansion is developed to compute the spectrum of the diocotron
instability in a non-neutral plasma column confined between two cylindrically
conducting walls. Moreover, the inner wall carries a given charge per unit
length in order to account for the presence of a charged neutron star at the
centre of the electrosphere. We show several eigenfunctions and eigenspectra
obtained for different initial density profiles and electromagnetic field
configurations useful for laboratory plasmas. The algorithm is very efficient
in computing the fastest growing modes. Applications to a cylindrical
electrosphere are also shown for several differential rotation profiles. It is
found that the growth rates of the diocotron instability are of the same order
of magnitude as the rotation rate.
