The linear theory stability of different collisionless plasma sheath structures, including the classic sheath, inverse sheath and space-charge limited (SCL) sheath, is investigated as a typical eigenvalue problem. The three background plasma sheaths formed between a Maxwellian plasma source and a dielectric wall with a fully self-consistent secondary electron emission condition are solved by recent developed 1D3V (one-dimensional space and three-dimensional velocities), steady-state, collisionless kinetic sheath model, within a wide range of Maxwellian plasma electron temperature
. Then, the eigenvalue equations of sheath plasma fluctuations through the three sheaths are numerically solved, and the corresponding damping and growth rates
are found: (i) under the classic sheath structure (i.e.
(the first threshold)), there are three damping solutions (i.e.
) for most cases, but there is only one growth-rate solution
due to the inhomogeneity of sheath being very weak; (ii) under the inverse sheath structure, which arises when
, there are no background ions in the sheath so that the fluctuations are stable; (iii) under the SCL sheath conditions (i.e.
, the second threshold), the obvious ion streaming through the sheath region again emerges and the three damping solutions are again found.