Stability of Electron Vortex Structures in Phase Space

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Abstract
The stability of one-dimensional, solitary vortex structures in the electron phase space (electron holes) is investigated. A linear eigenvalue problem is derived in the fluid limit and solved exactly, assuming that the normal mode is well represented by the lowest eigenstate of a properly chosen field operator. A new dispersion relation is obtained which exhibits purely growing solutions in two dimensions but only marginally stable solutions in one dimension. This explains the numerically well-known fact that vortex structures disappear in going from one to two dimensions.