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We study the stability of singly- and doubly-quantized vortex states ofharmonically trapped dipolar Bose-Einstein Condensates (BECs) by calculatingthe low-lying excitations of these condensates. We map the dynamical stabilityof these vortices as functions of the dipole-dipole interaction strength andtrap geometry by finding where their excitations have purely real energyeigenvalues. In contrast to BECs with purely contact interactions, we find thatdipolar BECs in singly-quantized vortex states go unstable to modes with anincreasing number of angular and radial nodes for more oblate trap aspectratios, corresponding to \emph{local} collapse that occurs on a characteristiclength scale. Additionally, we find that dipolar BECs in doubly-quantizedvortex states are unstable to decay into a different topological state (withtwo singly-quantized vortices) for all interaction strengths when the trapgeometry is sufficiently prolate to make the dipoles attractive, and in windowsof interaction strength when the trap geometry is sufficiently oblate to makethe dipoles repulsive.

Stability and Excitations of a Dipolar Bose-Einstein Condensate with a Vortex