Abstract: If the parameter ε², which measures the thickness-to-rise of the sliell, is small, the axismnnetrie polar dimpling oj.shallow.spherical.shell due to quadratic pressure distribution i.s dynamic instability, i.e., a small perturbation can change il to an asymmetric polar dimple mode.In two cases, the problem can be reduced to an eigenvalue problem Tw_n=c_n+w_n, where T can approximately be reduced to a Sturm-Liouvi/le operator if ε²≤1The existence of at least one real eigenvalue of T, which means that the axisyntmetric polar dimpling is dynamically unstable, i.s proved by spectral theorem or Hilbert theorem.Furthermore, an eigenfunction, which represents one of the asymmetric modes of the unstable dimple shell, belonging to an eigenvalue of T, is found.