On index stability of Noetherian differential operators in anisotropic Sobolev spaces

In this paper index stability is studied for differential operators perturbed by lower order terms. Conditions are established under which lower order terms do not affect the index of Noetherian operators, acting in anisotropic Sobolev spaces on $\mathbb{R}^m$. Obtained result is applied to investigation of Noethericity and index of semi-elliptic operators.