Motivated by the NLS and KdV linearizations near traveling waves, we study general forms of such operators. We prove a priori bounds on the unstable spectrum, by showing that if any unstable spectrum exists, it is contained in a strip around the real axis, with an explicit estimate of its width in terms of the potentials. To the best of our knowledge, this is the first result of this nature in the literature. We show that all sufficiently large (relative to the potential) pure imaginary eigenvalues are necessarily simple. In the case of spectral stability, we show optimal, at most polynomial in time, $L^2$ bounds for the associated semigroups generated such linearized operators. As it is for finite matrices, the power rate matches the maximal size of any Jordan block minus one.

受行波附近的NLS和KdV线性化的影响，我们研究了此类算子的一般形式。我们通过证明如果存在任何不稳定的光谱，将其包含在实轴周围的条带中，并通过势能明确估计其宽度，从而证明了该不稳定的光谱具有先验界限。据我们所知，这是这种性质在文献中的第一个结果。我们表明，所有足够大的（相对于潜在的）纯虚数特征值都必须是简单的。在频谱稳定的情况下，我们展示了最优的（最多为多项式）时间，${\mathrm{大号}}^2$关联的半群的边界生成了此类线性化算子。就像在有限矩阵中一样，功率比等于任何Jordan块的最大大小减去1。