We consider a slowly decaying oscillatory potential such that the corresponding 1D Schrdinger operator has a positive eigenvalue embedded into the absolutely continuous spectrum. This potential does not fall into a known class of initial data for which the Cauchy problem for the Korteweg–de Vries (KdV) equation can be solved by the inverse scattering transform. We nevertheless show that the KdV equation with our potential does admit a closed form classical solution in terms of Hankel operators. Comparing with rapidly decaying initial data our solution gains a new term responsible for the positive eigenvalue. To some extent this term resembles a positon (singular) solution but remains bounded. Our approach is based upon certain limiting arguments and techniques of Hankel operators.

我们考虑缓慢衰减的振荡势，使得相应的 1D Schrdinger 算子具有嵌入绝对连续谱的正特征值。这种势不属于已知的一类初始数据，对于 Korteweg-de Vries (KdV) 方程的柯西问题可以通过逆散射变换求解。尽管如此，我们证明了具有我们潜力的 KdV 方程确实承认了 Hankel 算子方面的封闭形式经典解。与快速衰减的初始数据相比，我们的解决方案获得了一个负责正特征值的新项。在某种程度上，该术语类似于位置（奇异）解，但仍然是有界的。我们的方法基于 Hankel 算子的某些限制参数和技术。