实轴附近Titchmarsh–Weyl函数的渐近性及其在KdV层级中的应用
Asymptotics of Titchmarsh--Weyl functions near the real axis andan application to the KdV hierarchy
AI总结：
本文推导了一维Schrödinger与Dirac算子的Titchmarsh–Weyl函数在边界趋近谱的多项式速率区域内的高能渐近展开，验证了Kotani构造KdV流的高能假设，证明了对应势可生成KdV层级指定成员的全局经典解。
AI中文摘要：
我们建立了一维Schrödinger算子和Dirac算子的Titchmarsh–Weyl函数在边界以规定多项式速率趋近于谱的区域内的高能渐近展开式。对于具有有界导数的有界势，这些展开式在这些边界上保持一致，余项的明确损失由趋近速率决定。我们同时处理自伴Dirac算子和具有斜自伴势矩阵的非自伴Dirac算子，追踪两个标量Weyl坐标自然定义的不同半平面。作为Schrödinger展开式的应用，我们验证了Kotani构造KdV流中的高能假设：对于每个奇数整数p≥3，每个实值q∈W^{2p-1,∞}(ℝ)都会生成KdV层级中由(p+1)/2索引的成员的全局经典解。
英文摘要：
We establish high-energy asymptotic expansions of Titchmarsh--Weyl functions for one-dimensional Schrödinger and Dirac operators in regions whose boundaries approach the spectrum at a prescribed polynomial rate. For bounded potentials with bounded derivatives, the expansions remain uniform up to these boundaries, with an explicit loss in the remainder determined by the rate of approach. We treat both self-adjoint Dirac operators and non-self-adjoint Dirac operators with skew-adjoint potential matrices, keeping track of the distinct half-planes in which the two scalar Weyl coordinates are naturally defined. As an application of the Schrödinger expansion, we verify the high-energy hypothesis in Kotani's construction of KdV flows: for every odd integer $p\geq3$, each real-valued $q\in W^{2p-1,\infty}(\mathbb{R})$ generates a global classical solution of the member of the KdV hierarchy indexed by $(p+1)/2$.
