Singular soliton operators and indefinite metrics
Singular soliton operators and indefinite metrics
复制标题
奇异孤子算子和不定度量
DOI：
10.1007/s00574-013-0035-5
复制
发表时间：
2011
期刊：
影响因子：
--
通讯作者：
S. Novikov
中科院分区：
文献类型：
--
作者：
P. Grinevich;S. Novikov
我们考虑奇异实二阶一维薛定谔算子，使得特征值问题的所有局部解对于所有 λ 都是 x 亚纯的。所有代几何势（即“奇异有限间隙”和“奇异孤子”）都满足这个条件。谱理论是针对具有奇点的函数类中的周期性和快速递减势而构建的：正如本作者发现的那样，相应的算子关于某些自然不定内积是对称的。在这两种情况（周期性和快速减少）中，它都有有限数量的负平方。 KdV 层次结构提供的时间动态保留了这个数字。具有良好乘性的黎曼曲面上的傅里叶变换（R-傅里叶变换）的正确模拟是该理论的部分案例。在这种情况下，势在 x = 0 处有一个极点，且渐进 u ∼ g(g + 1)/x2。这里g是光谱曲线的亏格。
We consider singular real second order 1D Schrödinger operators such that all local solutions to the eigenvalue problems are x-meromorphic for all λ. All algebrogeometrical potentials (i.e. “singular finite-gap” and “singular solitons”) satisfy to this condition. A Spectral Theory is constructed for the periodic and rapidly decreasing potentials in the classes of functionswith singularities: The corresponding operators are symmetric with respect to some natural indefinite inner product as it was discovered by the present authors. It has a finite number of negative squares in the both (periodic and rapidly decreasing) cases. The time dynamics provided by the KdV hierarchy preserves this number. The right analog of Fourier Transform on Riemann Surfaces with good multiplicative properties (the R-Fourier Transform) is a partial case of this theory. The potential has a pole in this case at x = 0 with asymptotics u ∼ g(g + 1)/x2. Here g is the genus of spectral curve.