Direkte und inverse Spektraltheorie periodischer Systeme

我们计划在以下情况下发展正反谱理论：A)矩阵Hill算子P=-d2/dx2 + Q，其中Q是一个对称的1周期N x N矩阵。在这种情况下，第一步是直接谱理论的构建：定义和分析Lyapunov函数，并将准动量（和态密度）理论发展为保角映射。第二步是根据谱数据获得势能的双面估计，并推导出反问题的结果。B)周期差分Schrödinger算子。我们计划根据谱数据获得势的双面估计，并解决表征问题：哪一组实区间可以是周期差的谱Schrödinger算子？C)加权周期算子T=-p-2d/dx(p2d/dx) + q为零或偶变号。这些问题与Camassa-Holm方程有关。D)紧支撑势的半线上的Schrödinger算子有无限多个共振。表征的问题是广泛开放的：哪一组点是这样一个算子的共振集？我们计划在某些特殊情况下解决这个问题。E)谱渐近性。我们计划确定某些特殊情况下伪微分（周期和规范周期）问题的半经典谱渐近性。

We plan to develop direct and inverse spectral theory in the following cases: A) The matrix Hill operator P=-d2/dx2 + Q, where Q is a symmetric 1-periodic N x N matrix. In this case a first step is the construction of the direct spectral theory: define and analyze the Lyapunov function and develop the theory of the quasimomentum (and the density of states) as a conformal mapping. A second step is to obtain double-sided estimates of the potential in terms of spectral data and to derive consequences for the inverse problem. B) The periodic difference Schrödinger operator. We plan to obtain double-sided estimates of the potential in terms of spectral data and to solve the problem of characterization: Which sets of real intervals can be the spectrum of periodic difference Schrödinger operator? C) The weighted periodic operator T=-p-2d/dx(p2d/dx) + q has zeroes or even changes sign. These problems are associated with the Camassa-Holm equation. D) The Schrödinger operator on the half-line with a compactly supported potential has infinitely many resonances. The problem of characterization is widely open: Which sets of points are the sets of resonances for such an operator? We plan to solve this problem for certain special situations. E) Spectral asymptotics. We plan to determine semiclassical spectral asymptotics for pseudodifferential (periodic and gauge-periodic) problems for certain special situations.