Direkte und inverse Spektraltheorie periodischer Systeme

周期系统的正谱理论和逆谱理论

• 批准号: 5433032
• 负责人: Professor Dr. Jochen Brüning
• 金额: --
• 依托单位: Institut für Mathematik
• 依托单位国家: 德国
• 项目类别: Research Grants
• 财政年份: 2004
• 资助国家: 德国
• 起止时间: 2003-12-31 至 2007-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/5433032?language=en
• 关键词: Direkte; und; inverse; Spektraltheorie; periodischer

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.

我们计划在以下情况下发展正谱理论和逆谱理论： A) 矩阵 Hill 算子 P=-d2/dx2 + Q，其中 Q 是对称 1-周期 N x N 矩阵。在这种情况下，第一步是构建直接谱理论：定义和分析李雅普诺夫函数，并将拟动量（和态密度）理论发展为共形映射。第二步是根据光谱数据获得潜力的双面估计，并得出反演问题的结果。 B) 周期性差分薛定谔算子。我们计划获得谱数据势的双面估计，并解决表征问题：哪些实区间集合可以是周期差薛定谔算子的谱？ C) 加权周期算子 T=-p-2d/dx(p2d/dx) + q 为零甚至改变符号。这些问题与 Camassa-Holm 方程相关。 D) 具有紧支撑势的半线上的薛定谔算子有无限多个共振。表征问题是一个广泛开放的问题：哪些点集是此类算子的共振集？我们计划针对某些特殊情况解决这个问题。 E) 谱渐进。我们计划在某些特殊情况下确定伪微分（周期和规范周期）问题的半经典谱渐近。