Spectral theory for non-self-adjoint differential operators

非自伴微分算子的谱理论

• 批准号: 387671260
• 负责人: Professor Dr. Carsten Trunk
• 金额: --
• 依托单位: Arbeitsgruppe Analysis und Systemtheorie
• 依托单位国家: 德国
• 项目类别: Research Grants
• 财政年份: 2017
• 资助国家: 德国
• 起止时间: 2016-12-31 至 2020-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/387671260?language=en
• 关键词: Spectral; theory; non; self; adjoint

A central topic in modern theoretical physics is the inconsistency between the standard model and general relativity. The quest for a Grand Unified Theory led to the development of new mathematical models, for instance in Quantum Mechanical non-self-adjoint operators were considered instead of self-adjoint.A prominent class of non-self-adjoint operators are self-adjoint operators in Krein spaces and PT-symmetric operators. Unlike self-adjoint operators in Hilbert spaces, the spectral properties of PT-symmetric operators are fundamentally different, for example, they may have non-real spectrum with accumulation points.In the present project we investigate the spectral properties of non-self-adjoint differential operators. The focus of this research program is on the spectral properties of indefinite singular Sturm-Liouville operators and indefinite elliptic differential operators. One aim is to localize the position of the non-real point spectrum. Furthermore, we want to study the accumulation of the point spectrum against the essential spectrum. In addition to the so-called WKB approximation for solutions of second order differential equations, we use oscillation techniques for Sturm-Liouville operators and perturbation results for operators in Krein spaces. Moreover, for PT--symmetric quantum mechanics we develop by means of the WKB approximation criteria for the limit point and the limit circle case for Sturm-Liouville operators with complex-valued coefficients.The present proposal is a continuation of the current Project (sign removed). Initially, the project (sign removed) was designed for 3 years, but finally it was approved for 18 month. Here we apply for another 18 month in order to complete all the intended research topics of the project (sign removed). We added two new research topics (classification of the limit point and limit circle case for Sturm-Liouville operators with complex coefficient, and indefinite elliptic differential operators), which seems to us very natural and closelyconnected to the results we already obtained in the first (18 month) period of the project (sign removed).

现代理论物理学的一个中心问题是标准模型和广义相对论之间的不一致。对大统一理论的追求导致了新的数学模型的发展，例如在量子力学中，非自伴随算子被认为是而不是自伴随算子。一类突出的非自伴随算子是Krein空间中的自伴随算子和pt对称算子。与Hilbert空间中的自伴随算子不同，pt对称算子的谱性质有本质上的不同，例如，它们可能具有带累加点的非实谱。本文研究了非自伴随微分算子的谱性质。研究了不定奇异Sturm-Liouville算子和不定椭圆微分算子的谱性质。一个目的是定位非实点谱的位置。此外，我们还想研究点谱对本质谱的累积。除了二阶微分方程解的WKB近似外，我们还使用了Sturm-Liouville算子的振荡技术和Krein空间中算子的微扰结果。此外，对于具有复值系数的Sturm-Liouville算子的极限点和极限环情况，我们利用WKB近似准则建立了PT对称量子力学的极限点和极限环情况。本提案是当前项目的延续（已删除符号）。最初，该项目（移除标志）设计了3年，但最终批准了18个月。在此，我们申请另外18个月，以完成项目的所有预期研究课题（删除签名）。我们增加了两个新的研究课题（复系数Sturm-Liouville算子的极限点和极限圆情况的分类，以及不定椭圆微分算子），这对我们来说是非常自然的，并且与我们在项目的第一个（18个月）期间（删除符号）已经获得的结果密切相关。