Spectral theory for non-self-adjoint differential operators

现代理论物理学的一个中心话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，非自伴算子被考虑而不是自伴算子。一类重要的非自伴算子是Krein空间中的自伴算子和PT-对称算子。与Hilbert空间中的自伴算子不同，PT-对称算子的谱性质有本质上的不同，例如，它们可能具有具有累积点的非实谱。本研究的重点是不定奇异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频谱的位置。此外，我们还想研究点谱相对于本质谱的累积。除了对二阶微分方程解的所谓WKB逼近，我们还使用了Sturm-Liouville算子的振动技巧和Krein空间中算子的摄动结果。此外，对于PT-对称量子力学，我们利用复系数Sturm-Liouville算子的极限点的WKB近似准则和极限圆情形来发展。本建议是当前计划(去掉符号)的继续。最初，该项目(移除标志)的设计期限为3年，但最终被批准为18个月。在这里，我们再申请18个月，以完成项目的所有预期研究课题(去掉符号)。我们增加了两个新的研究课题(具有复系数的Sturm-Liouville算子的极限点和极限圆的分类，以及不定椭圆微分算子)，这对我们来说是非常自然的，与我们在项目的第一个(18个月)期间(去掉符号)已经得到的结果密切相关。

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).