奇异非自伴哈密顿算子谱的研究

The theory of Hamiltonian systems is one of the important fields in mordern mathematics with profund theories and wide applications. Research on spectral thoeries is an important portion of nonlinear sciences. In this project, we consider the following problems related to the spectrum of singular non-selfadjoint Hamiltonian operators: combining with the limit point-limit circle classification of Hamiltonian systems with complex coefficients, we will give the descriptions of the J-selfadjoint extension and Friedrichs extension of the minimal Hamiltonian operator; by defining suitable Hilbert space, we will give the spectral properties of Hamiltonian operator with transmission condition, we major in the distribution of eigen-values, oscillation of eigen-functions and the completeness of the system of eigen-functions; we will give new criteria for limit point case （strong limit point case）and limit circle case; for the intermidiate case, we will give the characterization of selfadjoint domain using the solutions with real-parameter, and discuss the realtion between the number of solutions with real-parameter and the existence of continuous spectrum; we will also investgate the spectra for mixed order matrix differential operators and Schr?dinger operators.These problems are the forlands of research in this field, which have widely use in optimal control, singular purturbation, computational mechanics, quantum mechanics, etc.

哈密顿系统理论是现代数学中一个既有深刻理论又有广泛应用的研究领域, 对谱理论的研究是非线性研究领域的一个重要组成部分. 本项目将研究奇异非自伴哈密顿算子的谱理论：结合复系数哈密顿系统的点、圆分类方法, 建立复系数哈密顿算子的J-自伴扩张以及Friedrichs扩张的解析描述；通过定义合适的Hilbert空间, 建立具有转移条件的哈密顿算子的谱理论, 着重在特征值的分布，特征函数的振动性及特征函数系的完备性得到新成果；最小哈密顿算子的亏指数的判别, 包括极限点型（强极限点型）、极限圆型的判别；对于出现中间亏指数的情形, 将利用实参解构造分离边界条件, 给出自伴域的刻画, 并讨论实参解的个数与连续谱存在的关系；混和阶微分算子的谱问题以及Schr?dinger算子的谱问题等. 这些理论的建立和完善, 将在非线性边值问题、最优控制理论、奇异摄动理论、计算数学和量子力学等多门学科的研究中起重要作用.

本项目研究奇异非自伴哈密顿算子的谱理论。按照计划，我们主要研究了奇异哈密顿算子的谱的一些性质，包括自伴扩张的描述，亏指数的判别，点谱的刻画，具有转移条件的哈密顿算子的谱等问题；我们还研究了几类哈密顿系统的预备解的振动性；借助于积分不等式，我们还研究了一些非线性微分方程解的性质，得到解的渐近性态；对与分数阶微分方程，我们研究了弱奇异核的积分不等式以及分数阶微分方程的振动性和定性性质。这些工作的完成，将极大丰富微分算子的谱理论，并在非线性边值问题、最优控制、计算力学和量子力学等学科的研究中起到重要作用。