Spectral Theory of Differential Operators with Complex Coefficients

韦尔理论的成功以及物理学中的问题很快导致了将这一理论扩展到几个方向的尝试。它们是高阶对称算子、谱算子和复系数的Sturm-Liouville算子。谱微分算子的关键问题是证明给定的算子是谱的。因此，这一理论几乎没有超出常系数算子的范畴。Huige[7]推广了Schwartz的早期工作，利用傅里叶变换证明了半直线上常系数微分算子的谱性，并通过允许具有快速衰减系数的扰动进行了推广。俄罗斯学派、Neumark等人[8]和Sims[9]等人研究的具有复系数的Sturm-Liouville算子就是这种类型的算子。然而，这些算子主要是用傅里叶分析和复轮廓积分来分析本征函数展开的，困难主要来自高阶奇点及其幂零求和和在本质谱中的谱奇性。关于这些，我们一无所知。M-函数的作用同样不清楚，特别是m在奇点附近的行为。与Sturm-Liouville算子的大量例子相比，没有(！)已知的例子，展示了这种奇怪的行为。因此，这个研究方向从那时起就一直处于休眠状态。归根结底，是谱定理的缺失，使得非自伴算子理论变得复杂得多。然而，如果特征函数的形式近似地是已知的，则可以说更多。Behncke推广了Levinson提出的渐近积分理论，并使其适用于谱问题[1]。M-矩阵和渐近积分是分析对称微分算子谱理论的理想工具。在总共8篇论文中，Hinton和Behncke研究了高阶微分算子谱理论的许多方面。我们的主要结果表明，系数不太振荡的哈密顿系统只有绝对连续的本质谱，其特征值至多累加在特征多项式的双根处[4]。对于对称哈密顿系统，本质谱一般是区间序列，其边界由特征多项式的判别式确定。对于一般的C对称哈密顿量，谱将是一个相当一般的代数曲线。

The success of Weyls theory as well as problems from physics soon led to attempts to extend this theory into several directions. These were higher order symmetric operators, spectral operators and Sturm-Liouville operators with complex coefficients. The key problem with spectral differential operators is to show that a given operator is spectral. Thus the theory hardly developed beyond constant coefficient operators. Huige [7], extending earlier work of Schwartz, used the Fourier transform to prove the spectrality of constant coefficient differential operators on the half line and he extended this by allowing perturbations with rapidly decaying coefficients. The Sturm-Liouville operators with complex coefficients studied by the Russian school, Neumark et al [8] and Sims [9] and a few others were just of this type. However these operators were mostly analyzed in respect to an eigenfunction expansion by using Fourier analysis as well complex contour integrals.The difficulties stem mostly from the higher order singularities with its nilpotent summands and the spectral singularities in the essential spectrum. Of these nothing is known. The role of the m-function is likewise not clear, in particular the behavior of m near the singularities. Compared to a wealth of examples of Sturm-Liouville operators, there are no (!) examples known, which exhibit this strange behaviour. Thus this direction of study has been dormant ever since. Ultimately it is the absence of the spectral theorem, which makes the non self-adjoint operator theory so much more complicated. However, if the form of the eigenfunctions is approximately known, more can be said. Behncke extended the theory of asymptotic integration, introduced by Levinson and made it applicable to spectral problems [1]. The m-matrix as well as asymptotic integration is thus ideal tools to analyze the spectral theory of symmetric differential operators. In altogether 8 papers Hinton and Behncke have studied many aspects of the spectral theory of higher order differential operators. Our main result states that Hamiltonian systems with coefficients, which are not too oscillatory have only absolutely continuous essential spectrum with eigenvalues accumulating at the double roots of the characteristic polynomial at most [4]. For symmetric Hamiltonian systems the essential spectrum will in general be a sequence of intervals for which the boundary is determined by the discriminant of the characteristic polynomial. For general C-symmetric Hamiltonians the spectrum will be a rather general algebraic curve.