Spectral Theory of Differential Operators with Complex Coefficients

Weyls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。这些是高阶对称算子，谱算子和具有复系数的Sturm-Liouville算子。谱微分算子的关键问题是证明给定算子是谱算子。因此，该理论几乎没有超出常系数算符的范围。Huige[7]，扩展了Schwartz早期的工作，用傅里叶变换证明了常系数微分算子在半线上的频谱性，他通过允许具有快速衰减系数的微扰扩展了这一理论。俄罗斯学派、Neumark et al[8]和Sims[9]等人研究的复系数Sturm-Liouville算子就属于这种类型。然而，这些算子大多是用傅立叶分析和复轮廓积分来分析特征函数展开的。困难主要来自于具有幂零和的高阶奇异点和本质谱中的谱奇异点。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，表现出这种奇怪的行为。因此，从那时起，这个研究方向就一直处于休眠状态。最终是谱定理的缺失，使得非自伴随算子理论变得如此复杂。然而，如果特征函数的形式近似已知，则可以说得更多。Behncke推广了Levinson引入的渐近积分理论，并将其应用于谱问题[1]。因此，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的谱理论的许多方面。我们的主要结果表明，不太振荡的带系数哈密顿系统只有绝对连续的本质谱，特征值在特征多项式的二重根处最多累积到[4]。对于对称哈密顿系统，本质谱通常是一个区间序列，其边界由特征多项式的判别式决定。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.