Spectral Theory of Differential Operators Optimal Harvesting of Fish

项目：I微分算子谱理论II鱼的最优可持续收获I）在过去的10年里，教授D.B.欣顿和我已经发表了8篇关于高阶微分算子的论文[例如B.H. 2011年，2011年]。在这些文件中，它表明，谱是绝对连续的，如果系数满足温和的正则性假设。从Sturm-Liouville算子可知，这些假设几乎是可选的。主要技术是渐近积分和M-矩阵。最近Brown，Evans和Plum利用Weyl圆的方法构造了不一定自伴算子的M-矩阵。常系数算子的应用表明，这种技术几乎是有用的谱理论。因此，欣顿和我设计了一种新的方法，它也将允许确定光谱类型。对于几乎常系数的Hamilton系统，这种方法似乎是有前途的。然而，这需要详细分析特征多项式P（lambda，z）和代数曲线P（lambda，z）= 0。M矩阵的方法通常基于一个规则端点。然而，许多数学物理问题都有两个奇异的端点，无穷大和在0处的贝塞尔型奇点。一般来说，这个问题是处理的分解方法，导致光谱的更高的多重性[B.H. 2010年]。然而，如果左偏算子只有离散谱，这不再成立。在这种情况下，全M-矩阵不再是Nevanlinna型的。这类问题将单独研究。几年前，我在田纳西大学发表了一个关于鱼类最佳捕捞的演讲，其中考虑了渔网的宽度和鱼类种群的年龄结构。这导致了与S教授的联合工作。伦哈特和S博士。丁打算继续这项工作，特别是鳕鱼和鲱鱼的工作，确定最佳可持续产量，并分析相应的最佳控制问题。

Projekt: I Spectral Theory of Differential Operators II Optimal Sustainable Harvesting of Fish I) In the last 10 years Professor D.B. Hinton and I have published 8 papers on higher order differential operators [e.g. B.H. 2011, 2011]. In these papers it is shown that the spectra are absolutely continuous, if the coefficients satisfy mild regularity assumptions. It is known from Sturm-Liouville operators that these assumptions are almost optional. The main techniques are asymptotic integration and the M-matrix. Recently Brown, Evans, and Plum have constructed the M-matrix for not necessarily selfadjoint operators, by using the method of Weyl circles. Applications to constant coefficient operators show that this technique is hardly useful for spectral theory. Thus Hinton and Ihave devised a new method, which will also permit to determine the spectral type. For Hamiltonian systems with almost constant coefficients this method seems to be promising. This, however, requires a detailed analysis of the characteristic polynomial P(lambda, z) and the algebraic curves P(lambda, z) = 0. The method of the M-matrix is typically based on one regular endpoint. Many problems of mathematical physiscs, however, lead to two sigular endpoints, infinity and a Bessel type singularity at 0. Generally this problem is treated by the decomposition method, leading to spectra of higher multiplicity [B.H. 2010]. If, however, the left partial operator has only discrete spectrum, this does not hold any more. In this case to total M-matrix is not any more of Nevanlinna type. Problems of this type will be studied separately. II) Some years ago I presented a talk on optimal harvesting of fish at the University of Tennessee, which takes into account the width of the fishing nets and age structure of the fish population. This led to a joint work with Prof. S. Lenhart and Dr. S. Ding. it is intended to continue this work in particular for cod and herring and to determine the optimal sustainable yield and analyze the corresponding optimal control problems.