Spectral theory and optimal harvesting

谱理论在过去的10年里，D.B.Hinton教授和我发表了8篇关于高阶微分算子的论文[例如B.H.2011,2011]。在这些文献中，证明了如果系数满足温和的正则性假设，则谱是绝对连续的。从Sturm-Liouville算子得知，这些假设几乎是可选的。其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是自伴算子的M-矩阵。对常系数算符的应用表明，这种方法对谱理论几乎没有用处。因此，辛顿和我设计了一种新的方法，这种方法也可以确定光谱类型。对于系数几乎为常的哈密顿系统，这种方法似乎很有前途。然而，这需要对特征多项式P(lambda，z)和代数曲线P(lambda，z)=0进行详细分析。此外，我们需要确定预解式的良好表示，这允许进行精确估计，以便导出与M-矩阵的联系并导出谱型的性质。几年前，我在田纳西大学做了一次关于鱼的最佳收获的演讲，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.Lenhart教授和S.Ding博士的合作。其目的是继续这项工作，特别是鳕鱼和鲱鱼，并确定最佳可持续产量并分析相应的最优控制问题。

Spectral TheoryIn 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. Application to constant coefficient operators show that this technique is hardly useful for spectral theory. Thus Hinton and I have 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 characteristic polynomial P(lambda, z) and the algebraic curves P(lambda, z) = 0. Moreover we need to determine a good representation of the resolvent, which allows, precise estimates in order to derive the connection with the M-matrix and derive properties of the spectral type.Optimal harvestingSome years ago I presented a talk on optimal havesting 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.