Spectral theory and optimal harvesting

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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.