Mathematical Sciences: Spectral Problems and Inverse Spectral Problems

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This project focuses on research concerned with three related areas of mathematical analysis. In each case the work continues earlier investigations of a similar nature. The first objectives derive from efforts to apply inverse scattering results for selfadjoint ordinary differential operators. These operators arise naturally in the analysis of nonlinear wave equations which are known to be integrable. Although explicit solutions can be obtained by inverse methods, the long time behavior of solutions is not well understood, and is far from complete. The difficulty lies in the sign of the group velocity which does not stay fixed independent of energy, so disturbances move out in both directions. This means that free solutions are not good approximations for the full scattering solutions of the associated linear problem. A good parametrix for the problem with general initial data will be sought. Work will also be carried out on matrix inversion problems. The connection with differential equations is made through recent discoveries that diagonalization of self-adjoint matrices can be viewed in the context of integrable Hamiltonian systems. It was further observed that the QR, LU and Cholesky algorithms for nonsymmetric matrices also fit into this context. Moreover, the computational potential for this approach is most promising. Work will continue in studying the long term asymptotics of the Rayleigh shifted QR algorithm for nonsymmetric matrices. A third topic of concern deals with spectral problems for Schrodinger operators. An important question arising in the theory of crystal coloration revolves around the existence of eigenvalues of a perturbed Schrodinger operator. More precisely, one is interested in determining the extent to which eigenvalues of a perturbed operator can materialize within gaps in the spectrum of the original operator. The phenomenon is known to occur, the issue is to find suitable information on the distribution of the spectrum