Smith D.A. Spectral theory of ordinary and partial linear differential operators on finite intervals

Department of Mathematics & Statistics, University of Reading, UK. - 2011. - 167 pages.A new, uni ed transform method for boundary value problems on linear and integrable nonlinear partial di erential equations was recently introduced by Fokas. We consider initialboundary value problems for linear, constant-coe cient evolution equations of arbitrary order on a nite domain. We use Fokas' method to fully characterise well-posed problems. For odd order problems with non-Robin boundary conditions we identify su cient conditions that may be checked using a simple combinatorial argument without the need for any analysis. We derive similar conditions for the existence of a series representation for the solution to a well-posed problem.
We also discuss the spectral theory of the associated linear two-point ordinary di erential operator. We give new conditions for the eigenfunctions to form a complete system, characterised in terms of initial-boundary value problems.