| Abstract |
The natural frequencies of vibration of an elastic structure (bridge; airframe; bicycle frame) determine its mechanical behaviour. Frequencies which are likely to be excited by external loads should be designed out, in order to avoid mechanical failures. The yellow light emitted by sodium street-lamps (now mostly replaced by LCDs) was also an example of a natural frequency, determined by the law E = h f, where h is Planck's constant and E is determined by finding the allowed energy levels of electrons in a sodium atom: effectively, by solving something called an eigenvalue problem, for the Schroedinger equation. The importance of eigenvalue problems in so many different applications has resulted in large amounts of public and industrial money being spent on the development of appropriate software, such as ARPACK and LAPACK. These deal with problems in which the eigen-parameter appears linearly. Similarly, huge efforts have been made to develop theoretical tools to understand, at the level of pure mathematics, where eigenvalues lie. Are they real or complex? Do they lie inside the unit disc, or outside? Do they have positive or negative real part? The numerical range is one of the most widely used tools for making such estimates. However, eigen-parameters often appear polynomially or rationally. When this happens, a common approach has been to apply clever transformations to linearise the problem. In this proposal, we intend to show that this is generally not the best approach, by proving theorems and constructing estimates which treat the problem in its original form and get better estimates. In fact, we even propose that in most cases one should do the opposite of the usual approach: transform the original linear problem into a family of rational problems and, by considering estimates for the whole family together, obtain better estimates than can be obtained by direct treatment of the original. In fact, we shall show that this approach can, in theory, yield *all* information about the eigenvalues and, more generally, the spectrum, of the original problem. |
