A new paradigm for spectral localisation of operator pencils and analytic operator-valued functions

弹性结构（桥梁、机身、自行车架）的固有振动频率决定了其力学性能。为了避免机械故障，应该设计出可能被外部负载激发的频率。钠路灯发出的黄光（现在大部分被lcd取代）也是固有频率的一个例子，由E = h f定律决定，其中h是普朗克常数，E是通过找到钠原子中电子的允许能级来决定的：实际上，通过解决薛定谔方程的特征值问题。特征值问题在许多不同应用中的重要性导致大量的公共和工业资金被用于开发适当的软件，如ARPACK和LAPACK。这些方法处理特征参数线性出现的问题。同样，人们也付出了巨大的努力来开发理论工具，以便在纯数学层面上理解特征值的所在。它们是真实的还是复杂的？它们是在单位圆盘内部还是外部？它们的实部是正的还是负的？数值范围是进行此类估计最广泛使用的工具之一。然而，特征参数往往以多项式或理性的形式出现。当这种情况发生时，常用的方法是应用巧妙的变换将问题线性化。在这个建议中，我们打算通过证明定理和构造估计来证明这通常不是最好的方法，这些估计以其原始形式处理问题并获得更好的估计。事实上，我们甚至提出，在大多数情况下，人们应该做与通常方法相反的事情：将原始线性问题转化为一个有理问题族，并通过考虑整个族的估计，获得比直接处理原始问题更好的估计。事实上，我们将证明，这种方法在理论上可以产生关于特征值的所有信息，更一般地说，原始问题的谱。

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.