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

弹性结构(桥梁、机身、自行车车架)的固有振动频率决定其机械性能。为了避免机械故障，应设计出可能由外部载荷激励的频率。钠路灯发出的黄色光(现在主要被LCD取代)也是自然频率的一个例子，由定律E=hf确定，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确定的：对于薛定谔方程，有效地通过解决被称为本征值问题的东西。本征值问题在如此多不同的应用中的重要性导致了大量的公共和工业资金被花费在适当的软件的开发上，如ARPACK和LAPACK。它们处理本征参数呈线性出现的问题。同样，已经做出了巨大的努力来开发理论工具，以在纯数学的水平上理解本征值所在。它们是真实的还是复杂的？它们是在单位圆盘内，还是在单位圆盘外？他们有积极的真实部分还是消极的真实部分？数值范围是进行此类估计的最广泛使用的工具之一。然而，特征参数往往以多项式或有理形式出现。当这种情况发生时，一种常见的方法是应用巧妙的变换来线性化问题。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，这些估计以问题的原始形式处理并获得更好的估计。事实上，我们甚至提出，在大多数情况下，人们应该采取与通常方法相反的方法：将原始的线性问题转化为一族有理问题，并通过考虑整个家庭的估计，获得比直接处理原始估计更好的估计。事实上，我们将证明，在理论上，这种方法可以产生关于原始问题的特征值以及更广泛地说，关于谱的*所有*信息。

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.