Periodic Spectral Problems

周期微分算子存在于许多物理和数学领域，研究它的谱性质是非常重要的。周期算符的谱具有带隙结构，即它们由可能由间隙分隔的闭合区间的集合组成。有一个著名的假设，称为Bethe-Sommerfeld猜想，它声称空隙的数量是有限的。这对于有电场的薛定谔算符是合理的。本项目的一个目的是在更一般的情况下证明这一猜想。这一解决方案预计需要使用伪微积分、格点几何和几何组合学。在所谓的积分态密度中，作用于非紧流形上的微分算符的一个重要的量化特征。该函数是光谱计数函数的自然类比。我们计划研究这个函数在大能量值下的行为，特别是证明态密度在能量的(负的和正的)幂中具有完全的渐近展开式。我们还计划研究谱性质的更基本的性质，即谱是否绝对连续。我们计划给出一系列的充分条件来保证谱的绝对连续性。最后，我们计划研究极限周期问题。这些问题是周期性问题的自然概括。虽然极限周期算子类不如拟周期或概周期算子类广泛，但周期理论的一些方法适用于极限周期情形。我们打算在极限周期环境下证明Bethe-Sommerfeld猜想。

Periodic differential operators arise in many areas of physics and mathematics, and studying their spectral properties is very important. Spectra of periodic operators have a band-gap structure, that is, they consist of a collection of closed intervals possibly separated by gaps. There is a famous hypothesis, called the Bethe-Sommerfeld conjecture, which claims that the number of gaps is finite. It has been justified for the Schroedinger operator with an electric field. One aim of the present project is to prove the conjecture in a much more general setting. The solution is expected to require the use of the pseudo-differential calculus, geometry of lattices and geometrical combinatorics. An important quantitative characteristic of differential operators acting on a non-compact manifold in the so-called integrated density of states. This function is a natural analogue of the spectral counting function. We plan to study the behaviour of this function for large values of energy and, in particular, to prove that the density of states has a complete asymptotic expansion in the (negative as well as positive) powers of energy.We also plan to study more basic properties of the nature of the spectrum, namely whether the spectrum is absolutely continuous. We plan to give establish a wide range of sufficient conditions which guarantee the absolute continuity of the spectrum. Finally, we plan to study limit-periodic problems. These problems are natural generalisation of the periodic ones. While the class of limit-periodic operators is not as wide as the class of quasi-periodic or almost-periodic operators, some of the methods of the periodic theory are applicable to the limit-periodic case. We intend to prove the Bethe-Sommerfeld conjecture in the limit-periodic setting.