On distribution of scattering poles for several convex bodies

本课题的主要目的是解决与散射极分布有关的修正Lax-Phillips猜想。为了解决这个问题，我们首先考虑了三个严格凸体的情况，并试图找到一个突破口。总之，我们相信我们已经找到了突破口。为了证实我们的发现，我们还得再走几步。因此，即使可能需要两年多的时间才能达到最终目标，并且每一步都要写几篇准备论文，但可以肯定的是，我们已经找到了一种新的方法，使我们有可能攻击修正的拉克斯-菲利普斯猜想。修正拉克斯-菲利普斯猜想的一个典型问题是考虑几个严格凸体散射极点的分布问题。当我们按照WKB方法构造渐近解时，经典力学的zeta函数将作为上述渐近解的矩阵迹的主要部分出现。人们普遍认为散射矩阵与zeta函数之间有密切的关系。我们的新方法最重要的事实是，它使处理高频的情况成为可能。事实上，当我们用经典力学的近似值来考虑量子问题时，近似值有效的时间必须限制在频率的极限之内。另一方面，散射理论关注的是波在接近负无穷时的状态与接近正无穷时的状态的对应关系。因此，由于有效时间的限制，很难得到对散射问题有用的信息。我们的新想法无疑为解决这个困难开辟了突破口。

The main purpose of this research project is to solve the modified Lax-Phillips conjecture which is concerned with the distribution of scattering poles. To approach this problem, firstly we consider the case of three strictly convex bodies, and tried to find out a breakthrough.In conclusion, we believe that we have found out a breakthrough. We have to go forward several steps more in order to verify our discovery. Therefore, even if it may take more than two years to arrive the final goal and have to write several preparatory papers according each step, it is sure that we have found out a new method which makes us possible to attack the modified Lax-Phillips conjecture.A typical problem concerning the modified Lax-Phillips conjecture is the problem consider the distribution of the scattering poles for several strictly convex bodies. When we construct an asymptotic solution following WKB method, the zeta function for the classical mechanics will appear as the main part of the matrix trace of the above asymptotic solution. It is widely believed that there are close relationship between the scattering matrix and the zeta function. The most important fact of our new method is that it makes possible to treat the case of high frequencies. Indeed, when we consider quantum problems by using an approximation by the classical mechanics, the time for which the approximation is valid must be restricted within the limit of the frequency.On the other hand, the scattering theory is concerned with the correspondance from the situation of the wave for the time near the ninus infinity to the situation of the wave for the time near the plus infinity. Thus, the limitation for the valid time makes very difficult to derive useful informations for scattering problems.Our new idea surely opens a breakthrough for this difficulty.