权益分类	功能权益	普通用户	{{item.name}}会员
{{category.name}}	{{benefitItem.name}}

On distribution of scattering poles for several convex bodies

几种凸体散射极点的分布

基本信息

批准号：
16540189
负责人：
IKAWA Mitsuru
金额：
$ 1.28万
依托单位：
Kyoto University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2004
资助国家：
日本
起止时间：
2004 至 2005
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-16540189/
关键词：
wave equation scattering matrix modified Lax-Phillips conjecture zeta function system of the classical mechanics quantum mechanics WKB method 散乱理論修正版ラックス-ヒリップス予想散乱極

项目摘要

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.

本课题的主要目的是解决与散射极点分布有关的修正的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形，并试图找到一个突破口，最后我们认为我们已经找到了一个突破口。我们必须再前进几步，以证实我们的发现。因此，即使要花上两年多的时间才能达到最终目标，并且每一步都要写上几篇准备性的论文，但我们肯定已经找到了一种新的方法，使我们有可能攻击修正的Lax-Phillips猜想。修正的Lax-Phillips猜想的一个典型问题是考虑多个严格凸体的散射极点分布的问题。当我们用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，散射矩阵和zeta函数之间有着密切的关系。我们的新方法的最重要的事实是，它可以处理高频的情况。实际上，当我们用经典力学的近似来考虑量子问题时，近似有效的时间必须限制在频率的范围内。另一方面，散射理论关心的是从接近无穷远时间的波的情况到接近正无穷远时间的波的情况的对应。因此，有效时间的限制使我们很难从散射问题中得到有用的信息，而我们的新思想无疑为这一难题打开了一个突破口。