On distribution of scattering poles for several convex bodies

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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.