Singularity of the fundamental solution of Schroedinger equation

本文综述了具有周期系数的二阶椭圆型方程格林函数在无穷远处的渐近性结果。在[M.Murata，T Tuchida，Journal of Differential Equations，2002]一文中，M.Mutara教授和我发现了谱参数小于算子临界值的周期系数椭圆算子的格林函数的渐近性，并用简单的方法确定了Martin边界。受此结果的启发，2003年，我们研究了R^d，d[大于等于]2上具有周期系数的自共轭椭圆算子L的(L-λ-iO)^&lt；-1&gt；的积分核的渐近性，在谱参数λ大于且剂量位于谱的底部的情况下。我们研究了小拟标变量算子的Bloch变换的第一带函数的零点。然后，我们将解析的Fredholm理论、留数定理和驻相方法应用到Bloch逆变换的积分中，得到了渐近性。2004年，我们对上述方法进行了改进：我们直接计算了被积函数包含因子(x+o)^&lt；-1&gt；的积分的渐近性。作为SIMPLE方法的一个应用，我们可以给出极限吸收原理的一个比已知的其他证明(如Mourre方法)更简单的证明。在未来，我们的目标是在能量上表示格林函数，除了一维情况下的谱带的边缘。

This summary is concerning results for the asymptotics at infinity of Green function for second order elliptic differential equations with periodic coefficients. In the paper [M. Murata, T Tsuchida, Journal of differential equations, 2002], Prof. M.Mutara and I found the asymptotics of Green function for elliptic operators with periodic coefficients with the spectral parameter less than the critical value of the operator, and we determined the Martin boundary by using the asy nptotics. Motivated by this results, in 2003, we studied the asymptotics of the integral kernel of (L-λ-iO)^<-1> for the selfsdjoint elliptic operator L with periodic coefficients on R^d, d 【greater than or equal】 2, in the case that the spectral parameter λ is greater than and dose to the bottom of the spectrum. We investigated the zeros of the first band function of the Bloch transform of the operator for small quasirnomentum variables. Then we applied the analytic Fredholm theory, the residue theorem, and the stationary phase method to the integral of the inverse Bloch transform to obtain the asymptotics. In 2004, we improved the method stated above : we directly calculated asymptotics of an integral with an integrand including the factor (x+iO)^<-1>. As an application of the simple method we could give a simpler proof of the limiting absorption principle than known other proofs (e.g. Mourre method). In future we are aiming to represent the Green function at energy except for the edges of the band of the spectrum in the one dimensional case.