Singularity of the fundamental solution of Schroedinger equation

薛定谔方程基本解的奇异性

• 批准号: 15540187
• 负责人: TSUCHIDA Tetsuo
• 金额: $ 0.64万
• 依托单位: Meijo University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2003
• 资助国家: 日本
• 起止时间: 2003 至 2004
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-15540187/
• 关键词: Green function; elliptic equation; periodic coefficients; limiting absorption principle; Martin boundary

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.

本摘要涉及具有周期系数的二阶椭圆微分方程的格林函数无穷远渐近结果。在论文[M. Murata，T Tsuchida，微分方程杂志，2002]，M.Mutara教授和我发现了具有周期性系数且谱参数小于算子临界值的椭圆算子的Green函数的渐近性，并利用渐近性确定了Martin边界。受此结果的启发，我们于2003年研究了在谱参数λ大于且接近谱底的情况下，周期系数为R^d，d【大于或等于】2的自联椭圆算子L的(L-λ-iO)^<-1>积分核的渐近性。我们研究了小准动变量算子布洛赫变换的第一带函数的零点。然后我们将解析的Fredholm理论、留数定理和固定相法应用到布洛赫逆变换的积分中以获得渐近方程。 2004年，我们改进了上述方法：直接计算被积函数包含因子(x+iO)^<-1>的积分的渐近性。作为简单方法的应用，我们可以给出比已知的其他证明（例如莫雷方法）更简单的限制吸收原理证明。将来，我们的目标是在一维情况下表示除光谱带边缘之外的能量格林函数。

项目成果

Murata Minoru, Tetsuo Tsuchida: "Asymptotics of Green functions and Martin boundaries for elliptic operators with periodic coefficients"Journal of Differential Equations. 195・1. 82-118 (2003)

村田稔、土田哲夫：“具有周期系数的椭圆算子的格林函数和马丁边界的渐近”《微分方程杂志》195・1（2003）。