Singularity of the fundamental solution of Schroedinger equation

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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.