In the recent preprint, I study the wave equation for the elliptic operator in divergence form. In \mathbb{R}^3 , define

H=-{\rm div }(1+V)\nabla,

where V oscillates and decays at infinity. More precisely, V={\rm div} \,Q where \|V\|<\infty, \|Q\|<\infty and the norm \|f\| is defined as

\|f\|=\left(\sum_{n=0}\max_{|x|\in [n,n+1]}|f|^2\right)^{1/2}.

I also assume \|V\|_\infty<1 to make sure that H is non-negative operator. The wave equation for H is

u_{tt}+Hu=0, \, u(x,0)=f_1,\, u_t(x,0)=f_2.

The main result of the paper states that the following wave operators

\lim_{t\to\pm \infty} e^{it\sqrt{H}}e^{-it\sqrt{-\Delta}}f=W^{\pm}f

exist for every f\in L^2(\mathbb{R}^3) and the limit is understood in L^2(\mathbb{R}^3) norm. The condition on V is optimal in some sense, i.e., the rate of decay is sharp and the oscillation is necessary if the potential is not short-range.

The proof is based on the analysis of the asymptotical behavior of the Green’s function G(x,y,k^2) where k\in \mathbb{C}^+ , y is fixed, and |x|\to\infty . The result about asymptotics is similar to that for the orthogonal polynomials on the circle in the Szego case. The main difference with the one-dimensional situation is that the resulting “Szego” function belongs to the vector-valued Hardy space. The operator e^{it\sqrt{H}} can be written through the resolvent (H-z)^{-1} by the contour integral and this is how the Green’s function enters the proof. The method is quite general and can be adapted to wave equations for the Schrodinger equation and other problems.

 

 

 

 

 

 

 

Advertisements