Scattering estimates for a time dependent matricial Klein-Gordon operator. (Estimations de diffusion pour un opérateur de Klein-Gordon matriciel dépendant du temps.) (French) Zbl 0919.35099

Benchaou, Mohammed

Bull. Soc. Math. Fr. 126, No. 2, 273-294 (1998).

Summary: We study the scattering theory for a time dependent \(2\times 2\) matricial Klein-Gordon operator, of the type: \[ P= (\sqrt{1- h^2\Delta_x}){\mathbf I}_2+ V(t,x)+ hR(t,x) \] on \(L^2(\mathbb{R}^n)\oplus L^2(\mathbb{R}^n)\), where \(V(t, x)\) is a real diagonal matrix, the eigenvalues of which are never equal when \((t,x)\) varies in \(\mathbb{R}^{n+ 1}\). One also assumes that \(V\) and \(R\) extend holomorphically in a complex strip around \(\mathbb{R}^{n+ 1}\), and satisfy to some decay properties at infinity. Then, denoting \(S= (S_{i,j})_{1\leq i,j\leq 2}\) the scattering operator associated to \(P\), we show that its off-diagonal coefficients \(S_{1,2}\) and \(S_{2,1}\) have an exponentially small norm as \(h\) tends to \(0_+\). More precisely, we obtain an estimate of the type \({\mathcal O}(e^{-\Sigma/h})\), where \(\Sigma>0\) is a constant which is explicitely related to the behaviour of \(V(t, x)\) in the complex domain.

MSC:

35P25	Scattering theory for PDEs
81U05	\(2\)-body potential quantum scattering theory

Keywords:

diffusion; semi-classical theory; exponential decay

References:

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