RESOLVENT KERNEL ESTIMATES NEAR THRESHOLDS 1 ...

Differential and Integral Equations

Volume 19, Number 1 (2006), 1–14

RESOLVENT KERNEL ESTIMATES NEAR THRESHOLDS Matania Ben-Artzi Institute of Mathematics, Hebrew University Jerusalem 91904, Israel Yves Dermenjian and Anton Monsef Universit´e de Provence, CMI 39, Rue Joliot Curie, 13453 Marseille cedex 13, France (Submitted by: Reza Aftabizadeh) Abstract. The paper deals with the spectral structure of the operator H = −∇ · b∇ in Rn where b is a stratified matrix-valued function. Using a partial Fourier transform, it is represented as a direct integral of a family of ordinary differential operators Hp , p ∈ Rn . Every operator Hp has two thresholds and the kernels are studied in their (spectral) neighborhoods, uniformly in compact sets of p . As in [3], such estimates lead to a limiting absorption principle for H. Furthermore, estimates of the resolvent of H near the bottom of its spectrum (“low energy” estimates) are obtained.

1. Introduction In [3], estimates of resolvent kernels near thresholds have been given. Such estimates are essential in the study of the acoustic propagator H0 = 1 −c2 (y)ρ(y)∇. ρ(y) ∇ where ∇ = ∇x,y , (x, y) ∈ Rn−1 × R, n ≥ 2, with ρ ∈ C 1 (R) and c piecewise continuous. In this paper, we get the same estimates for a more general operator H = −∇ · b∇,

(1.1)

where ∇ = ∇x,y , (x, y) ∈ Rn−1 ×R, n ≥ 2, b(x, y) = b(y), b ∈ L∞ (R, M (n, n)) (M (n, n) is the algebra of complex square matrices of order n). We suppose that there exist positive constants yc , c+ , c− and c, c+ < c− , such that ⎧ ⎨ b(y) = c2± Id if ± y ≥ yc , b(y) is a Hermitian matrix for a.e. y, (1.2) ⎩  p p b ≥ c|p|2 , ∀p ∈ C n l , for a.e. y. j jl l j,l Accepted for publication: July 2005. AMS Subject Classifications: 35P05, 47F05. 1

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Matania Ben-Artzi, Yves Dermenjian, and Anton Monsef

We first reduce the study of H to a study of a direct integral of operators Hp acting in L2 (R) (see [13], Chapter XIII). The positive selfadjoint operator H acting in L2 (Rn ) is associated with the continuous, coercive and Hermitian form a : H 1 (Rn ) × H 1 (Rn ) → C l defined by  a(u, v) = bjl (y)∂j u∂l vdx dy. Rn

j,l

If we take the partial Fourier transform with respect to the x-coordinates  1 −ip.x u(x, y)dx), we define a Hermitian form a (ˆ u(p, y) = ˆ on Vˆ n−1 Rn−1 e (2π)

2

such that a(u, v) = a ˆ(ˆ u, vˆ) where   a ˆ(ˆ u, vˆ) = bnn (y)∂n u ˆ∂n vˆdpdy −  +

Rn

Rn

Rn

(i



pj bjn )ˆ u∂n vˆdpdy +

(



Rn j,l