Scattering theory for perturbed stratified media
A trace formula for the operator of acoustic propagation in a perturbed stratified medium is proved. The operator of acoustic propagation in the stratified medium is of the form \(H_0= -c^2_0(y)/\mu_0(y) \nabla_z\mu^{-1}_0(y) \nabla_z\), \(z= (x,y)\in \mathbb{R}^n\times \mathbb{R}\), where the speed of propagation \(c_0\) and the density of the medium \(\mu_0\) are bounded positive piecewise smooth functions with a finite number of singularities and are rapidly convergent to some constants \(c_\pm\), \(\mu_\pm\), \(0<c_+\leq c_-\), when \(y\to\pm\infty\) in the sense that \(c_0(y)- c_\pm\), \(\partial^j_y(\mu_0(y)- \mu_\pm)\), \(j= 0,1\) converge to zero, when \(y\to\pm\infty\) faster than any negative power of \(| y|\). The operator of acoustic propagation in the perturbed stratified medium is \(H_0= -c^2(z)\mu(z)\nabla_z \mu^{-1}(z) \nabla_z\), where \(c- c_0\), \(\mu- \mu_0\) and \(|\nabla_z(\mu- \mu_0)|\) converge to zero when \(| z|\to\pm\infty\) faster than any negative power of \(| z|\). Other additional assumptions on \(c_0\), \(\mu_0\), \(c\) and \(\mu\), which we shall not mention here, are made. The proof of the trace formula is based on the knowledge of the asymptotic expansions of the generalized eigenfunctions of \(H\). A blow-up space is introduced in order to better describe this asymptotic expansions. The main section of the paper is devoted to the proof of asymptotic expansion formulas for the pullbacks to the blow-up space of \((H-(\lambda+ i0)^2)^{-1}f\), for \(f\) in \(L^2_{\text{comp}}(\mathbb{R}^{n+1})+ {\mathcal S}\left(\mathbb{R}^n; \bigcap_{k\in\mathbb{N}}(1+| y|^2)^{-k/2} L^2(\mathbb{R})\right)\). The asymptotic expansion formulas for the generalized eigenfunctions are obtained from this more general result. Then a relative scattering matrix \(S^r(\lambda)\) is defined and is related to the asymptotics of the generalized eigenfunctions. It is proved that the scattering matrix is unitary and of the form \(S^r(\lambda)= I+ A(\lambda)\) with \(A(\lambda)\) a trace class operator. The paper ends with the proofs of the following two theorems: Theorem 1 (the trace formula). If \(n>1\) then \[ \text{Tr} \int(\cos(t\sqrt P)- \cos(t\sqrt{P_0})) f(t)dt= -{1\over 4\pi i} \int \widehat f(\lambda){d\over d\lambda}\log\text{det }S^r(\lambda)d\lambda \] for \(f\) in \({\mathcal S}(\mathbb{R})\), where \(P= c^{-1}\mu^{-1/2} Hc\mu^{1/2}\), \(P_0= c^{-1}_0\mu^{-1/2}_0 H_0c_0\mu^{1/2}_0\). The same equality holds for \(n= 1\) if \(\widehat f(0)= 0\). Theorem 2 (Weyl asymptotic of the scattering phase). Let \(c_0(y)= c_\pm\) for \(\pm y>0\), \(c_+< c_-\) and \(\mu_0(y)\equiv \mu_+\). Let \(c\), \(\mu\) be compactly supported perturbations of \(c_0\), \(\mu_+\) respectively with \(\mu\in C^\infty(\mathbb{R}^{n+1})\) and \(c\) smooth except at \(y=0\), but with \(\lim_{y\uparrow 0} D^\alpha_z c(z)\), \(\lim_{y\downarrow 0} D^\alpha_z c(z)\) smooth functions of \(x\) and \(c\) and \(\mu\) bounded and bounded away from zero. Let \(\sigma(\lambda)= (2\pi i)^{-1}(\log\text{det }S^r(\lambda)- \log\text{det }S^r(0))\) be the scattering phase. Then \[ \sigma(\lambda)= \gamma_{n+1} \int_{\mathbb{R}^{n+1}} (c^{-n-1}_0- c^{-n-1}) \lambda^{n+1}+ O(\lambda^n\log\lambda), \] where \(\gamma_{n+1}= (2\pi)^{-1}\) the volume of the unit ball in \(\mathbb{R}^{n+1}\). If the support of \(c-c_0\), \(\mu- \mu_0\) does not intersect \(y= 0\), then the remainder term can be improved to \(O(\lambda^n)\).
asymptotic of generalized eigenfunctions
scientific article; zbMATH DE number 1274822
scientific article