##
**Spectral and scattering theory for wave propagation in perturbed stratified media.**
*(English)*
Zbl 0711.76083

Applied Mathematical Sciences, 87. New York etc.: Springer-Verlag. 188 p. DM 74.00 (1991).

A medium is called stratified if its physical properties depend on a single coordinate. The stratified medium is a mathematical model for ocean acoustics, integrated optics, geophysics.

In the book the author considers the same mathematical problems of wave propagation in media whose physical characteristics depend upon all of the coordinates and they stabilize at infinity to the functions which depend only on a single coordinate. This condition means these media are the perturbed stratified media. It should be noted that in the classical scattering theory the medium is the perturbation of the homogeneous medium.

In the first part of the book the author considers the acoustic equation in \({\mathbb{R}}^{n+1}:\) \((\partial^ 2u/\partial t^ 2)-c^ 2(x,y)\Delta u=0\), \(x\in {\mathbb{R}}^ n\), \(y\in {\mathbb{R}}\), \(t\in {\mathbb{R}}^ 1\), where c(x,y) is the sound speed, the fluid density \(\rho =const.\); c(x,y) is a real valued measurable function on \({\mathbb{R}}^{n+1}\) such that \(0<c_ m\leq c(x,y)\leq c_ M<\infty\), for almost every (x,y) and some positive constant \(c_ m\) and \(c_ M\), and \(| c(x,y)-c_ 0(y)| \leq c(1+| (x,y)|)^{-1-\epsilon}\), \(\epsilon >0\), where \(c_ 0(y)\) is a real valued measurable function on \({\mathbb{R}}\) which satisfies the same conditions of stabilization at \(y\to \pm \infty.\)

The author considers the following problems: (1) the limiting absorption principle for acoustic operators in unperturbed (stratified) and perturbed media; 2) the absence of the positive eigenvalues on acoustic operators for perturbed media; 3) the acoustic scattering theory in which the unperturbed operator is an acoustic operator for the stratified media; 4) the generalized Fourier transforms; 5) the structure of the scattering matrix.

In the second part of the book the author considers the Maxwell system of equations

\[ \left\{\begin{aligned} \nabla \times \bar E=-\mu (x,z)\partial \bar H/\partial t, \\ \nabla \times \bar H=\epsilon (x,z)\partial \bar E/\partial t, \\ \nabla \cdot (\epsilon (x,z)\bar E)=0,\\ \nabla \cdot (\mu (x,z)\bar H)=0,\end{aligned}\right.\tag{1} \] where \(x=(x_ 1,x_ 2)\in {\mathbb{R}}^ 2\), \(z\in {\mathbb{R}}\), \(t\in {\mathbb{R}}\), \(\bar E=(E_ 1,E_ 2,E_ 3)\), \(\bar H=(H_ 1,H_ 2,H_ 3)\) are the functions from \({\mathbb{R}}^ 4\) into \({\mathbb{R}}^ 3\) that correspond, respectively, to the electric and magnetic fields, \(\epsilon(x,z)\), \(\mu(x,z)\) are real valued measurable and bounded function defined on \({\mathbb{R}}^ 3\) that represent, respectively, the electric and magnetic susceptibilities.

The author assumes that \(0<c_ m\leq \epsilon (x,z)\leq c_ M<\infty\), \(0<c_ m\leq \mu (x,z)\leq c_ M<\infty\), and \(| \epsilon (x,z)- \epsilon_ 0(z)| \leq c(1+| (x,z)|)^{-1-\epsilon}\), \(| \mu (x,z)-\mu_ 0(z)| \leq c(1+| (x,z)|)^{-1-\epsilon}\), \(\epsilon >0\), where \(\epsilon_ 0(z)\), \(\mu_ 0(z)\) are the real valued measurable functions defined on \({\mathbb{R}}\), which satisfy the same conditions of stabilization at \(z\to \pm \infty.\)

The author considers the limiting absorption principle, the spectral properties and the scattering theory for the Maxwell system (1).

Reviewer’s remark: In the case when the fluid density \(\rho\) (x,y) is the function depending upon all of the coordinates, similar problems for acoustic operators were considered in the works of M. Ben-Artzi, Y. Dermenjian and J.-C. Guillot [Commun. Partial Differ. Equations 14, No.4, 479-517 (1989; Zbl 0675.35065)] and V. S. Rabinovich [On the solvability of acoustic problems in an open waveguide (in Russian), Differ. Uravn. 26, No.12, 2178-2180 (1990)].

In the book the author considers the same mathematical problems of wave propagation in media whose physical characteristics depend upon all of the coordinates and they stabilize at infinity to the functions which depend only on a single coordinate. This condition means these media are the perturbed stratified media. It should be noted that in the classical scattering theory the medium is the perturbation of the homogeneous medium.

In the first part of the book the author considers the acoustic equation in \({\mathbb{R}}^{n+1}:\) \((\partial^ 2u/\partial t^ 2)-c^ 2(x,y)\Delta u=0\), \(x\in {\mathbb{R}}^ n\), \(y\in {\mathbb{R}}\), \(t\in {\mathbb{R}}^ 1\), where c(x,y) is the sound speed, the fluid density \(\rho =const.\); c(x,y) is a real valued measurable function on \({\mathbb{R}}^{n+1}\) such that \(0<c_ m\leq c(x,y)\leq c_ M<\infty\), for almost every (x,y) and some positive constant \(c_ m\) and \(c_ M\), and \(| c(x,y)-c_ 0(y)| \leq c(1+| (x,y)|)^{-1-\epsilon}\), \(\epsilon >0\), where \(c_ 0(y)\) is a real valued measurable function on \({\mathbb{R}}\) which satisfies the same conditions of stabilization at \(y\to \pm \infty.\)

The author considers the following problems: (1) the limiting absorption principle for acoustic operators in unperturbed (stratified) and perturbed media; 2) the absence of the positive eigenvalues on acoustic operators for perturbed media; 3) the acoustic scattering theory in which the unperturbed operator is an acoustic operator for the stratified media; 4) the generalized Fourier transforms; 5) the structure of the scattering matrix.

In the second part of the book the author considers the Maxwell system of equations

\[ \left\{\begin{aligned} \nabla \times \bar E=-\mu (x,z)\partial \bar H/\partial t, \\ \nabla \times \bar H=\epsilon (x,z)\partial \bar E/\partial t, \\ \nabla \cdot (\epsilon (x,z)\bar E)=0,\\ \nabla \cdot (\mu (x,z)\bar H)=0,\end{aligned}\right.\tag{1} \] where \(x=(x_ 1,x_ 2)\in {\mathbb{R}}^ 2\), \(z\in {\mathbb{R}}\), \(t\in {\mathbb{R}}\), \(\bar E=(E_ 1,E_ 2,E_ 3)\), \(\bar H=(H_ 1,H_ 2,H_ 3)\) are the functions from \({\mathbb{R}}^ 4\) into \({\mathbb{R}}^ 3\) that correspond, respectively, to the electric and magnetic fields, \(\epsilon(x,z)\), \(\mu(x,z)\) are real valued measurable and bounded function defined on \({\mathbb{R}}^ 3\) that represent, respectively, the electric and magnetic susceptibilities.

The author assumes that \(0<c_ m\leq \epsilon (x,z)\leq c_ M<\infty\), \(0<c_ m\leq \mu (x,z)\leq c_ M<\infty\), and \(| \epsilon (x,z)- \epsilon_ 0(z)| \leq c(1+| (x,z)|)^{-1-\epsilon}\), \(| \mu (x,z)-\mu_ 0(z)| \leq c(1+| (x,z)|)^{-1-\epsilon}\), \(\epsilon >0\), where \(\epsilon_ 0(z)\), \(\mu_ 0(z)\) are the real valued measurable functions defined on \({\mathbb{R}}\), which satisfy the same conditions of stabilization at \(z\to \pm \infty.\)

The author considers the limiting absorption principle, the spectral properties and the scattering theory for the Maxwell system (1).

Reviewer’s remark: In the case when the fluid density \(\rho\) (x,y) is the function depending upon all of the coordinates, similar problems for acoustic operators were considered in the works of M. Ben-Artzi, Y. Dermenjian and J.-C. Guillot [Commun. Partial Differ. Equations 14, No.4, 479-517 (1989; Zbl 0675.35065)] and V. S. Rabinovich [On the solvability of acoustic problems in an open waveguide (in Russian), Differ. Uravn. 26, No.12, 2178-2180 (1990)].

Reviewer: V.S.Rabinovich

### MSC:

76Q05 | Hydro- and aero-acoustics |

35P05 | General topics in linear spectral theory for PDEs |

76-02 | Research exposition (monographs, survey articles) pertaining to fluid mechanics |