##
**Geometry of reflecting rays and inverse spectral problems.**
*(English)*
Zbl 0761.35077

Pure and Applied Mathematics. A Wiley-Interscience Series of Texts, Monographs and Tracts. Chichester etc.: John Wiley & Sons Ltd. (ISBN 0-471-93174-8). vi, 313 p. (1992).

The book under review is devoted to certain classical inverse problems concerning either the spectrum of the Laplace operator \(\Delta\) in a bounded domain \(\Omega\) with smooth boundary \(\partial\Omega\) and Dirichlet (Neumann) boundary conditions, or suitable scattering data for unbounded domains. The inverse spectral problem for the Laplacian in bounded domains was clearly formulated by M. Kac. Although one can not always hear the shape of the drum, as it was recently shown by C. Gordon, D. Webb and S. Wolpert, one of the main problems in the inverse spectral theory is still to find out what kind of geometric information about the domain is encoded in the spectrum of the Laplacian. It is well known that the Euclidean volume of \(\Omega\) as well as the area of the boundary and certain integrals of polynomials of the curvature of \(\partial\Omega\) and its derivatives, are spectral invariants, which are encoded in the Taylor series at \(t=0\) of the trace of the parabolic semigroup \(\exp(t\Delta)\).

Another approach to this problem, which relies on the “geometric optics” for the wave equation and the propagation of singularities phenomenon for hyperbolic equations, turns out to be more fruitful. It relates the spectrum of \(\Delta\) to the length spectrum of the domain \(\Omega\), which is defined as the set of lengths of all closed billiard trajectories in \(\Omega\). This approach is based on the singularities of the trace of the wave group \(\sigma(t)\), which is a tempered distribution on \(\mathbb{R}\) given by \(\sum\exp(it\lambda)\) where the sum is taken over all \(\lambda\) with \(\lambda^ 2\) in the spectrum of \(-\Delta\) counted with multiplicity. The Poisson relation proved by Anderson, Melrose and Sjöstrand, says that the singular support of \(\sigma(t)\) is contained in the set of all \(T\) such that either \(T=0\) or the absolute value of \(T\) belongs to the length spectrum of \(\Omega\). Unfortunately, singularities created by different closed generalized geodesics of the same length can cancel each other, and it seems that the Poisson relation does not always turn into an equality.

One of the main topics of the book is the Poisson relation for generic domains. The analysis of the generic properties of the billiard trajectories is the second topic. The authors prove that for generic domains any closed billiard trajectory \(\gamma\) is either ordinary, that is it reflects transversally at the boundary, or it is a closed geodesic on the boundary. Moreover, any such \(\gamma\) is non-degenerated, and the periods of any two billiard trajectories are rationally independent. One of the main results in the book is that for generic planar domains and for generic convex domains in \(\mathbb{R}^ n\), \(n\geq 3\), both the length spectrum and the spectra of the linear Poincaré maps are encoded in the spectrum of the Laplacian. In particular, the Poisson relation turns into equality for generic domains. The third topic concerns the singularities of the scattering kernel corresponding to the acoustic scattering from a bounded obstacle with smooth boundary. The authors establish a Poisson relation for the scattering kernel and prove that for generic obstacles the inverse relation also holds. Moreover, the leading singularity is examined. Special attention is paid to the case of finitely many strictly convex obstacles. The hyperbolicity of the periodic reflecting rays allows the authors to prove that the “scattering length spectrum” is always encoded in the singular support of the scattering kernel in this case.

This highly recommended book builds a bridge between the spectral and scattering theory for the Laplace operator from one side and the classical dynamics and the symplectic properties of the billiard trajectories from the other side. It is self contained although it requires some basic knowledge of differential geometry and differential topology.

The book is divided into 10 chapters. The first one has preliminary character. Various generic properties of the billiard trajectories are proved in Chapter 3. An analogy of the classical bumpy metric theorem of Abraham-Klingenberg-Takens-Anosov for hypersurfaces in the Euclidean space is proved in Chapter 4. Next chapter is devoted to the Poisson relation for bounded domains. To make the exposition more transparent the authors give a separate proof for convex domains which is much simpler than the general one. The Poisson formula of Guillemin and Melrose concerning the singularities of the distribution \(\sigma(t)\) is studied in Chapter 6. The main result in Chapter 7 is that the length spectrum is a subset of the singular support of \(\sigma(t)\) for generic planar domains and for generic convex domains in \(\mathbb{R}^ n\), \(n\geq 3\). Chapters 8 and 9 are devoted to the singularities of the scattering kernel while Chapter 10 is concerned with certain inverse scattering problems.

Another approach to this problem, which relies on the “geometric optics” for the wave equation and the propagation of singularities phenomenon for hyperbolic equations, turns out to be more fruitful. It relates the spectrum of \(\Delta\) to the length spectrum of the domain \(\Omega\), which is defined as the set of lengths of all closed billiard trajectories in \(\Omega\). This approach is based on the singularities of the trace of the wave group \(\sigma(t)\), which is a tempered distribution on \(\mathbb{R}\) given by \(\sum\exp(it\lambda)\) where the sum is taken over all \(\lambda\) with \(\lambda^ 2\) in the spectrum of \(-\Delta\) counted with multiplicity. The Poisson relation proved by Anderson, Melrose and Sjöstrand, says that the singular support of \(\sigma(t)\) is contained in the set of all \(T\) such that either \(T=0\) or the absolute value of \(T\) belongs to the length spectrum of \(\Omega\). Unfortunately, singularities created by different closed generalized geodesics of the same length can cancel each other, and it seems that the Poisson relation does not always turn into an equality.

One of the main topics of the book is the Poisson relation for generic domains. The analysis of the generic properties of the billiard trajectories is the second topic. The authors prove that for generic domains any closed billiard trajectory \(\gamma\) is either ordinary, that is it reflects transversally at the boundary, or it is a closed geodesic on the boundary. Moreover, any such \(\gamma\) is non-degenerated, and the periods of any two billiard trajectories are rationally independent. One of the main results in the book is that for generic planar domains and for generic convex domains in \(\mathbb{R}^ n\), \(n\geq 3\), both the length spectrum and the spectra of the linear Poincaré maps are encoded in the spectrum of the Laplacian. In particular, the Poisson relation turns into equality for generic domains. The third topic concerns the singularities of the scattering kernel corresponding to the acoustic scattering from a bounded obstacle with smooth boundary. The authors establish a Poisson relation for the scattering kernel and prove that for generic obstacles the inverse relation also holds. Moreover, the leading singularity is examined. Special attention is paid to the case of finitely many strictly convex obstacles. The hyperbolicity of the periodic reflecting rays allows the authors to prove that the “scattering length spectrum” is always encoded in the singular support of the scattering kernel in this case.

This highly recommended book builds a bridge between the spectral and scattering theory for the Laplace operator from one side and the classical dynamics and the symplectic properties of the billiard trajectories from the other side. It is self contained although it requires some basic knowledge of differential geometry and differential topology.

The book is divided into 10 chapters. The first one has preliminary character. Various generic properties of the billiard trajectories are proved in Chapter 3. An analogy of the classical bumpy metric theorem of Abraham-Klingenberg-Takens-Anosov for hypersurfaces in the Euclidean space is proved in Chapter 4. Next chapter is devoted to the Poisson relation for bounded domains. To make the exposition more transparent the authors give a separate proof for convex domains which is much simpler than the general one. The Poisson formula of Guillemin and Melrose concerning the singularities of the distribution \(\sigma(t)\) is studied in Chapter 6. The main result in Chapter 7 is that the length spectrum is a subset of the singular support of \(\sigma(t)\) for generic planar domains and for generic convex domains in \(\mathbb{R}^ n\), \(n\geq 3\). Chapters 8 and 9 are devoted to the singularities of the scattering kernel while Chapter 10 is concerned with certain inverse scattering problems.

Reviewer: G.Popov (Darmstadt)

### MSC:

35P25 | Scattering theory for PDEs |

35-02 | Research exposition (monographs, survey articles) pertaining to partial differential equations |

58J50 | Spectral problems; spectral geometry; scattering theory on manifolds |

81U40 | Inverse scattering problems in quantum theory |

35R30 | Inverse problems for PDEs |

47A10 | Spectrum, resolvent |