Inverse boundary spectral problems (Q2756692)

Let \(M\) be a compact smooth and connected Riemannian manifold with a smooth boundary \(\partial M\), and let \(q\) be a smooth real-valued function on \(M\). Then the Dirichlet realization of the Schrödinger operator on \(M\), \(P=-\Delta+q\), has a discrete spectrum consisting of the eigenvalues \(\lambda_j\), and the corresponding eigenfunctions \(\varphi_j\) form an orthonormal basis in \(L^2(M)\). By the {boundary spectral data} associated to \(P\) one means the collection NEWLINE\[NEWLINE \{\partial M,\, \lambda_j,\, \partial_{\nu} \varphi_j | _{\partial M},\, j=1,2\ldots\}. NEWLINE\]NEWLINE Here \(\partial_{\nu}\) is the derivative with respect to the outward unit normal \(\nu\) to \(\partial M\). One of the main problems studied in the book under review is the inverse problem of reconstructing the potential \(q\) and an isometric copy of the Riemannian manifold \(M\) from the boundary spectral data. In fact, in the book one studies elliptic self-adjoint second order operators in the divergence form that are more general than the operator \(P\) above, and matters are reduced to the Schrödinger operator by means of an appropriate gauge transformation. The main result is then that the potential \(q\) can indeed be recovered uniquely, and that the manifold \(M\) can also be reconstructed, up to an isometry. The book under review gives a thorough and detailed presentation of this result as well as of its various generalizations and extensions.NEWLINENEWLINEThe inverse boundary spectral problem, the book under review is devoted to, has a very long tradition in the area of inverse problems. A study of the inverse spectral problem for the Schrödinger operator on a bounded interval on the real line was pioneered by Borg in the late 1940's, and further progress was achieved shortly thereafter in the works by Gelfand and Levitan, Krein and others. Passing to the case of higher dimensions, a multi-dimensional version of the Borg-Levinson theorem, stating that on a bounded domain in \({\mathbb R}^n\), \(n\geq 2\), a bounded potential is uniquely determined by the boundary spectral data, was obtained in the late 1980's by \textit{A. Nachman}, \textit{J. Sylvester} and \textit{G. Uhlmann} [Commun. Math. Phys. 115, No. 4, 595--605 (1988; Zbl 0644.35095)], and, independently, by \textit{R. G. Novikov} [Funct. Anal. Appl. 22, No. 4, 263--272 (1988; Zbl 0689.35098)]. The problem of recovering a Riemannian manifold \(M\) from the boundary spectral data of the corresponding Dirichlet Laplacian was then studied by \textit{M. I. Belishev} and \textit{Y. V. Kurylev} [Commun. Partial Differ. Equations 17, 767--804 (1992; Zbl 0812.58094)], using the {boundary control method} developed by Belishev. The latter is based upon control theory for the wave equation. Further progress in the field of dynamical inverse problems is due, in particular, to the authors of the present book.NEWLINENEWLINEThe book comprises four chapters, organized as follows. Chapter 1 is devoted to a study of the one-dimensional case -- here the techniques to be used in the subsequent chapters are explained in a relatively simple situation. Chapter 2 contains the relevant background material, required in the main part of the book. Included are sections on basic tools of Riemannian geometry, initial-boundary value problems for the wave equation, construction of the Gaussian beams, and a theorem of Tataru on unique continuation for the wave equation with partially analytic coefficients. It is the opinion of the reviewer that, in particular, the last two sections should prove most useful, since he knows of no other source where this material is treated in a book form. The main uniqueness result for the inverse boundary spectral problem is then presented in Chapter 3. The final Chapter 4 contains various generalizations of the basic inverse problem of the previous chapter. Considered, for example, is the inverse problem when the boundary spectral data are given only along a portion of the boundary. Here too, the methods used are of dynamical nature.NEWLINENEWLINEIn conclusion, the book under review contains a wealth of important methods and ideas, and the presentation is always very clear. The present book is a very interesting and valuable contribution to the literature on inverse problems for partial differential equations.