##
**Mathematical scattering theory. Analytic theory.**
*(English)*
Zbl 1197.35006

Mathematical Surveys and Monographs 158. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-0331-8/hbk). xiii, 444 p. (2010).

The main subject of this very interesting book of a well-known specialist in spectral theory is applications of methods of scattering theory to differential operators, primarily the Schrödinger operators. In fact the book can be considered as the second volume of the author’s monograph [Mathematical scattering theory: General theory. Transl. Math. Monographs 105, Providence, AMS (1992; Zbl 0761.47001)], but it is presumably possible to read it independently. A consistent use of the stationary approach (which gives formula representations for the basic objects of the theory), as well as the choice of concrete material (most of it due to the author) distinguishes this book from others such as the third volume of M. Reed and B. Simon [Methods of modern mathematical physics. III: Scattering theory. New York, San Francisco, London: Academic Press (1979; Zbl 0405.47007)].

There are two different trends in scattering theory for differential operators. The first one relies on the abstract scattering theory. The second one is almost independent of it; in this approach the abstract theory is replaced by a concrete investigation of the corresponding differential equation. In this book, which is structured in twelve chapters, an introduction and a review of the literature, both of these trends are presented. The first half of this book begins (Chapter 0) with the summary of the main results of the general scattering theory from the author [loc. cit.]. The next three chapters illustrate basic theorems of abstract scattering theory, presenting, in particular, their applications to scattering theory of perturbations of differential operators with constant coefficients (smooth method) and to the analysis of the trace class method. So, Chapter 1 “Smooth theory. The Schrödinger operator”, deals with the Schrödinger operator \(H_v= -\Delta+ v(x)\) in the space \(H= L^2(\mathbb{R}^d)\). Under the “short-range” assumption \(v(x)= O(|x|^{-\rho})\), \(\rho> 1\), \(|x|\to\infty\), the existence of the wave operators (WO) is verified with the help of the book criterion. The proof of the completeness of WO relies on the stationary method, which requires the strong \(H_0\)-smoothness of the operator \(\langle x\rangle^{-\alpha}\) for \(\alpha> 1/2\). One proves the absence of the singular continuous spectrum and one gives representations of the scattering matrix (SM) in Chapter 2 “Smooth theory. General differential operators”, the results of Chapter 1 are carried over to a rather general class of differential operators, where the coefficients of an imperturbed operator \(H_0\) do not depend on \(x\) while a perturbation is a differential operator with coefficients bounded by \(C|x|^{-\rho}\), \(\rho> 1\), at infinity. Chapter 3 “Scattering for perturbations of trace class type” presents various methods of verification of trace class conditions, which allow the proof of the existence and completeness of WO for more general classes of unperturbed operators \(H_0\), but imposes more stringent assumptions on the perturbations. A stationary representation of SM and properties of the spectral shift function (SSF) are also discussed.

The second half of the book begins with two chapters devoted to the one-dimensional problem, where specific methods of ordinary differential equations can be used. Chapter 4 “Scattering on the half-line” deals with the Hamiltonian \(H_v=-d^2/dx^2+ v(x)\), \(v=\overline v\), with the boundary condition \(u(0)= 0\) in the space \(H= L^2(\mathbb{R}_+)\). In this case a scattering theory can be constructed directly, avoiding general abstract results; a study of the resolvent can be performed with the help of Volterra integral equations. Some more special questions are also studied: low- and high-energy asymptotics, trace identities, new information on the SSF. Chapter 5 “One-dimensional scattering” is a direct continuation of Chapter 4; the Schrödinger operator \(H_v=-d^2/dx^2+ v(x)\) is considered in the space \(L^2(\mathbb{R})\), but the methods are the same as in the previous chapter. The main difference is that now \(H_v\) has a continuous spectrum of multiplicity 2, whereas its spectrum is simple in \(L^2(\mathbb{R}_+)\).

In the following chapters the author returns to the multidimensional problem and discusses various analytical methods and tools appropriate for the analysis of differential operators. In Chapter 6 “The limiting absorption principle (LAP), the radiation conditions and the expansion theorem”, which can be considered as a direct continuation of Chapter 1, scattering theory is formulated in terms of solutions of the Schrödinger equation satisfying some “boundary conditions” (radiation conditions) at infinity. Also the Mourre method is used to establish the LAP for long-range potentials. Chapter 7 “High- and low-energy asymptotics” deals with the behaviour of the resolvent \(R(z)= (H- z)^{-1}\) of the Schrödinger operator for high (as \(|z|\to\infty\)) and low (as \(|z|\to 0\)) energies. In Chapter 8 “The scattering matrix (SM) and the scattering cross section” the author continues the study of the SM started in chapters 1 and 6. Some asymptotic methods, such as the ray expansion and eikonal expansion, are also discussed. Chapter 9 “Spectral shift function (SSF) and trace formulas” is devoted to the construction and study of the SSF: high-energy asymptotics and trace identities. Basic results on long-range scattering can be found in Chapter 10 “The Schrödinger operator with a long-range potential”. The author gives a sketch of a proof of the asymptotic completeness, but the main goal is a description of diagonal singularities of the corresponding scattering amplitude. The last chapter “The LAP and radiation estimates revisited” deals with the Schrödinger equation \(\Delta u+ vu=\lambda u+ f\), \(\lambda> 0\), where the potential \(v\) has a long-range part. The main goal is to show the existence and unicity of solutions satisfying the radiation condition at infinity. In particular, a new proof of the LAP, independent of the Mourre method, is given.

This book fills in numerous gaps present in the monographic literature and is really very useful for a reader (for example, a graduate student in mathematical physics) interested in a deeper study of scattering theory.

There are two different trends in scattering theory for differential operators. The first one relies on the abstract scattering theory. The second one is almost independent of it; in this approach the abstract theory is replaced by a concrete investigation of the corresponding differential equation. In this book, which is structured in twelve chapters, an introduction and a review of the literature, both of these trends are presented. The first half of this book begins (Chapter 0) with the summary of the main results of the general scattering theory from the author [loc. cit.]. The next three chapters illustrate basic theorems of abstract scattering theory, presenting, in particular, their applications to scattering theory of perturbations of differential operators with constant coefficients (smooth method) and to the analysis of the trace class method. So, Chapter 1 “Smooth theory. The Schrödinger operator”, deals with the Schrödinger operator \(H_v= -\Delta+ v(x)\) in the space \(H= L^2(\mathbb{R}^d)\). Under the “short-range” assumption \(v(x)= O(|x|^{-\rho})\), \(\rho> 1\), \(|x|\to\infty\), the existence of the wave operators (WO) is verified with the help of the book criterion. The proof of the completeness of WO relies on the stationary method, which requires the strong \(H_0\)-smoothness of the operator \(\langle x\rangle^{-\alpha}\) for \(\alpha> 1/2\). One proves the absence of the singular continuous spectrum and one gives representations of the scattering matrix (SM) in Chapter 2 “Smooth theory. General differential operators”, the results of Chapter 1 are carried over to a rather general class of differential operators, where the coefficients of an imperturbed operator \(H_0\) do not depend on \(x\) while a perturbation is a differential operator with coefficients bounded by \(C|x|^{-\rho}\), \(\rho> 1\), at infinity. Chapter 3 “Scattering for perturbations of trace class type” presents various methods of verification of trace class conditions, which allow the proof of the existence and completeness of WO for more general classes of unperturbed operators \(H_0\), but imposes more stringent assumptions on the perturbations. A stationary representation of SM and properties of the spectral shift function (SSF) are also discussed.

The second half of the book begins with two chapters devoted to the one-dimensional problem, where specific methods of ordinary differential equations can be used. Chapter 4 “Scattering on the half-line” deals with the Hamiltonian \(H_v=-d^2/dx^2+ v(x)\), \(v=\overline v\), with the boundary condition \(u(0)= 0\) in the space \(H= L^2(\mathbb{R}_+)\). In this case a scattering theory can be constructed directly, avoiding general abstract results; a study of the resolvent can be performed with the help of Volterra integral equations. Some more special questions are also studied: low- and high-energy asymptotics, trace identities, new information on the SSF. Chapter 5 “One-dimensional scattering” is a direct continuation of Chapter 4; the Schrödinger operator \(H_v=-d^2/dx^2+ v(x)\) is considered in the space \(L^2(\mathbb{R})\), but the methods are the same as in the previous chapter. The main difference is that now \(H_v\) has a continuous spectrum of multiplicity 2, whereas its spectrum is simple in \(L^2(\mathbb{R}_+)\).

In the following chapters the author returns to the multidimensional problem and discusses various analytical methods and tools appropriate for the analysis of differential operators. In Chapter 6 “The limiting absorption principle (LAP), the radiation conditions and the expansion theorem”, which can be considered as a direct continuation of Chapter 1, scattering theory is formulated in terms of solutions of the Schrödinger equation satisfying some “boundary conditions” (radiation conditions) at infinity. Also the Mourre method is used to establish the LAP for long-range potentials. Chapter 7 “High- and low-energy asymptotics” deals with the behaviour of the resolvent \(R(z)= (H- z)^{-1}\) of the Schrödinger operator for high (as \(|z|\to\infty\)) and low (as \(|z|\to 0\)) energies. In Chapter 8 “The scattering matrix (SM) and the scattering cross section” the author continues the study of the SM started in chapters 1 and 6. Some asymptotic methods, such as the ray expansion and eikonal expansion, are also discussed. Chapter 9 “Spectral shift function (SSF) and trace formulas” is devoted to the construction and study of the SSF: high-energy asymptotics and trace identities. Basic results on long-range scattering can be found in Chapter 10 “The Schrödinger operator with a long-range potential”. The author gives a sketch of a proof of the asymptotic completeness, but the main goal is a description of diagonal singularities of the corresponding scattering amplitude. The last chapter “The LAP and radiation estimates revisited” deals with the Schrödinger equation \(\Delta u+ vu=\lambda u+ f\), \(\lambda> 0\), where the potential \(v\) has a long-range part. The main goal is to show the existence and unicity of solutions satisfying the radiation condition at infinity. In particular, a new proof of the LAP, independent of the Mourre method, is given.

This book fills in numerous gaps present in the monographic literature and is really very useful for a reader (for example, a graduate student in mathematical physics) interested in a deeper study of scattering theory.

Reviewer: Viorel Iftimie (Bucureşti)

### MSC:

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

35P10 | Completeness of eigenfunctions and eigenfunction expansions in context of PDEs |

35P25 | Scattering theory for PDEs |

47A40 | Scattering theory of linear operators |

34L25 | Scattering theory, inverse scattering involving ordinary differential operators |

81U05 | \(2\)-body potential quantum scattering theory |

35J10 | Schrödinger operator, Schrödinger equation |

35Q41 | Time-dependent Schrödinger equations and Dirac equations |