##
**Scattering theory.**
*(English)*
Zbl 0186.16301

Aus dem Vorwort der Verff.:

“Scattering theory compares the asymptotic behavior of an evolving system as \(t\) tends to \(-\infty\) with its asymptotic behavior as \(t\) tends to \(+infty\). It is especially fruitful for studying systems constructed from a simpler system by the imposition of a disturbance (also called perturbation or scatterer) provided that the influence of the disturbance on motions at large \(\vert t\vert\) is negligible, i.e., if any motion of the perturbed system for large \(\vert t\vert\) is indistinguishable from a motion of the unperturbed system. Thus, if \(U(t)\) and \(U_0(t)\) denote the operators relating the states of the perturbed and unperturbed systems at time zero to their respective states at time \(t\), then to each state \(f\) of the perturbed system there correspond two states \(f_-\) and \(f_+\) of the unperturbed system such that \(U(t)f\) behaves like \(U_0(t)f_-\) as \(t\to -\infty\) and like \(U_0(t)f_+\) as \(t\to +\infty\). The scattering operator is defined as the mapping: \(S\colon f_-\to f_+\). The aim of scattering theory is to prove the existence of such a scattering operator and to link its properties to the nature of the scatterer. In situations where the scattering operator constitutes the only physically observable data of motion the main task is the inverse problem of reconstructing the scatterer from the scattering operator...

In our approach we deal with systems described by a group of unitary operators \(\{U(t)\}\) acting on a Hilbert space \(H\) in which there are two distinguished subspaces \(D_-\) and \(D_+\), with the property that, as \(t\) varies from \(-\infty\) to \(+\infty\), the subspaces \(U(t)D_-\) and \(U(t)D_+\) increase (decrease) monotonically from the zero subspace to the whole space \(H\); we call \(D_-\) and \(D_+\) the incoming and outgoing subspaces, respectively. It is not difficult to show that with each subspace \(D_-\) and \(D_+\) we can associate a special spectral representation of the group \(\{U(t)\}\); in the one \(D_-\) is represented by functions analytic in the lower half-plane, in the second \(D_+\) is represented by functions analytic in the upper half-plane. The two representations are related by a unitary, operator-valued multiplicative factor \(\mathcal S(\sigma)\), \(-\infty<\sigma<\infty\), which we call the scatte-ing matrix. If \(D_-\) and \(D_+\) are orthogonal then \(\mathcal S(\sigma)\) is the restriction to the real axis of a bounded operator-valued analytic function holomorphic in the lower half-plane.”

Dieses Problem behandeln die Verff. für die Wellengleichung, für symmetrische hyperbolische Systeme und für die Schrödingergleichung. Die Darstellung ist klar, die Lektüre des Buches kann jedem Interessenten empfohlen werden.

Es sei noch das Inhaltsverzeichnis angeführt:

I. Introduction. 1. The dynamic approach. 2. Scattering theory formulated in terms of representation theory. 3. A semigroup of operators related to the scattering matrix. 4. The form of the scattering matrix. 5. A simple example. 6. Scattering theory for transport phenomena. 7. Notes and remarks.

II. Representation theory and the scattering operator. 1. The discrete case. 2. The scattering operator in the discrete case. 3. The continuous case. 4. The scattering operator in the continuous case. 5. Notes and remarks.

III. A semigroup of operators related to the scattering matrix. 1. The related semigroups. 2. On semi-groups of contraction operators. 3. Spectral theory. 4. A spectral mapping theorem. 5. Applications of the spectral theory. 6. Equivalent incoming and outgoing representations. 7. Notes and remarks.

IV. The translation representation for the solution of the wave equation in free space. 1. The Hilbert space \(H_0\) and the group \(\{U_0(t)\}\). 2. Spectral and translation representations of \(\{U_0(t)\}\). 3. The operator \(\mathcal J\) extended to distributions. 4. Translation representation for outgoing and incoming data with finite energy. 5. Notes and remarks.

V. The solution of the wave equation in an exterior domain. 1. The Hilbert space \(H\) and the group (U(t)). 2. Energy decay and translation representations. 3. The semigroup \(\{Z(t)\}\). 4. The relation between the semigroup \(\{Z(t)\}\) and the solution of the reduced wave equation. 5. The scattering matrix. 6. Notes and remarks.

VI. Symmetric hyperbolic systems, the acoustic equation with an indefinite energy form, and the Schrödinger equation. Part 1. Symmetric hyperbolic systems. 1. Translation representation in free space. 2. Solution of hyperbolic systems in an exterior domain. Part 2. The acoustic equation with an indefinite energy form and the Schrödinger equation. 3. Scattering for the acoustic equation with an indefinite energy form. 4. The Schrödinger scattering matrix. 5. Notes and remarks.

App. 1. Semigroups of operators. App. 2. Energy decay. App. 3 (by C. S. Morawetz). Energy decay for star-shaped obstacles. App. 4 (by G. Schmidt), Scattering theory for Maxwell’s equations.

“Scattering theory compares the asymptotic behavior of an evolving system as \(t\) tends to \(-\infty\) with its asymptotic behavior as \(t\) tends to \(+infty\). It is especially fruitful for studying systems constructed from a simpler system by the imposition of a disturbance (also called perturbation or scatterer) provided that the influence of the disturbance on motions at large \(\vert t\vert\) is negligible, i.e., if any motion of the perturbed system for large \(\vert t\vert\) is indistinguishable from a motion of the unperturbed system. Thus, if \(U(t)\) and \(U_0(t)\) denote the operators relating the states of the perturbed and unperturbed systems at time zero to their respective states at time \(t\), then to each state \(f\) of the perturbed system there correspond two states \(f_-\) and \(f_+\) of the unperturbed system such that \(U(t)f\) behaves like \(U_0(t)f_-\) as \(t\to -\infty\) and like \(U_0(t)f_+\) as \(t\to +\infty\). The scattering operator is defined as the mapping: \(S\colon f_-\to f_+\). The aim of scattering theory is to prove the existence of such a scattering operator and to link its properties to the nature of the scatterer. In situations where the scattering operator constitutes the only physically observable data of motion the main task is the inverse problem of reconstructing the scatterer from the scattering operator...

In our approach we deal with systems described by a group of unitary operators \(\{U(t)\}\) acting on a Hilbert space \(H\) in which there are two distinguished subspaces \(D_-\) and \(D_+\), with the property that, as \(t\) varies from \(-\infty\) to \(+\infty\), the subspaces \(U(t)D_-\) and \(U(t)D_+\) increase (decrease) monotonically from the zero subspace to the whole space \(H\); we call \(D_-\) and \(D_+\) the incoming and outgoing subspaces, respectively. It is not difficult to show that with each subspace \(D_-\) and \(D_+\) we can associate a special spectral representation of the group \(\{U(t)\}\); in the one \(D_-\) is represented by functions analytic in the lower half-plane, in the second \(D_+\) is represented by functions analytic in the upper half-plane. The two representations are related by a unitary, operator-valued multiplicative factor \(\mathcal S(\sigma)\), \(-\infty<\sigma<\infty\), which we call the scatte-ing matrix. If \(D_-\) and \(D_+\) are orthogonal then \(\mathcal S(\sigma)\) is the restriction to the real axis of a bounded operator-valued analytic function holomorphic in the lower half-plane.”

Dieses Problem behandeln die Verff. für die Wellengleichung, für symmetrische hyperbolische Systeme und für die Schrödingergleichung. Die Darstellung ist klar, die Lektüre des Buches kann jedem Interessenten empfohlen werden.

Es sei noch das Inhaltsverzeichnis angeführt:

I. Introduction. 1. The dynamic approach. 2. Scattering theory formulated in terms of representation theory. 3. A semigroup of operators related to the scattering matrix. 4. The form of the scattering matrix. 5. A simple example. 6. Scattering theory for transport phenomena. 7. Notes and remarks.

II. Representation theory and the scattering operator. 1. The discrete case. 2. The scattering operator in the discrete case. 3. The continuous case. 4. The scattering operator in the continuous case. 5. Notes and remarks.

III. A semigroup of operators related to the scattering matrix. 1. The related semigroups. 2. On semi-groups of contraction operators. 3. Spectral theory. 4. A spectral mapping theorem. 5. Applications of the spectral theory. 6. Equivalent incoming and outgoing representations. 7. Notes and remarks.

IV. The translation representation for the solution of the wave equation in free space. 1. The Hilbert space \(H_0\) and the group \(\{U_0(t)\}\). 2. Spectral and translation representations of \(\{U_0(t)\}\). 3. The operator \(\mathcal J\) extended to distributions. 4. Translation representation for outgoing and incoming data with finite energy. 5. Notes and remarks.

V. The solution of the wave equation in an exterior domain. 1. The Hilbert space \(H\) and the group (U(t)). 2. Energy decay and translation representations. 3. The semigroup \(\{Z(t)\}\). 4. The relation between the semigroup \(\{Z(t)\}\) and the solution of the reduced wave equation. 5. The scattering matrix. 6. Notes and remarks.

VI. Symmetric hyperbolic systems, the acoustic equation with an indefinite energy form, and the Schrödinger equation. Part 1. Symmetric hyperbolic systems. 1. Translation representation in free space. 2. Solution of hyperbolic systems in an exterior domain. Part 2. The acoustic equation with an indefinite energy form and the Schrödinger equation. 3. Scattering for the acoustic equation with an indefinite energy form. 4. The Schrödinger scattering matrix. 5. Notes and remarks.

App. 1. Semigroups of operators. App. 2. Energy decay. App. 3 (by C. S. Morawetz). Energy decay for star-shaped obstacles. App. 4 (by G. Schmidt), Scattering theory for Maxwell’s equations.

Reviewer: Klaus Habetha (Berlin)

### MSC:

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

47-02 | Research exposition (monographs, survey articles) pertaining to operator theory |

35P25 | Scattering theory for PDEs |

47A40 | Scattering theory of linear operators |

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

11F72 | Spectral theory; trace formulas (e.g., that of Selberg) |