##
**Spectral theory of random Schrödinger operators.**
*(English)*
Zbl 0717.60074

Probability and Its Applications. Basel etc.: Birkhäuser Verlag. xxvi, 587 p. DM 98.00; sFr 58.00 (1990).

This volume is the first self-contained account in book form (about 600 pages!) of the mathematical results of the spectral theory of random Schrödinger operators. This book is about wave propagation in random media and localization for disordered systems, the mathematics involved ranging from the spectral theory of selfadjoint operators to the theory of stochastic processes and of random matrices. The book includes the general facts of the abstract theory of deterministic and random selfadjoint operators on Hilbert spaces but mainly emphasizes the study of the specific models for which a detailed analysis is possible.

The selfadjoint operators in the study are of the form \(H=H_ 0+V\), where \(H_ 0\) is a deterministic quantization of the kinetic energy and V is the operator of multiplication by a potential function, regarded as a perturbation of \(H_ 0\). It is assumed that V is random; more precisely V(\(\cdot,\omega)\) is a sample from a stationary ergodic stochastic process (or random field) defined on some probability space. The full Hamiltonian H becomes a function of the random parameter \(\omega\), but essentially all the spectral sets (spectrum, absolutely continuous spectrum, singular continuous spectrum,...) are nonrandom. He has a tendency to have pure point spectrum, especially in low dimension or for large disorder. In some cases \(H_ 0\) is bounded and the potential energy may be much larger, and it is more convenient in these situations to regard the operator \(H_ 0\) as a perturbation of the potential operator; this is the approach to localization used for almost periodic potentials in one dimension and for multidimensional lattice systems.

The book is divided into nine chapters (plus a short introductory one); each of them is followed by a set of problems, notes and complements. The first three chapters are devoted to the functional analysis of selfadjoint operators in general, of Schrödinger operators in particular and of one- (or quasi-one-)dimensional systems. The general theory of products of random matrices is presented in Chapter IV and that of ergodic selfadjoint operators in Chapter V. The last four chapters deal with the actual spectral characteristics of random Schrödinger operators; except for some results on the integrated density of states. They can be read independently.

Chapter I: Spectral theory of selfadjoint operators. It is devoted to the introduction of the notations, the definitions and most of the results from functional analysis needed in the sequel.

Chapter II: Schrödinger operators. There are discussed their definitions and some of their spectral properties. The models considered are the so-called two-body Hamiltonians and the proofs are probabilistic whenever such a proof is available. Path integral methods are used to show the regularity properties of the generated semigroups and the decay of \(L^ 2\)-eigenfunctions, properties which are important in the investigation of the so-called generalized eigenfunction expansions.

Chapter III: One dimensional Schrödinger operators. It is devoted to the investigation of the spectral properties of one-dimensional Schrödinger operators. They deserve a special treatment for the theory of ordinary differential equations and its lattice analog; the main new input is the systematic use of the properties of the propagators of the ordinary differential operators or the transfer matrices in the lattice case.

Chapter IV: Products of random matrices. It is needed in the one- dimensional or quasi-one-dimensional theory of localization. One of the most important results in this direction is the extension to matrix- valued random variables of the strong law of large numbers.

Chapter V: Ergodic families of selfadjoint operators. The first part of this chapter deals with the definition of the random Schrödinger operators H(\(\omega\)) on some probability space. The second part of the chapter deals with the various spectra of the operators H(\(\omega\)). Finally, some regularity properties of the Lyapunov exponents of one- dimensional and quasi-one-dimensional Schrödinger operators are discussed.

Chapter VI: The integrated density of states. It is proved its existence for fairly general families of stationary ergodic random potentials and that its topological support is almost surely equal to the spectrum of the operator. In particular, it has to vanish at the left most point of this spectrum. The authors study the following question: How fast does it vanish? In the one-dimensional cases and the lattice cases many smoothness results are presented.

Chapter VII: Absolutely continuous spectrum and inverse theory. The aim of this chapter is to study the essential support of the absolutely continuous component \(E_{ac}(\cdot,\omega)\) of the resolution of the identity of \(H(\omega)=H_ 0+V(\cdot,\omega)\). The case of the one- dimensional periodic potentials is completely analysed using the ordinary differential equation approach developed in Chapter III and the Floquet theory of periodic differential equations. The authors prove that in the one-dimensional case it is possible to identify the (almost sure) absolutely continuous spectrum and the set where the Lyapunov exponent vanishes. A section is devoted to the basic elements of an inverse spectral theory for one-dimensional random Schrödinger operators.

Chapter VIII: Localization in one dimension. It is proved that for one- dimensional random Schrödinger operators the spectrum is pure point with exponentially decaying eigenfunctions (three different proofs are given).

Chapter IX: Localization in any dimension. It is devoted to the extension to quasi-one-dimensional systems of the results of Chapter VIII.

The book is a valuable reference and learning tool for researchers and graduate students in probability theory and mathematical physics.

The selfadjoint operators in the study are of the form \(H=H_ 0+V\), where \(H_ 0\) is a deterministic quantization of the kinetic energy and V is the operator of multiplication by a potential function, regarded as a perturbation of \(H_ 0\). It is assumed that V is random; more precisely V(\(\cdot,\omega)\) is a sample from a stationary ergodic stochastic process (or random field) defined on some probability space. The full Hamiltonian H becomes a function of the random parameter \(\omega\), but essentially all the spectral sets (spectrum, absolutely continuous spectrum, singular continuous spectrum,...) are nonrandom. He has a tendency to have pure point spectrum, especially in low dimension or for large disorder. In some cases \(H_ 0\) is bounded and the potential energy may be much larger, and it is more convenient in these situations to regard the operator \(H_ 0\) as a perturbation of the potential operator; this is the approach to localization used for almost periodic potentials in one dimension and for multidimensional lattice systems.

The book is divided into nine chapters (plus a short introductory one); each of them is followed by a set of problems, notes and complements. The first three chapters are devoted to the functional analysis of selfadjoint operators in general, of Schrödinger operators in particular and of one- (or quasi-one-)dimensional systems. The general theory of products of random matrices is presented in Chapter IV and that of ergodic selfadjoint operators in Chapter V. The last four chapters deal with the actual spectral characteristics of random Schrödinger operators; except for some results on the integrated density of states. They can be read independently.

Chapter I: Spectral theory of selfadjoint operators. It is devoted to the introduction of the notations, the definitions and most of the results from functional analysis needed in the sequel.

Chapter II: Schrödinger operators. There are discussed their definitions and some of their spectral properties. The models considered are the so-called two-body Hamiltonians and the proofs are probabilistic whenever such a proof is available. Path integral methods are used to show the regularity properties of the generated semigroups and the decay of \(L^ 2\)-eigenfunctions, properties which are important in the investigation of the so-called generalized eigenfunction expansions.

Chapter III: One dimensional Schrödinger operators. It is devoted to the investigation of the spectral properties of one-dimensional Schrödinger operators. They deserve a special treatment for the theory of ordinary differential equations and its lattice analog; the main new input is the systematic use of the properties of the propagators of the ordinary differential operators or the transfer matrices in the lattice case.

Chapter IV: Products of random matrices. It is needed in the one- dimensional or quasi-one-dimensional theory of localization. One of the most important results in this direction is the extension to matrix- valued random variables of the strong law of large numbers.

Chapter V: Ergodic families of selfadjoint operators. The first part of this chapter deals with the definition of the random Schrödinger operators H(\(\omega\)) on some probability space. The second part of the chapter deals with the various spectra of the operators H(\(\omega\)). Finally, some regularity properties of the Lyapunov exponents of one- dimensional and quasi-one-dimensional Schrödinger operators are discussed.

Chapter VI: The integrated density of states. It is proved its existence for fairly general families of stationary ergodic random potentials and that its topological support is almost surely equal to the spectrum of the operator. In particular, it has to vanish at the left most point of this spectrum. The authors study the following question: How fast does it vanish? In the one-dimensional cases and the lattice cases many smoothness results are presented.

Chapter VII: Absolutely continuous spectrum and inverse theory. The aim of this chapter is to study the essential support of the absolutely continuous component \(E_{ac}(\cdot,\omega)\) of the resolution of the identity of \(H(\omega)=H_ 0+V(\cdot,\omega)\). The case of the one- dimensional periodic potentials is completely analysed using the ordinary differential equation approach developed in Chapter III and the Floquet theory of periodic differential equations. The authors prove that in the one-dimensional case it is possible to identify the (almost sure) absolutely continuous spectrum and the set where the Lyapunov exponent vanishes. A section is devoted to the basic elements of an inverse spectral theory for one-dimensional random Schrödinger operators.

Chapter VIII: Localization in one dimension. It is proved that for one- dimensional random Schrödinger operators the spectrum is pure point with exponentially decaying eigenfunctions (three different proofs are given).

Chapter IX: Localization in any dimension. It is devoted to the extension to quasi-one-dimensional systems of the results of Chapter VIII.

The book is a valuable reference and learning tool for researchers and graduate students in probability theory and mathematical physics.

Reviewer: V.Iftimie

### MSC:

60H25 | Random operators and equations (aspects of stochastic analysis) |

47B80 | Random linear operators |