Regularity and approximability of electronic wave functions.

*(English)*Zbl 1204.35003
Lecture Notes in Mathematics 2000. Dordrecht: Springer (ISBN 978-3-642-12247-7/pbk; 978-3-642-12248-4/ebook). viii, 182 p. (2010).

The electronic Schrödinger equation describes the motion of \(N\)-electrons under Coulomb interaction forces in a field of clamped nuclei. The solutions of this equation, the electronic wave functions, depend on \(3N\) variables, with three spatial dimensions for each electron. Approximating these solutions is thus inordinately challenging, and it is generally believed that a reduction to simplified models, such as those of the Hartree-Fock method or density functional theory, is the only tenable approach.

This book seeks to show that this conventional wisdom need not be ironclad: the regularity of the solutions, which increases with the number of electrons, the decay behavior of their mixed derivatives, and the antisymmetry enforced by the Pauli principle contribute properties that allow these functions to be approximated with an order of complexity which comes arbitrarily close to that for a system of one or two electrons. The text is accessible to a mathematical audience at the beginning graduate level as well as to physicists and theoretical chemists with a comparable mathematical background and requires no deeper knowledge of the theory of partial differential equations, functional analysis, or quantum theory.

The organization of the book is as follows: The first chapter reviews the foundations of quantum mechanics and summarizes a reasonably comprehensive overview of the various methods and strategies associated with the solution of the electronic Schrödinger equation. The second chapter is a short but concise one and it is devoted to Fourier analysis and spaces of weakly differentiable functions. Chapter three introduces various concepts of quantum mechanics. Chapter four presents the quantum mechanical description of atoms, molecules, and ions by employing the electronic Schrödinger equation for a system of charged particles that interact by Coulomb attraction and repulsion forces. Chapter five studies the solutions of the electronic Schrödinger equation and compile and prove some basic facts about its solutions in suitable form. Chapter six presents topics, such as the existence and decay of mixed derivatives of the electronic wave functions, and investigate the regularity of these functions. Chapter seven derives the discrete counterparts of the regularity theorems from the previous chapter similar to how smoothness can be characterized for periodic functions in terms of the decay rate of their Fourier coefficients. Chapter eight discusses the convergence rates and complexity bounds associated with the existing algorithms. Finally, the last chapter applies the symmetry principles in quantum mechanics for the decomposition of \(N\)-particle wave functions and the radial Schrödinger equation.

This book seeks to show that this conventional wisdom need not be ironclad: the regularity of the solutions, which increases with the number of electrons, the decay behavior of their mixed derivatives, and the antisymmetry enforced by the Pauli principle contribute properties that allow these functions to be approximated with an order of complexity which comes arbitrarily close to that for a system of one or two electrons. The text is accessible to a mathematical audience at the beginning graduate level as well as to physicists and theoretical chemists with a comparable mathematical background and requires no deeper knowledge of the theory of partial differential equations, functional analysis, or quantum theory.

The organization of the book is as follows: The first chapter reviews the foundations of quantum mechanics and summarizes a reasonably comprehensive overview of the various methods and strategies associated with the solution of the electronic Schrödinger equation. The second chapter is a short but concise one and it is devoted to Fourier analysis and spaces of weakly differentiable functions. Chapter three introduces various concepts of quantum mechanics. Chapter four presents the quantum mechanical description of atoms, molecules, and ions by employing the electronic Schrödinger equation for a system of charged particles that interact by Coulomb attraction and repulsion forces. Chapter five studies the solutions of the electronic Schrödinger equation and compile and prove some basic facts about its solutions in suitable form. Chapter six presents topics, such as the existence and decay of mixed derivatives of the electronic wave functions, and investigate the regularity of these functions. Chapter seven derives the discrete counterparts of the regularity theorems from the previous chapter similar to how smoothness can be characterized for periodic functions in terms of the decay rate of their Fourier coefficients. Chapter eight discusses the convergence rates and complexity bounds associated with the existing algorithms. Finally, the last chapter applies the symmetry principles in quantum mechanics for the decomposition of \(N\)-particle wave functions and the radial Schrödinger equation.

Reviewer: Ömer Kavaklioglu (Izmir)

##### MSC:

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

35Q60 | PDEs in connection with optics and electromagnetic theory |

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