Regularity and Approximability of Electronic Wave Functions

By: Harry Yserentant

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Paperback | 4 June 2010

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The electronic Schrodi ] nger equation describes the motion of N electrons under Coulomb interaction forces in a eld of clamped nuclei. Solutions of this equation depend on 3N variables, three spatial dimensions for each electron. Approxim- ing the solutions is thus inordinately challenging, and it is conventionally believed that a reduction to simpli ed models, such as those of the Hartree-Fock method or density functional theory, is the only tenable approach. This book seeks to c- vince the reader 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 present notes arose from lectures that I gave in Berlin during the academic year 2008/09 to introduce beginning graduate students of mathematics into this subject. They are kept on an intermediate level that should be accessible to an audience of this kind as well as to physicists and theoretical chemists with a c- responding mathematical training.

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"This monograph discusses the possible direct approximation of the wavefunction itself, in view of several nice properties of the functions at hand (spatial exponential decay, symmetries, regularity, etc.). ... the book is very pleasant to read, having been written very carefully, and with many proofs provided explicitly. The text should be understandable to any graduate student in (applied) mathematics, as well as students and researchers from chemistry and physics ... ." (Gabriel Stoltz, Mathematical Reviews, Issue 2011 d)

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Non-Fiction Mathematics Calculus & Mathematical Analysis Functional Analysis & Transforms Science Physics Classical Mathematics Wave Mechanics including Vibration & Acoustics Differential Calculus & Equations Numerical Analysis Electricity

Introduction and Outline	p. 1
Fourier Analysis	p. 13
Rapidly Decreasing Functions	p. 13
Integrable and Square Integrable Functions	p. 17
Spaces of Weakly Differentiable Functions	p. 20
Fourier and Laplace Transformation	p. 25
The Basics of Quantum Mechanics	p. 27
Waves, Wave Packets, and Wave Equations	p. 28
The Schrödinger Equation for a Free Particle	p. 29
The Mathematical Framework of Quantum Mechanics	p. 32
The Harmonic Oscillator and Its Eigenfunctions	p. 37
The Weak Form of the Schrödinger Equation	p. 44
The Quantum Mechanics of Multi-Particle Systems	p. 46
The Electronic Schrödinger Equation	p. 51
The Hardy Inequality and the Interaction Energy	p. 52
Spin and the Pauli Principle	p. 55
Spectrum and Exponential Decay	p. 59
The Minimum Energy and the Ionization Threshold	p. 60
Discrete and Essential Spectrum	p. 61
The Rayleigh-Ritz Method	p. 71
The Lower Bound of the Essential Spectrum	p. 78
The Exponential Decay of the Eigenfunctions	p. 82
Existence and Decay of Mixed Derivatives	p. 87
A Modified Eigenvalue Problem	p. 88
Spaces of Functions with High-Order Mixed Derivatives	p. 91
Estimates for the Low-Order Terms, Part 1	p. 93
Estimates for the Low-Order Terms, Part 2	p. 97
The Regularity of the Weighted Eigenfunctions	p. 103
Atoms as Model Systems	p. 111
The Exponential Decay of the Mixed Derivatives	p. 115
Eigenfunction Expansions	p. 117
Discrete Regularity	p. 118
Antisymmetry	p. 123
Hyperbolic Cross Spaces	p. 124
Convergence Rates and Complexity Bounds	p. 127
The Growth of the Eigenvalues in the 3d-Case	p. 128
A Dimension Estimate for Hyperbolic Cross Spaces	p. 131
An Asymptotic Bound	p. 134
A Proof of the Estimate for the Partition Numbers	p. 135
The Complexity of the Quantum N-Body Problem	p. 139
The Radial-Angular Decomposition	p. 141
Three-Dimensional Spherical Harmonics	p. 142
The Decomposition of N-Particle Wave Functions	p. 151
The Radial Schrödinger Equation	p. 154
An Excursus to the Coulomb Problem	p. 159
The Harmonic Oscillator	p. 162
Eigenfunction Expansions Revisited	p. 164
Approximation by Gauss Functions	p. 165
The Effect of Scaling	p. 167
Appendix: The Standard Basis of the Spherical Harmonics	p. 170
References	p. 177
Index	p. 181
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Regularity and Approximability of Electronic Wave Functions

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