Regularity and Approximability of Electronic Wave Functions

1. Front Matter

   Pages i-viii

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2. Introduction and Outline

   • Harry Yserentant

   Pages 1-11

3. Fourier Analysis

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   Pages 13-26

4. The Basics of Quantum Mechanics

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   Pages 27-50

5. The Electronic Schrödinger Equation

   • Harry Yserentant

   Pages 51-58

6. Spectrum and Exponential Decay

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   Pages 59-85

7. Existence and Decay of Mixed Derivatives

   • Harry Yserentant

   Pages 87-116

8. Eigenfunction Expansions

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   Pages 117-125

9. Convergence Rates and Complexity Bounds

   • Harry Yserentant

   Pages 127-140

10. The Radial-Angular Decomposition

    • Harry Yserentant

    Pages 141-176

11. Back Matter

    Pages 177-188

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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.

• differential equation
• fourier analysis
• functional analysis
• partial differential equation
• partial differential equations

From the reviews:

“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)

• Institut für Mathematik, TU Berlin, Berlin, Germany

  Harry Yserentant

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