On the regularity of the electronic Schrödinger equation in Hilbert spaces of mixed derivatives

Summary.The electronic Schrödinger equation describes the motion of electrons under Coulomb interaction forces in the field of clamped nuclei and forms the basis of quantum chemistry. The present article is devoted to the regularity properties of the corresponding wavefunctions that are compatible with the Pauli principle. It is shown that these wavefunctions possess certain square integrable mixed weak derivatives of order up to N+1 with N the number of electrons, across the singularities of the interaction potentials. The result is of particular importance for the analysis of approximation methods that are based on the idea of sparse grids or hyperbolic cross spaces. It indicates that such schemes could represent a promising alternative to current methods for the solution of the electronic Schrödinger equation and that it may even be possible to reduce the computational complexity of an N-electron problem to that of a one-electron problem.

[1]  Wolfgang Hackbusch.  Wavelet approximation of correlated wavefunctions � , 2003 .

[2]  Barry Simon,et al.  Schrödinger operators in the twentieth century , 2000 .

[3]  Israel Michael Sigal,et al.  The quantum N-body problem , 2000 .

[4]  A. Szabó,et al.  Modern quantum chemistry : introduction to advanced electronic structure theory , 1982 .

[5]  J. Neumann.  Mathematische grundlagen der Quantenmechanik , 1935 .

[6]  Wolfgang Hackbusch,et al.  Wavelet approximation of correlated wave functions. II. Hyperbolic wavelets and adaptive approximation schemes , 2002 .

[7]  Tosio Kato.  Fundamental properties of Hamiltonian operators of Schrödinger type , 1951 .

[8]  Reinhold Schneider,et al.  Wavelet approximation of correlated wave functions. I. Basics , 2002 .

[9]  M. Griebel,et al.  On the computation of the eigenproblems of hydrogen helium in strong magnetic and electric fields with the sparse grid combination technique , 2000 .

[10]  R. DeVore,et al.  Hyperbolic Wavelet Approximation , 1998 .

[11]  R. Parr.  Density-functional theory of atoms and molecules , 1989 .

[12]  Wolfgang Hackbusch,et al.  The Efficient Computation of Certain Determinants Arising in the Treatment of Schrödinger's Equations , 2001, Computing.

[13]  M. Ratner.  Molecular electronic-structure theory , 2000 .

[14]  M. Griebel,et al.  Optimized Tensor-Product Approximation Spaces , 2000 .

[15]  T. Hoffmann-Ostenhof,et al.  Local properties of Coulombic wave functions , 1994 .

[16]  T. Hoffmann-Ostenhof,et al.  Electron Wavefunctions and Densities for Atoms , 2000, math/0005018.

[17]  Søren Fournais,et al.  The Electron Density is Smooth Away from the Nuclei , 2002 .