Existence and Decay of Mixed Derivatives

Abstract

A primary aim of this work, and the decisive step to our analysis of the complexity of electronic wave functions, is to study the regularity of these functions. We want to show that they possess certain high-order square integrable weak derivatives and that these derivatives even decay exponentially, in the same way as the wave functions themselves. This goal is reached in the present chapter. A central idea of the proof is to examine instead of the solutions of the original Schrödinger equation the solutions of a modified equation for the correspondingly exponentially weighted wave functions. This equation is set up in the first section of this chapter and is based on the result on the exponential decay of the wave functions from Sect. 5.5. The study of the regularity in isotropic Hölder spaces in [32] is based on a similar idea. In Sect. 6.2 we introduce the high-order solution spaces and the corresponding norms. The actual proof relies on a mixture of variational techniques and Fourier analysis. The key is the estimates for the arising low-order terms, particularly for the nucleus-electron and the electron-electron interaction potential. These estimates are proven in Sect. 6.3 and Sect. 6.4. The estimates for the nucleus-electron interaction potential and an additional term coming from the exponential weights are in the end based on the Hardy inequality from Sect. 4.1, whose central role is reflected here again. In contrast to these estimates the estimates for the electron-electron interaction potential require that the considered functions satisfy the Pauli principle, that is, are antisymmetric with respect to the exchange of the positions of electrons with the same spin. The reason is that such functions vanish at the places where electrons with the same spin meet, which counterbalances the singularities of the electron-electron interaction potential. To derive these estimates and to master the arising singularities a further three-dimensional Hardy-type estimate is needed that holds only for functions vanishing at the origin. In Sect. 6.5 the regularity theorem for the exponentially weighted wave functions is stated and proven. This result serves then to derive bounds for the exponential decay of the mixed derivatives of the original wave functions. The present chapter is partly based on two former papers [92,94] of the author in which the existence of the mixed derivatives has been proven and estimates for their L₂-norms were given. The result on the exponential decay of these derivatives [95] was up to now only available on the author’s website.