The density in a one-dimensional potential
In [ibid. 95, No. 2, 145-305 (1992; Zbl 0797.34083)] we proved precise estimates for the eigenvalues \(E_ k\) and normalized eigenfunctions \(\varphi_ k\) of an ordinary differential operator \(H = - d^ 2/dx^ 2 + V(x)\) on an interval \(I_{\text{BVP}}\). In this paper, we apply these results to study the density \[ \rho (x) = \sum_{E_ k \leq 0} | \varphi_ k (x) |^ 2. \] We introduce a simple approximation \(\rho_{sc} (x)\) and estimate \(\rho-\rho_{sc}\). Our goal is proving the asymptotic formula announced in [Bull. Am. Math. Soc., New Ser. 23, No. 2, 525-530 (1990; Zbl 0722.35072)] for the ground-state energy of an atom. This requires estimates for \(\rho - \rho_{sc}\) in weighted Sobolev norms of order \(-1\), which we derive here. The complete proofs of the results are given here and in a series of further papers.
scientific article; zbMATH DE number 703896