Asymptotic formulas with sharp remainder estimates for bound states of Schrödinger operators. II (Q801482)

This paper is a sequel to part I [ibid. 40, 166-182 (1981; Zbl 0501.35063)]. It completes the proof of Theorem A in the previous paper on the asymptotic formula for the bound states (i.e. negative eigenvalues) of Schrödinger operators \(H=-\Delta -V\) in \(L^ 2({\mathbb{R}}^ n)\). The assumptions made on the potential V are as in the first paper. If N(\(\lambda)\), \(\lambda >0\), denotes the number of eigenvalues of H which are less than -\(\lambda\) and \(H_ 0(x,\xi)=| \xi |^ 2-V(x)\), Theorem A is the following result: if \(n\geq 2\) \[ N(\lambda)=(2\pi)^{- n}\iint_{H_ 0(x,\xi)<-\lambda}d\xi dx(1+O(\lambda^{[1/m- 1/2]}))\quad as\quad \lambda \to 0. \] In part I it was shown that this theorem can be extended to hold for a wide class of singular potentials which includes Coulomb potential 1/\(| x|\) for which the remainder estimate for N(\(\lambda)\) is attained.