The asymptotic formulas for the number of bound states in the strong coupling limit (Q1088070)

The Schrödinger operators \(-\Delta-\lambda V\) in \(L^ 2({\mathbb{R}}^ 3)\) with the following potentials: \(V>0\), \(V\in C^{\infty}({\mathbb{R}}^ 3\setminus \{0\})\); for \(| x| \leq 1\) \(V(x)\sim | x|^{- d}\), \(0\leq d<2\), \(| \partial^{\alpha}V| \leq C_{\alpha}V(x)| x|^{-\alpha}\), \(\forall \alpha\); for \(| x| \geq 1\) V(x)\(\leq c(1+| x|^ 2)^{-m}\), \(m>2\), \(| \partial^{\alpha}V(x)\leq C_{\alpha}V(x)(1+| x|^ 2)^{-\ell \alpha}\), \(1\geq \ell >3-m\), \(\forall \alpha\) are considered. Using Tauberian arguments the author proves the following asymptotic formulas for the number of bound states of \(-\Delta-\lambda V:\) \[ (6\pi^ 2)^{-1}\int V(x)^{3/2}dx\quad \lambda^{3/2}(1+O(\lambda^{- 1/2})),\quad \lambda \to \infty. \] In the concluding remarks the author states that the restrictions on the space dimension \(n=3\) and the positivity of V are not essential.