On the Schrödinger equation and the eigenvalue problem

Abstract

If λ _k is thek ^th eigenvalue for the Dirichlet boundary problem on a bounded domain in ℝⁿ, H. Weyl's asymptotic formula asserts that$\lambda _k \sim C_n \left( {\frac{k}{{V(D)}}} \right)^{2/n}$, hence$\sum\limits_{i = 1}^k {\lambda _i \sim \frac{{nC_n }}{{n + 2}}k^{\frac{{n + 2}}{n}} V(D)^{ - 2/n} }$. We prove that for any domain and for all$\sum\limits_{i = 1}^k {\lambda _i \geqq \frac{{nC_n }}{{n + 2}}k^{\frac{{n + 2}}{n}} V(D)^{ - 2/n} }$. A simple proof for the upper bound of the number of eigenvalues less than or equal to -α for the operator Δ−V(x) defined on ℝⁿ (n≧3) in terms of$\int\limits_{\mathbb{R}^n } {(V + \alpha )_ - ^{n/2} }$ is also provided.