On the Schrödinger equation and the eigenvalue problem

  title={On the Schr{\"o}dinger equation and the eigenvalue problem},
  author={Peter Li and Shing-Tung Yau},
  journal={Communications in Mathematical Physics},
  • Peter Li, S. Yau
  • Published 1 September 1983
  • Mathematics
  • Communications in Mathematical Physics
AbstractIf λk is thekth 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… 
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