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Communications in Mathematical Physics

, Volume 88, Issue 3, pp 309–318 | Cite as

On the Schrödinger equation and the eigenvalue problem

  • Peter Li
  • Shing-Tung Yau
Article

Abstract

If λ k is thekth eigenvalue for the Dirichlet boundary problem on a bounded domain in ℝ n , 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 ℝ n (n≧3) in terms of\(\int\limits_{\mathbb{R}^n } {(V + \alpha )_ - ^{n/2} } \) is also provided.

Keywords

Neural Network Statistical Physic Complex System Nonlinear Dynamics Eigenvalue Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag 1983

Authors and Affiliations

  • Peter Li
    • 1
  • Shing-Tung Yau
    • 2
  1. 1.Department of MathematicsStanford UniversityStanfordUSA
  2. 2.School of MathematicsInstitute for Advanced StudyPrincetonUSA

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