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


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.


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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Aubin, T.: Problèmes isopérimétriques et espaces de Sobolev. J. Diff. Geom.11, 573–598 (1976)Google Scholar
  2. 2.
    Cheng, S.Y., Li, P.: Heat kernel estimates and lower bound of eigenvalues. Commun. Math. Helv. (56)3, 327–338 (1981)Google Scholar
  3. 3.
    Cwikel, W.: Weak type estimates for singular values and the number of bound states of Schrödinger operators. Ann. Math.106, 93–100 (1977)Google Scholar
  4. 4.
    Lieb, E.: The number of bound states of one-body Schrödinger operators and the Weyl problem. Proc. Sym. Pure Math.36, 241–252 (1980)Google Scholar
  5. 5.
    Lieb, E., Thirring, W.: Inequalities for the moments of the eigenvalues of the Schrödinger equation and their relation to Sobolev inequalities. Studies in Math. Phys.: Essay in Honor of Valentine Bargmann. Princeton, NJ: Princeton University Press 1976Google Scholar
  6. 6.
    Pólya, G.: On the eigenvalues of vibrating membranes. Proc. London Math. Soc. (3)11, 419–433 (1961)Google Scholar
  7. 7.
    Rosenbljum, G.V.: Distribution of the discrete spectrum of singular operator. Dokl. Akad. Nauk SSSR202, 1012–1015 (1972)Google Scholar
  8. 8.
    Simon, B.: Weak trace ideals and the number of bound states of Schrödinger operators. Trans. Am. Math. Soc.224, 367–380 (1976)Google Scholar
  9. 9.
    Lieb, E.: Bounds on the eigenvalues of the Laplace and Schroedinger operators. Bull. AMS82, 751–753 (1976)Google Scholar

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

Personalised recommendations