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On the Schrödinger equation and the eigenvalue problem

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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.

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Communicated by B. Simon

Research partially supported by a Sloan Fellowship and NSF Grant No. 81-07911-A1

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Li, P., Yau, ST. On the Schrödinger equation and the eigenvalue problem. Commun.Math. Phys. 88, 309–318 (1983).

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