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The Number of Bound States of One-Body Schroedinger Operators and the Weyl Problem

  • Elliott H. Lieb

Abstract

If Ñ( Ω, λ) is the number of eigenvalues of A in a domain Ω, in a suitable Riemannian manifold of dimension n, we derive bounds of the form Ñ(Ω, λ)< D λn/2}Ω{ for all Ω, λ , n , Likewise, if Nα(V) is the number of nonpositive eigenvalues of -Δ + V(x) which are ≤ α ≤ 0, then \( {N_\alpha }(V) \leqslant {L_n}{\int\limits_M {\left[ {V - \alpha } \right]} _ - }^{n/2} \) for all α and V and n ≥ 3.

Keywords

Pure Math Semigroup Property Sharp Constant Weak Type Estimate Classical Constant 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Elliott H. Lieb
    • 1
  1. 1.Departments of Mathematics and PhysicsPrinceton University Jadwin HallPrincetonUSA

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