Abstract
If Ñ(Ω,λ) is the number of eigenvalues of -Δ in a domain Ω in a suitable Riemannian manifold of dimension n, we derive bounds of the form Ñ(Ω,λ)≤ Dn 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 Nα(V≤ Ln∫M [V − α]- n/2 for all α and V and n ≥ 3.
Work supported by U.S.National Foundation grants PHYS-7825390 and INT 78-01160.
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Lieb, E.H. (1997). The Number of Bound States of One-Body Schroedinger Operators and the Weyl Problem. In: Thirring, W. (eds) The Stability of Matter: From Atoms to Stars. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-03436-1_19
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