Inequalities for the Moments of the Eigenvalues of the Schrödinger Hamiltonian and their Relation to Sobolev Inequalities

Elliott H. Lieb and Walter E. Thirring
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Elliott H. Lieb: Princeton University, Departments of Mathematics and Physics
Walter E. Thirring: Universität Wien, Institut für Theoretische Physik

A chapter in Inequalities, 2002, pp 203-237 from Springer

Abstract: Abstract Estimates for the number of bound states and their energies, ej ≤ 0, are of obvious importance for the investigation of quantum mechanical Hamiltonians. If the latter are of the single particle form H = ≤ Δ + V(x) in Rn, we shall use available methods to derive the bounds 1 $${\sum\limits_j {\left| {{e_j}} \right|} ^y} \le {L_{y,n}}\int {{d^n}} x\left| {V(x)} \right|_ - ^{y + n/2},y > \max (0,1 - n/2)$$ Here, $$\left| {V(x)} \right|\_ = - V(x){\rm{ if V(x)}} \le {\rm{0}}$$ and is zero otherwise.

Keywords: Mathematical Physic; Sobolev Inequality; Bare Matrice; Single Particle Form; Bound State Eigenvalue (search for similar items in EconPapers)
Date: 2002
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-642-55925-9_20

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DOI: 10.1007/978-3-642-55925-9_20

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