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Inequalities pp 203-237 | Cite as

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

  • Elliott H. Lieb
  • Walter E. Thirring

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
$${\sum\limits_j {\left| {{e_j}} \right|} ^y} \le {L_{y,n}}\int {{d^n}} x\left| {V(x)} \right|_ - ^{y + n/2},y > \max (0,1 - n/2)$$
(1)
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 
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.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Elliott H. Lieb
    • 1
  • Walter E. Thirring
    • 2
  1. 1.Departments of Mathematics and PhysicsPrinceton UniversityPrincetonUSA
  2. 2.Institut für Theoretische PhysikUniversität WienAustria

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