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An uncertainty principle for fermions with generalized kinetic energy

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Abstract

We derive semiclassical upper bounds for the number of bound states and the sum of negative eigenvalues of the one-particle Hamiltoniansh=f(−i∇)+V(x) acting onL 2(ℝn). These bounds are then used to derive a lower bound on the kinetic energy\(\sum\limits_{j = 1}^N {\left\langle {\psi ,f( - i\nabla _j )\psi } \right\rangle }\) for anN-fermion wavefunction ψ. We discuss two examples in more detail:f(p)=|p| andf(p)=(p 2+m 2)1/2−m, both in three dimensions.

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References

  1. Reed, M., Simon, B.: Methods of modern mathematical physics, Vol. IV: Analysis of operators. New York: Academic Press 1978

    Google Scholar 

  2. Lieb, E., Thirring, W.: A bound for the kinetic energy of fermions which proves the stability of matter. Phys. Rev. Lett.35, 687–689 (1975). More details are given in Lieb, E., Thirring, W.: Inequalities for the moments of the eigenvalues of the Schrödinger Hamiltonian and their relation to Sobolev inequalities. In: Studies in Mathematical Physics: Essays in Honor of Valentine Bargmann, Lieb, E. H., Simon, B., Wightman, A. S., (eds.). Princeton: Princeton University Press 1976

    Google Scholar 

  3. Rosenbljum, G.: The distribution of the discrete spectrum for singular differential operators. Dokl. Akad. Nauk SSSR202 (1972) (Transl. Sov. Math. Dokl.13, 242–249 (1972))

  4. Lieb, E.: Bounds on the eigenvalues of the Laplace and Schrödinger operators. Bull. Am. Math. Soc.82, 751–753 (1976). More details can be found in Lieb, E.: The number of bound states of one-body Schrödinger operators and the Weyl problem. Proc. Am. Math. Soc.36, 241–252 (1980)

    Google Scholar 

  5. Cwikel, M.: Weak type estimates and the number of bound states of Schrödinger operators. Ann. Math.106, 93–102 (1977)

    Google Scholar 

  6. Lieb, E.: The stability of matter. Rev. Mod. Phys.48, 553–569 (1976)

    Google Scholar 

  7. Daubechies, I., Lieb, E.: One-electron relativistic molecules with Coulomb interaction. Commun. Math. Phys.90, (1983)

  8. Bratteli, O., Kishimoto, A., Robinson, D.: Positivity and monotonicity ofC 0-semigroups, I. Commun. Math. Phys.75, 67–84 (1980)

    Google Scholar 

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Authors and Affiliations

  1. Physics Department, Princeton University, 08544, Princeton, NJ, USA

    Ingrid Daubechies

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  1. Ingrid Daubechies
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Communicated by B. Simon

Work parially supported by US National Science Foundation Grant PHY-8116101-A01

On leave from Vrije Universiteit Brussel, and Interuniversitair Instituut voor Kernwetenschappen, Belgium

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Daubechies, I. An uncertainty principle for fermions with generalized kinetic energy. Commun.Math. Phys. 90, 511–520 (1983). https://doi.org/10.1007/BF01216182

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