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An uncertainty principle for fermions with generalized kinetic energy. (English) Zbl 0946.81521
Summary: We derive semiclassical upper bounds for the number of bound states and the sum of negative eigenvalues of the one-particle Hamiltonians $$h=f(-i\nabla)+V(x)$$, acting on $$L^2({R}^n)$$. These bounds are then used to derive a lower bound on the kinetic energy $$\sum^N_{j=1}\langle\psi,f(-i\nabla_j)\psi\rangle$$ for an $$N$$-fermion wavefunction $$\psi$$. We discuss two examples in more detail: $$f(0)=|p|$$ and $$f(p)=\break (p^2+m^2)^{1/2}-m$$, both in three dimensions.”.

##### MSC:
 81V55 Molecular physics
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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 · Zbl 0401.47001 [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 · doi:10.1103/PhysRevLett.35.687 [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) · Zbl 0329.35018 · doi:10.1090/S0002-9904-1976-14149-3 [5] Cwikel, M.: Weak type estimates and the number of bound states of Schrödinger operators. Ann. Math.106, 93-102 (1977) · Zbl 0362.47006 · doi:10.2307/1971160 [6] Lieb, E.: The stability of matter. Rev. Mod. Phys.48, 553-569 (1976) · doi:10.1103/RevModPhys.48.553 [7] Daubechies, I., Lieb, E.: One-electron relativistic molecules with Coulomb interaction. Commun. Math. Phys.90, (1983) · Zbl 0946.81522 [8] Bratteli, O., Kishimoto, A., Robinson, D.: Positivity and monotonicity ofC 0-semigroups, I. Commun. Math. Phys.75, 67-84 (1980) · Zbl 0453.47020 · doi:10.1007/BF01962592
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