zbMATH — the first resource for mathematics

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.”.

81V55 Molecular physics
Full Text: DOI
[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.