Journal of Mathematical Physics

ℓP interpolation and optimized bounds on pairwise interacting fermion systems

Jean-Louis Basdevant, André Martin

Published Dec 1, 1996• 3 citations

Abstract

We derive a set of inequalities which relate the translation invariant problem of N identical particles with pairwise interactions to an independent particle problem. These inequalities apply to attractive power law potentials V(r)=rq, 1≤q≤∞, and superpositions of such potentials; they become identities in the harmonic oscillator case q=2. We use the inequalities to derive new upper and lower bounds for the ground state energies of fermion systems, which interact through these potentials. These bounds improve all previous results in the range 1≤q≤∞; they reduce to the exact answer in the harmonic oscillator case.

DOI: 10.1063/1.531756

Related papers

Energy lower bounds for translation-invariantN-fermion systems

Richard L. Hall

A restriction on the choice of translation-invariant coordinates for a system of N identical fermions, interacting by pair forces, allows a lower bound to be established on the ground-state binding energy. This bound is the sum of the first N -1 eigenvalues of a two-particle problem. The quality of the method is found to depend on the choice of coordinates consistent with the initial restraint, the optimum set giving a bound which, in the case of Hooke's law pair forces, is [radical]3( N -1)/2( N +1) times the exact energy.

Proceedings of the Physical SocietyMay 1, 196732 citations

Lower energy bounds for translation-invariant atomlike systems

Richard L. Hall

A system of N identical particles with masses m1 interact with each other via a pair potential V and with an additional particle of mass m0 by the pair potential U, and the system obeys nonrelativistic quantum mechanics. A general lower bound formula is derived for the ground-state energy, which reduces to Coleman’s result in the special case m0=∞. The lower bound is compared with the exact energy for the harmonic oscillator problem V (x) =k21x2 and U (x) =k02x2, which has recently been solved exactly (Hall).

Journal of Mathematical PhysicsSep 1, 19785 citations

Lower bound on the ground-state energy and one-dimensionalN-fermion problem

Srećko Kilić, K. Mikelić

We derive an expression for a lower bound of the ground-state energy of N identical fermions. In principle, this relation can describe any order of multiplicity of many-body forces and represents a generalization of a Hall inequality. Applying the Hall lower-bound inequality to the Calogero and Sutherland potentials, we show the great importance of spin states. Furthermore, using a generalized formula, we obtain a lower bound for a system with three-body forces.

Physical Review AOct 1, 19933 citations

A set of inequalities between the ground state energies of many-body systems

John Lekner

The energy EN of a system of N identical pairwise-interacting particles of mass M is shown to satisfy the inequalities where N > n ⩾ 2. These inequalities generalize the Stenschke result E 3(M) ⩾ 3E 2(3M/2). They hold for Bose or Fermi systems in any number of dimensions, and are shown to be the strongest possible of this type. Further extensions are made to the case where the potential is varied instead of the mass, or where both are varied together.

Molecular PhysicsFeb 1, 19733 citations

Exact first-order effect of interactions on the ground-state energy of harmonically-confined fermions

Pierre Le Doussal, Naftali R. Smith, Nathan Argaman

We consider a system of N N spinless fermions, interacting with each other via a power-law interaction \epsilon/r^n ϵ / r n , and trapped in an external harmonic potential V(r) = r^2/2 V ( r ) = r 2 / 2 , in d=1,2,3 d = 1 , 2 , 3 dimensions. For any 0 < n < d+2 0 < n < d + 2 , we obtain the ground-state energy E_N E N of the system perturbatively in \epsilon ϵ , E_{N}=E_{N}^{≤ft(0)}+\epsilon E_{N}^{≤ft(1)}+O≤ft(\epsilon^{2}) E N = E N ≤ f t ( 0 ) + ϵ E N ≤ f t ( 1 ) + O ≤ f t ( ϵ 2 ) . We calculate E_{N}^{≤ft(1)} E N ≤ f t ( 1 ) exactly, assuming that N N is such that the “outer shell” is filled. For the case of n=1

arXiv:2311.09013SciPost PhysicsAug 7, 20241 citation

Many-particle systems VI. N-fermion systems: derivation of a lower-bound shell model

R J M Carr, H R Post

The programme of reducing N -particle problems to one-particle problems is extended to include the Pauli principle. To calculate the ground-state energy of a system of N fermions interacting by pair forces, a rigorous lower-bound shell model is derived. This shell model follows a building-up principle, and tends to improve with increasing number of particles as well as with increasing strength of interaction. For twenty particles antisymmetric in ordinary space interacting by square-well interaction of strength V 0 a 2 = 200 hslash 2 /2 m the shell model differs by less than 8% from the calculated upper bound, and hence a fortiori from the exact energy. In order to test the quality of the approximation for various interactions, it was necessary to calculate upper bounds.

Journal of Physics A Mathematical and GeneralSep 1, 196810 citations