Analytic Solution for the Ground State Energy of the Extensive Many-Body Problem

Details Analytic Solution for the Ground State Energy of the Extensive Many-Body Problem Analytic Solution for the Ground State Energy of the Extensive Many-Body Problem

1996-02-16

arxiv.org/abs/cond-mat/9602091v1

Physics

Condensed Matter

10 pages REVTex3.0, 3 PS figures

Scientific paper

10.1103/PhysRevB.54.16309

A closed form expression for the ground state energy density of the general extensive many-body problem is given in terms of the Lanczos tri-diagonal form of the Hamiltonian. Given the general expressions of the diagonal and off-diagonal elements of the Hamiltonian Lanczos matrix, $\alpha_n(N)$ and $\beta_n(N)$, asymptotic forms $\alpha(z)$ and $\beta(z)$ can be defined in terms of a new parameter $z\equiv n/N$ ($n$ is the Lanczos iteration and $N$ is the size of the system). By application of theorems on the zeros of orthogonal polynomials we find the ground-state energy density in the bulk limit to be given in general by ${\cal E}_0 = {\rm inf}\,\left[\alpha(z) - 2\,\beta(z)\right]$.

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