Abstract

This study examines the well-known Thomas-Fermi equation as a Euler-Lagrange equation associated with the Fermi energy. The first integral of Thomas-Fermi equation and the behaviour of the solution near the saddle point of the equation has been determined. Then, drawing upon advanced ingredients of Sobolev spaces and weak solutions, an exact methodology is presented for the quantum correction near the origin of Thomas-Fermi equation. By this approach, the existence and uniqueness of the minimizer for the energy functional of the Thomas-Fermi equation have been proved. It has been demonstrated that by the definition of such a functional and the relevant Sobolev spaces, the Thomas-Fermi equation, particularly of a neutral atom, extends to the nonlinear Poisson equation. Accordingly, weak solutions for the more general Euler-Lagrange equation with more singularities are proposed.

Keywords:

Euler-lagrange equation, fermi energy, sobolev space, thomas-fermi equation, weak solution,

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Competing interests

The authors have no competing interests.

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