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.
Highlights
Thomas-Fermi equation as a special case of nonlinear Poisson equations arises from a statistical model of many-electron atoms
One special Euler-Lagrange equation as a minimizer of energy functional gives a nonlinear Poisson equation which is an extended version of Thomas-Fermi equation
Proof: it will be shown that the weak solution of the Lagrangian which is a minimizer of the Euler-Lagrange equation is the key to this enigma
Summary
Thomas-Fermi equation as a special case of nonlinear Poisson equations arises from a statistical model of many-electron atoms. The intersections of these loci are the singular Theorem 1: The first integral of Thomas-Fermi Eq (3) points of the differential equation. They are a node at the with boundary conditions (4) and (5), the path origin O and a saddle point at p (144, −432). In the which satisfies the boundary condition (5) and the behaviour of the solution near the saddle point of the equation can be determined In effect, it can be given an algorithm which defines the integration of the problem (3) with boundary conditions (4) and (5). This problem can be solved in Theorem 1 by analytic ingredients, especially weak solutions
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