The exactly-solvable position-dependent mass Schrodinger equation (PDMSE) for the Thomas-Fermi potential is presented. The PDMSE is transformed into a standard Schrodinger equation (SSE) with constant mass by means of a point canonical transformation scheme. By proposing an exponential type potential of the SSE it is possible to determine a PDMSE with the Thomas-Fermi potential. The resulting PDMSE is carried to the Sturm- Liouville form and the corresponding theory is developed for the particular resulting problem in order to obtain closed solutions in terms of Bessel functions of the first kind. Beyond the case considered in this work, the approach is general and can be useful in the study of electronic properties of non-uniform materials in which the carrier effective mass depends on the position as well as in the search of new solvable potentials suitable for quantum systems.The exactly-solvable position-dependent mass Schrodinger equation (PDMSE) for the Thomas-Fermi potential is presented. The PDMSE is transformed into a standard Schrodinger equation (SSE) with constant mass by means of a point canonical transformation scheme. By proposing an exponential type potential of the SSE it is possible to determine a PDMSE with the Thomas-Fermi potential. The resulting PDMSE is carried to the Sturm- Liouville form and the corresponding theory is developed for the particular resulting problem in order to obtain closed solutions in terms of Bessel functions of the first kind. Beyond the case considered in this work, the approach is general and can be useful in the study of electronic properties of non-uniform materials in which the carrier effective mass depends on the position as well as in the search of new solvable potentials suitable for quantum systems.

Journal of Physics Conference SeriesJan 21, 20150 citations

The Thomas-Fermi Energy Density Functional and Its Generalizations

Eugene S. Kryachko, Eduardo V. Ludeña

Springer eBooksJan 1, 19900 citations

A sharp gradient-corrected Lieb--Thirring inequality in one dimension with applications to density functional theory

Thiago Carvalho Corso, Louis Garrigue

We first prove a sharp Lieb-Thirring inequality with semi-classical constant and gradient correction in one dimension. In the periodic setting, the corresponding inequality also includes a parity dependent shell-correction term, which appears to be new. Moreover, we also characterize the optimizers of these inequalities. In particular, the optimizing density profiles form a two-dimensional manifold, which we call here the Poincaré density manifold. We then apply these results to study the Thomas--Fermi--von Weizsäcker (TFW) approximation of the kinetic energy in Density Functional Theory. More precisely, we first obtain an explicit and non-asymptotic bound on the global and local error made by the TFW approximation. These bounds are controlled by the distance of the normalized density profile from the Poincaré density manifold times the number of particles. In particular, these bounds show that the TFW approximation is exact on the Poincaré manifold. We then show that, in the periodic setting, the distance between the Kohn-Sham potential and the inverse TFW potential in the natural dual Sobolev space can also be controlled by explicit density dependent constants times the distance to the Poincaré manifold. As a corollary, we prove that for a fixed density profile, the Kohn-Sham potential converges towards the inverse Thomas--Fermi potential in the high-density limit $N \rightarrow \infty$ with asymptotic rate of order $N^{-2}$ . This rate is shown to be optimal.We first prove a sharp Lieb-Thirring inequality with semi-classical constant and gradient correction in one dimension. In the periodic setting, the corresponding inequality also includes a parity dependent shell-correction term, which appears to be new. Moreover, we also characterize the optimizers of these inequalities. In particular, the optimizing density profiles form a two-dimensional manifold, which we call here the Poincaré density manifold. We then apply these results to study the Thomas--Fermi--von Weizsäcker (TFW) approximation of the kinetic energy in Density Functional Theory. More precisely, we first obtain an explicit and non-asymptotic bound on the global and local error made by the TFW approximation. These bounds are controlled by the distance of the normalized density profile from the Poincaré density manifold times the number of particles. In particular, these bounds show that the TFW approximation is exact on the Poincaré manifold. We then show that, in the periodic setting, the distance between the Kohn-Sham potential and the inverse TFW potential in the natural dual Sobolev space can also be controlled by explicit density dependent constants times the distance to the Poincaré manifold. As a corollary, we prove that for a fixed density profile, the Kohn-Sham potential converges towards the inverse Thomas--Fermi potential in the high-density limit $N \rightarrow \infty$ with asymptotic rate of order $N^{-2}$. This rate is shown to be optimal.

math-phCross-listedarXiv:2610.30887Oct 23, 2026

N \rightarrow \infty

with asymptotic rate of order

N^{-2}

. This rate is shown to be optimal.