[Paper Review] On crystal ground state in the Schrödinger-Poisson model

Alexander Komech|arXiv (Cornell University)|Oct 11, 2013

Spectral Theory in Mathematical Physics29 references4 citations

TL;DR

This paper establishes the existence of a space-periodic ground state in the Schrödinger-Poisson model for 1D, 2D, and 3D lattices of smeared ions, using energy minimization per unit cell. It proves the existence of a stationary, $̲$-periodic solution satisfying the coupled Schrödinger and Poisson equations, with rigorous treatment of infrared divergences in lower dimensions via modified functional analytic techniques.

ABSTRACT

A space-periodic ground state is shown to exist for lattices of smeared ions in $\R^3$ coupled to the Schrödinger and scalar fields. The elementary cell is necessarily neutral. The 1D, 2D and 3D lattices in $\R^3$ are considered, and a ground state is constructed by minimizing the energy per cell. The case of a 3D lattice is rather standard, because the elementary cell is compact, and the spectrum of the Laplacian is discrete. In the cases of 1D and 2D lattices, the energy functional is differentiable only on a dense set of variations, due to the presence of the continuous spectrum of the Laplacian that causes the infrared divergence of the Coulomb bond. Respectively, the construction of electrostatic potential and the derivation of the Schrödinger equation for the minimizer in these cases require an extra argument. The space-periodic ground states for 1D and 2D lattices give the model of the nanostructures similar to the carbon nanotubes and graphene respectively.

Motivation & Objective

• To establish the existence of a space-periodic ground state for lattices of smeared ions in the Schrödinger-Poisson model in $\mathbb{R}^3$.
• To address the challenge of infrared divergence in the Coulomb interaction for 1D and 2D lattices due to the continuous spectrum of the Laplacian.
• To construct a ground state via energy minimization per elementary cell, ensuring neutrality of the unit cell.
• To provide rigorous existence and regularity results for the electron wave function and electrostatic potential in 1D and 2D cases.
• To model nanostructures such as carbon nanotubes (1D) and graphene (2D) using periodic solutions of the coupled Schrödinger-Poisson system.

Proposed method

• Formulates the Schrödinger-Poisson system for $d$-dimensional ion lattices in $\mathbb{R}^3$, with $d=1,2,3$, using periodic boundary conditions on the elementary cell $T_d$.
• Defines the energy functional per unit cell and minimizes it over the set of normalized wave functions in $M_1$ for the 1D case.
• Uses a modified version of Lemma 3.13 to construct the electrostatic potential $\phi^0$ by splitting the solution into $\phi_1 + \phi_2$, with $\phi_2 \in H^2$ and $\phi_1$ derived from a contour integral of a smooth vector field.
• Applies Sobolev embedding and $L^2$-based estimates to control the growth of $\phi^0$, proving the bound $|\phi^0(\mathbf{x})| \leq C(1 + |x_2| + |x_3|)^{1/2}$ in 1D.
• Imposes Condition III: $\frac{\hat{\mu}^{\rm per}_1(0,\xi) + eZ_1}{|\xi|} \in L^2(D)$ for $D = \{\xi \in \mathbb{R}^2 : |\xi| \leq 1\}$, ensuring integrability near the origin.
• Establishes regularity of the minimizer $\psi^0$ via Sobolev embedding and bootstrapping, showing $\psi^0 \in H^2_{\rm loc}(T_1)$ and smoothness under $C^\infty$ ion density.

Experimental results

Research questions

• RQ1 Does a space-periodic ground state exist for 1D and 2D lattices of smeared ions in the Schrödinger-Poisson model, despite infrared divergences?
• RQ2 Can the energy minimization approach be extended to 1D and 2D lattices where the Laplacian has a continuous spectrum?
• RQ3 What conditions on the ion charge density ensure the existence and boundedness of the electrostatic potential in 1D and 2D?
• RQ4 How does the electrostatic potential grow with distance in 1D, and is the derived bound $|\phi^0(\mathbf{x})| \leq C(1 + |x_2| + |x_3|)^{1/2}$ optimal?
• RQ5 Can the minimizer $\psi^0$ be shown to satisfy the stationary Schrödinger-Poisson equations in the distributional sense and possess sufficient regularity?

Key findings

• A ground state exists for 1D, 2D, and 3D lattices of smeared ions in the Schrödinger-Poisson model, constructed via energy minimization per unit cell.
• The elementary cell must be neutral, i.e., $\hat{\mu}^{\rm per}_1(0) + eZ_1 = 0$, to ensure the existence of a finite-energy solution.
• For the 1D case, the electrostatic potential satisfies $|\phi^0(\mathbf{x})| \leq C(1 + |x_2| + |x_3|)^{1/2}$, a sub-logarithmic growth bound.
• The minimizer $\psi^0$ belongs to $H^2_{\rm loc}(T_1)$ and satisfies the stationary Schrödinger equation with a real, bounded potential $\phi^0$.
• Under the condition $\mu_1^{\rm per} \in C^\infty(T_1)$, the solution $\psi^0$ and $\phi^0$ are smooth functions.
• The construction resolves infrared divergences in 1D and 2D by imposing a condition on the low-frequency behavior of the ion density Fourier transform, ensuring integrability of the Coulomb interaction.