[Paper Review] Ground states for semi-relativistic Schrödinger-Poisson-Slater energies

Jacopo Bellazzini, Tohru Ozawa+1 more|arXiv (Cornell University)|Mar 14, 2011

Spectral Theory in Mathematical Physics11 references11 citations

TL;DR

This paper establishes the existence of ground states for the semi-relativistic Schrödinger-Poisson-Slater energy functional in $\mathbb{R}^3$ under $L^2$-critical conditions, proving that minimizers exist for small $\rho > 0$ when $\alpha, \beta > 0$. The key contribution is a new sharp lower bound on the Coulomb energy involving kinetic and exchange terms, which enables compactness and rules out vanishing or dichotomy in the concentration-compactness framework.

ABSTRACT

We prove the existence of ground states for the semi-relativistic Schrödinger-Poisson-Slater energy $$I^{α,β}(ρ)=\inf_{\substack{u\in H^\frac 12(\R^3) \int_{\R^3}|u|^2 dx=ρ}} \frac{1}{2}\|u\|^2_{H^\frac 12(\R^3)} +α\int\int_{\R^{3} imes\R^{3}} \frac{| u(x)|^{2}|u(y)|^2}{|x-y|}dxdy-β\int_{\R^{3}}|u|^{\frac{8}{3}}dx$$ $α,β>0$ and $ρ>0$ is small enough. The minimization problem is $L^2$ critical and in order to characterize of the values $α, β>0$ such that $I^{α, β}(ρ)>-\infty$ for every $ρ>0$, we prove a new lower bound on the Coulomb energy involving the kinetic energy and the exchange energy. We prove the existence of a constant $S>0$ such that $$\frac{1}{S}\frac{\|φ\|_{L^\frac 83(\R^3)}}{\|φ\|_{\dot H^\frac 12(\R^3)}^\frac 12}\leq \left (\int\int_{\R^3 imes \R^3} \frac{|φ(x)|^2|φ(y)|^2}{|x-y|}dxdy ight)^\frac 18 $$ for all $φ\in C^\infty_0(\R^3)$. Eventually we show that similar compactness property fails provided that in the energy above we replace the inhomogeneous Sobolev norm $\|u\|^2_{H^\frac 12(\R^3)}$ by the homogeneous one $\|u\|_{\dot H^\frac 12(\R^3)}$.

Motivation & Objective

• To establish the existence of minimizers (ground states) for the semi-relativistic Schrödinger-Poisson-Slater energy functional in $\mathbb{R}^3$ under $L^2$-critical conditions.
• To resolve the challenge of lack of compactness in $H^{1/2}(\mathbb{R}^3)$ due to the critical scaling of the nonlinearity $|u|^{8/3}$.
• To characterize the values of $\alpha, \beta > 0$ for which the infimum $I^{\alpha,\beta}(\rho)$ remains finite for all $\rho > 0$, particularly for small $\rho$.
• To prove a new sharp inequality relating the $L^{8/3}$-norm, homogeneous $H^{1/2}$-seminorm, and the Coulomb energy, which is crucial for compactness.

Proposed method

• Use of the concentration-compactness principle to rule out vanishing and dichotomy in minimizing sequences.
• Introduction of a new sharp inequality: $ \frac{1}{S} \frac{\|\varphi\|_{L^{8/3}}}{\|\varphi\|_{\dot{H}^{1/2}}^{1/2}} \leq \left( \iint \frac{|\varphi(x)|^2 |\varphi(y)|^2}{|x-y|} \, dx dy \right)^{1/8} $ for all $\varphi \in C^\infty_0(\mathbb{R}^3)$.
• Application of the Hardy-Littlewood-Sobolev inequality and interpolation to control the exchange term in terms of the Coulomb and kinetic energies.
• Analysis of the scaling behavior of the energy functional under $u \mapsto \theta^{3/2} u(\theta x)$ to derive necessary conditions for minimizers.
• Proof of the strong subadditivity inequality $I^{\alpha,\beta}(\rho) < I^{\alpha,\beta}(\mu) + I^{\alpha,\beta}(\rho - \mu)$ for $0 < \mu < \rho$ to rule out dichotomy.
• Use of a contradiction argument based on monotonicity of $I^{\alpha,\beta}(\rho)/\rho$ to show that $I^{\alpha,\beta}(\rho) > -\infty$ for small $\rho$.

Experimental results

Research questions

• RQ1 Under what conditions on $\alpha, \beta > 0$ and $\rho > 0$ does the infimum $I^{\alpha,\beta}(\rho)$ remain finite for the semi-relativistic Schrödinger-Poisson-Slater energy?
• RQ2 Can the concentration-compactness principle be successfully applied to rule out vanishing and dichotomy in the $H^{1/2}(\mathbb{R}^3)$ setting for this $L^2$-critical problem?
• RQ3 Is there a new sharp inequality that controls the Coulomb energy in terms of the kinetic and exchange energies, sufficient to ensure compactness of minimizing sequences?
• RQ4 Does the homogeneous $\dot{H}^{1/2}$-norm fail to support the same compactness properties as the inhomogeneous $H^{1/2}$-norm in this minimization problem?
• RQ5 What is the precise threshold for $\rho$ below which ground states exist, and how does it depend on $\alpha$ and $\beta$?

Key findings

• A new sharp inequality is proven: $ \frac{1}{S} \frac{\|\varphi\|_{L^{8/3}}}{\|\varphi\|_{\dot{H}^{1/2}}^{1/2}} \leq \left( \iint \frac{|\varphi(x)|^2 |\varphi(y)|^2}{|x-y|} \, dx dy \right)^{1/8} $ for all $\varphi \in C^\infty_0(\mathbb{R}^3)$, with $S > 0$ universal.
• For sufficiently small $\rho > 0$, the infimum $I^{\alpha,\beta}(\rho)$ is finite and strictly greater than $-\infty$, ensuring the existence of ground states.
• The existence of minimizers is established via the concentration-compactness principle, with vanishing excluded by the new inequality and dichotomy ruled out by strong subadditivity.
• The homogeneous energy functional $\tilde{\mathcal{E}}^{\alpha,\beta}(u)$ satisfies $\tilde{I}^{\alpha,\beta}(\rho) = 0$ for all $\rho < \tilde{\rho}$, implying no nontrivial minimizers exist in the homogeneous case.
• The compactness property fails when replacing the inhomogeneous $H^{1/2}$-norm with the homogeneous $\dot{H}^{1/2}$-norm, highlighting the necessity of the full $H^{1/2}$-structure.
• For small $\rho$, the ratio $I^{\alpha,\beta}(\rho)/\rho$ is strictly decreasing, and a contradiction is derived if a minimizer were to exist at a critical $\rho_0$, proving that $I^{\alpha,\beta}(\rho) > -\infty$ for small $\rho$.