Exact asymptotic behavior of the Pekar–Tomasevich functional

• Published in: Journal of mathematical physics 2011-05, Vol.52 (5), p.052110-052110-7
• Main Authors: Benguria, Rafael D., Bley, Gonzalo A.
• Format: Article
• Language: English
• Subjects: Asymptotic methods; Electrons; Exact sciences and technology; Mathematical methods in physics; Mathematics; Physics; Quantum physics; Sciences and techniques of general use
• ISSN: 0022-2488; 1089-7658
• DOI: 10.1063/1.3587117

An explicit asymptotic expression for the ground-state energy of the Pekar–Tomasevich functional for the N-polaron is found, when the repulsion parameter U of the electrons satisfies the inequality 0 ⩽ U ⩽ 2α, where α is the coupling constant of the polaron. If \documentclass[12pt]{minimal}\begin{document}$\mathcal {E}_U^N$\end{document} E U N denotes this ground-state energy for the case of N electrons and repulsion parameter U, we prove that \documentclass[12pt]{minimal}\begin{document}$\mathcal {E}_U^N/N^3 \rightarrow -c_p(U - 2\alpha )^2/4$\end{document} E U N / N 3 → − c p ( U − 2 α ) 2 / 4 as N → ∞, where, c p = 0.10851…. Moreover, we show that \documentclass[12pt]{minimal}\begin{document}$\mathcal {E}_0^N = -c_p\alpha ^2N^3$\end{document} E 0 N = − c p α 2 N 3 , for all N.