Contact interactions II; Gross-Pitaewskii equation, Bose-Einstein condensate, Fermi sea

Author:

Dell’Antonio G. F.

Abstract

AbstractIn Dell’Antonio (Eur Phys J Plus 13:1–20, 2021), we explored the possibility to analyse contact interaction in Quantum Mechanics using a variational tool, Gamma Convergence. Here, we extend the analysis in Dell’Antonio (Eur Phys J Plus 13:1–20, 2021) of joint weak contact of three particles to the non-relativistic case in which the free one particle Hamiltonian is $$ H_0 = - \frac{\Delta }{2M} $$ H 0 = - Δ 2 M . We derive the Gross–Pitaevskii equation for a system of three particles in joint weak contact. We then define and study strong contact and show that the Gross–Pitaevskii equation is also the variational equation for the energy of the Bose–Einstein condensate (strong contact in a four-particle system). We add some comments on Bogoliubov’s theory. In the second part, we use the non-relativistic Pauli equation and weak contact to derive the spectrum of the conduction electrons in an infinite crystal. We prove that the spectrum is pure point with multiplicity two and eigenvalues that scale as $$ \frac{1}{log {n}}$$ 1 logn .

Funder

Scuola Internazionale Superiore di Studi Avanzati - SISSA

Publisher

Springer Science and Business Media LLC

Subject

General Physics and Astronomy

Cited by 1 articles. 订阅此论文施引文献 订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献

1. Gamma convergence and renormalization group: Two sides of a coin?;The European Physical Journal Plus;2022-06-28

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