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
Reference13 articles.
1. G.F. Dell’Antonio, Contact interaction and Gamma convergence: new tools in Q.M. Eur. Phys. J. Plus 13, 1–20 (2021)
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