| Summary: | 博士 === 國立臺灣大學 === 數學研究所 === 101 === Spin-1 Bose-Einstein condensate (BEC) is a special three-component system, written as
$Psi=(psi_1,psi_0,psi_{-1})$. Its behavior is described
by an energy functional $E[Psi]$ with two constraints:
the conservation of the number of atoms and the conservation of total magnetization. That is $int |Psi|^2$ and $intlt(|psi_1|^2-|psi_{-1}|^2t)$ are fixed numbers.
And a ground state is a minimizer of $E$ under the constraints. To explain what we do in this thesis, we remark that according to the sign of a specific parameter in the energy $E$, spin-1 BECs are classified into two groups: ferromagnetic ones and antiferromagnetic ones.
The works in this thesis are motivated by the following two phenomena:
1. Any ground state of a ferromagnetic system is of the form
begin{align*}
Psi = (gamma_1 psi,gamma_0 psi,gamma_{-1} psi),
end{align*}
where $gamma_j$ are constants and $psi$ a function. This is called single-mode approximation.
2. When an external magnetic field is applied, the ground state of an antiferromagnetic system undergoes a bifurcation from $psi_0 equiv 0$
to $psi_0 ne 0$ as the strength of the magnetic field surpasses a critical value.
Although these phenomena have been well-known from numerical simulations for
quite a long time, there were no rigorous mathematical justifications before
our investigations. In this thesis, our works [16,17]
on their proofs are given, with more details.
The proofs rely on a principle which says that a redistribution of the mass densities
(i.e. $|psi_1|^2$, $|psi_0|^2$ and $|psi_{-1}|^2$) will decrease the kinetic energy. This principle can be regarded as a simple generalization of a well-known convexity inequality for gradients. We will show how this principle can give a rather unified approach toward our problems.
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