零射程相互作用N费米子加不同粒子系统的稳定性
Stability for a System of N Fermions Plus a Different Particle with Zero-Range Interactions
摘要：
研究了由单位质量$ N \geq 2 $全同费米子与质量为$ m $的不同粒子通过强度为$ α\in \R $的零程相互作用组成的三维非相对论量子系统的稳定性问题.我们构造了相应的重整化二次（或能量）形式$ \form $和所谓的Skornyakov-Ter-Martirosyan对称扩张$ H_α $，这是系统的哈密顿量的自然候选者。我们找到一个质量$ m^*(N) $的值，使得对于$ m > m^*(N)$，形式$ \form $是封闭的，并且从下面有界。因此，$ \form $定义了一个唯一的自伴和有界的$ H_α$的下扩张，因此系统是稳定的。另一方面，我们也证明了形式$ \form $是从下无界的$ m < m^*(2)$。与众所周知的玻色子情况类似，这表明系统对于$ m < m^*(2)$是不稳定的，并且发生了所谓的托马斯效应。
英文摘要：
We study the stability problem for a non-relativistic quantum system in dimension three composed by $ N \geq 2 $ identical fermions, with unit mass, interacting with a different particle, with mass $ m $, via a zero-range interaction of strength $ α\in \R $. We construct the corresponding renormalised quadratic (or energy) form $ \form $ and the so-called Skornyakov-Ter-Martirosyan symmetric extension $ H_α $, which is the natural candidate as Hamiltonian of the system. We find a value of the mass $ m^*(N) $ such that for $ m > m^*(N)$ the form $ \form $ is closed and bounded from below. As a consequence, $ \form $ defines a unique self-adjoint and bounded from below extension of $ H_α$ and therefore the system is stable. On the other hand, we also show that the form $ \form $ is unbounded from below for $ m < m^*(2)$. In analogy with the well-known bosonic case, this suggests that the system is unstable for $ m < m^*(2)$ and the so-called Thomas effect occurs.
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