J. I. Abdullaev, A. M. Khalkhuzhaev, T. H. Rasulov
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引用次数: 0
Abstract
We consider three-particle Schrödinger operator \({{H}_{{\mu ,\gamma }}}({\mathbf{K}})\), \({\mathbf{K}} \in {{\mathbb{T}}^{3}}\), associated to a system of three particles (two of them are bosons with mass 1 and one is an arbitrary with mass \(m = {\text{1/}}\gamma < 1\)), interacting via zero-range pairwise potentials \(\mu > 0\) and λ > 0 on the three dimensional lattice \({{\mathbb{Z}}^{3}}\). It is proved that there exist critical value of ratio of mass γ = γ1 and γ = γ2 such that the operator \({{H}_{{\mu ,\gamma }}}(\mathbf{0})\)0 = (0, 0, 0), has a unique eigenvalue for \(\gamma \in (0,{{\gamma }_{1}})\), has two eigenvalues for \(\gamma \in ({{\gamma }_{1}},{{\gamma }_{2}})\) and four eigenvalues for \(\gamma \in ({{\gamma }_{2}}, + \infty )\), located on the left-hand side of the essential spectrum for large enough µ > 0 and fixed λ > 0.
Abstract We consider three-particle Schrödinger operator \({{H}_{\mu ,\gamma }}}({\mathbf{K}})\), \({\mathbf{K}} \ in {\{mathbb{T}}^{3}}), associated to a system of three particles (two of them are bosons with mass 1 and one is an arbitrary with mass \(m = {\text{1/}}\gamma <；1）），在三维晶格({\{mathbb{Z}}^{3}}\)上通过零距离对偶势（\(\mu > 0\) and λ > 0）相互作用。研究证明，存在质量比临界值 γ = γ1 和 γ = γ2，使得算子 \({{H}_{\mu ,\gamma }}}(\mathbf{0})\)0 = (0, 0, 0), \(\gamma \in (0,{{\gamma }_{1}})\) 有一个唯一的特征值, \(\gamma \in ({{\gamma }_{1}}、({{\gamma}_{2}})有两个特征值，而(\gamma \in ({{\gamma }_{2}}, + \infty )\) 有四个特征值，位于足够大的 µ >；0 和固定的 λ > 0.
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