The number of eigenvalues of three-particle Schrödinger operators on lattices
The number of eigenvalues of three-particle Schrödinger operators on lattices
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晶格上三粒子薛定谔算子的特征值个数
DOI：
10.1088/1751-8113/40/49/015
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发表时间：
2007
期刊：
影响因子：
--
通讯作者：
S. Lakaev
中科院分区：
文献类型：
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作者：
S. Albeverio;G. Dell' Antonio;S. Lakaev
本文考虑三维晶格上三个量子力学粒子（两个全同费米子和一个玻色子）通过零程吸引势相互作用的系统的哈密顿量。本文描述了三粒子离散薛定谔算符Hγ（K）的本征谱的位置和结构，K是属于三维环面的总准动量，γ > 0是费米子与玻色子的质量之比.当γ > 0时，我们选择相互作用μ（γ），使得由一个费米子和一个玻色子组成的系统具有零能量共振。对于所有非零的拟动量，证明了Hγ（K）在本质谱底τγ（K）以下的本征值个数N（K，γ; τγ（K））的有限性，并给出了N（K，γ; 0）在K → 0时的渐近性.证明了算子Hγ（0）的无穷多个特征值的存在性，并对小于z < 0的特征值个数N（0，γ; z）给出了当z → 0时的渐近性.
We consider the Hamiltonian of a system of three quantum mechanical particles (two identical fermions and a boson) on the three-dimensional lattice interacting by means of zero-range attractive potentials. We describe the location and structure of the essential spectrum of the three-particle discrete Schrödinger operator Hγ(K), K being the total quasi-momentum belonging to the three-dimensional torus and γ > 0 the ratio of the mass of fermion to boson. We choose for γ > 0 the interaction μ(γ) in such a way that the system consisting of one fermion and one boson has a zero-energy resonance. For all nonzero values of the quasi-momentum , we prove the finiteness of the number N(K, γ; τγ(K)) of eigenvalues of Hγ(K) below the bottom τγ(K) of the essential spectrum and we give for N(K, γ; 0) an asymptotics as K → 0. Moreover, we prove the existence of infinitely many eigenvalues of the operator Hγ(0) and give for the number N(0, γ; z) of eigenvalues lying below z < 0 an asymptotics as z → 0.