The Number of Eigenvalues of the Three-Particle Schrödinger Operator on Three Dimensional Lattice

Abstract

We consider the three-particle discrete Schrödinger operator $$H_{\mu,\gamma}(\mathbf{K}),$$ $$\mathbf{K}\in\mathbb{T}^{3}$$ associated to a system of three particles (two fermions and one another particle) interacting through zero range pairwise potential $$\mu>0$$ on the three-dimensional lattice $$\mathbb{Z}^{3}.$$ It is proved that there exist positive numbers $$\gamma_{2}>\gamma_{1}$$ that the operator $$H_{\mu,\gamma}(\boldsymbol{\pi}),\boldsymbol{\pi}=(\pi,\pi,\pi)$$ for $$\gamma\in(0,\gamma_{1})$$ has no eigenvalue, for $$\gamma\in(\gamma_{1},\gamma_{2})$$ has a simple eigenvalue and for $$\gamma>\gamma_{2}$$ it has three eigenvalues lying below the essential spectrum for sufficiently large $$\mu.$$

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DOI

10.1134/s1995080222150112

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