Integral Operators and Some Their Applications to Discrete Schrödinger Operators of a Three-particle System

In this paper, the invariant subspaces of the integral operator $$A$$ with kernel $$K(\mathbf{x,y})$$ in the Hilbert space $$L^{2}\left(\mathbb{T}^{3}\right)$$ are studied. If the kernel $$K(\mathbf{x,y})$$ is even with respect to the arguments $$(x_{i},y_{i})$$ , $$i=1,2,3$$ , then it is proved that operator $$A$$ has eight invariant subspaces. Using these properties, it is proved that the Schrödinger operator $$H_{\mu}(\mathbf{0})$$ , $$\mathbf{0}=(0,0,0)\in\mathbb{T}^{3}$$ , corresponding to a system of three bosons with pair zero range potentials $$\mu>0$$ on the three dimensional cubic lattice $$\mathbb{Z}^{3}$$ , has a unique simple eigenvalue for sufficiently large $$\mu$$ .

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