Infiniteness of the Discrete Spectrum of Two-Particle Discrete Schrödinger Operators

Annotatsiya

We consider a family of discrete Schrödinger operators $$h^{d}(k),$$ where $$k\in\mathbb{T}^{d}=(-\pi,\pi]^{d}$$ is the two-particle quasi-momentum associated to a system of two particles on the $$d$$ -dimensional lattice $$\mathbb{Z},\,d\geq 1$$ . When the pre-image of the two-particle dispersion relation at the right edge $$E_{k}^{-1}(\mathfrak{e}_{\mathrm{M}}(k))$$ of the essential spectrum of $$h^{d}(k)$$ is of dimension $$d-1$$ or $$d-2$$ , we obtain the necessary and sufficient conditions for the existence of an infinite number of eigenvalues of $$h^{d}(k)$$ to the right of the essential spectrum, while in the case $$\textrm{dim}E_{k}^{-1}(\mathfrak{e}_{\textrm{M}}(k))\leq d-3$$ , the number of eigenvalues $$h^{d}(k)$$ is finite.

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Identifikatorlar

DOI

10.1134/s1995080223070272

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