On the structure of the essential spectrum for three-particle discrete SchroЁdinger operators on the four dimensional lattices

DOI10.29229/UZMJ.2020-4-10zbMATH Open1488.81023arXivmath-ph/0312050OpenAlexW4251530162MaRDI QIDQ5011593

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Publication date: 27 August 2021

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Abstract: A system of three quantum particles on the three-dimensional lattice $$ with arbitrary "dispersion functions" having non-compact support and interacting via short-range pair potentials is considered. The energy operators of the systems of the two-and three-particles on the lattice $$ in the coordinate and momentum representations are described as bounded self-adjoint operators on the corresponding Hilbert spaces. For all sufficiently small nonzero values of the two-particle quasi-momentum $kin(-pi,pi{]}^3$ the finiteness of the number of eigenvalues of the two-particle discrete Schr"odinger operator $h_{a}lpha(k)$ below the continuous spectrum is established. A location of the essential spectrum of the three-particle discrete Schr"odinger operator $H(K),Kin(-pi,pi{]}^3$ the three-particle quasi-momentum, by means of the spectrum of $h_{a}lpha(k)$ is described. It is established that the essential spectrum of $H(K),Kin(-pi,pi{]}^3$ consists of a finitely many bounded closed intervals.

Full work available at URL: https://arxiv.org/abs/math-ph/0312050

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