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The Essential Spectrum of a Three Particle Schrödinger Operator on Lattices

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Abstract

We consider the Hamiltonian \(\textrm{H}_{\mu\lambda},\mu,\lambda\in\mathbb{R}\) of a system of three-particles (two identical bosons and one different particle) moving on the lattice \({\mathbb{Z}}^{d},\,d=1,2\) interacting through zero-range pairwise potentials \(\mu\neq 0\) and \(\lambda\neq 0\). The essential spectrum of the three-particle discrete Schrödinger operator \(H_{\mu\lambda}(K),\,K\in\mathbb{T}^{d}\), being the three-particle quasi-momentum, is described by means of the spectrum of non-perturbed three-particle operator \(H_{0}(K)\) and the two-particle discrete Schrödinger operator \(h_{\mu}(k),h_{\lambda,\gamma}(k),k\in\mathbb{T}^{d},\gamma>0\). It is established that the essential spectrum of the three-particle discrete Schrödinger operator \(H_{\mu\lambda}(K),\,K\in\mathbb{T}^{d}\) consists of no more than three bounded closed intervals.

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Funding

The authors acknowledge support from the Foundation for Basic Research of the Republic of Uzbekistan (grant no. FZ-20200929224).

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Authors and Affiliations

  1. Samarkand State University, 140104, Samarkand, Uzbekistan

    S. N. Lakaev

  2. Romanovskii Institute of Mathematics, Academy of Sciences of Uzbekistan, 100174, Tashkent, Uzbekistan

    A. T. Boltaev

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  1. S. N. Lakaev
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  2. A. T. Boltaev
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Correspondence to S. N. Lakaev or A. T. Boltaev.

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(Submitted by T. K. Yuldashev)

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Lakaev, S.N., Boltaev, A.T. The Essential Spectrum of a Three Particle Schrödinger Operator on Lattices. Lobachevskii J Math 44, 1176–1187 (2023). https://doi.org/10.1134/S1995080223030198

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  • DOI: https://doi.org/10.1134/S1995080223030198

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