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On the essential and discrete spectrum of a model operator related to three-particle discrete Schrödinger operators

Sergio Albeverio·Saidakhmat N. Lakaev·Ramiza Kh. Djumanova
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Abstract

A model operator $H$ corresponding to a three-particle discrete Schrödinger operator on a lattice $\Z^3$ is studied. The essential spectrum is described via the spectrum of two Friedrichs models with parameters $h_α(p),$ $α=1,2,$ $p \in \T^3=(-π,π]^3.$ The following results are proven: 1) The operator $H$ has a finite number of eigenvalues lying below the bottom of the essential spectrum in any of the following cases: (i) both operators $h_α(0), α=1,2,$ have a zero eigenvalue; (ii) either $h_1(0)$ or $h_2(0)$ has a zero eigenvalue. 2) The operator $H$ has infinitely many eigenvalues lying below the bottom and accumulating at the bottom of the essential spectrum, if both operators $h_α(0),α=1,2,$ have a zero energy resonance.

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Sergio Albeverio, Saidakhmat N. Lakaev, Ramiza Kh. Djumanova. 2005-01-11. On the essential and discrete spectrum of a model operator related to three-particle discrete Schrödinger operators. https://arxiv.org/abs/math-ph/0501024

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