Geodesic
Universality of the discrete spectrum asymptotics of the three-particle Schr\"odinger operator on a lattice
Nanosistemy: fizika, himiâ, matematika, Tome 6 (2015) no. 2, pp. 280-293.

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In the present paper, we consider the Hamiltonian $H(K)$, $K\in\mathbb T^3:=(-\pi,\pi]^3$ of a system of three arbitrary quantum mechanical particles moving on the three-dimensional lattice and interacting via zero range potentials. We find a finite set $\Lambda\subset \mathbb T^3$ such that for all values of the total quasi-momentum $K\in\Lambda$ the operator $H(K)$ has infinitely many negative eigenvalues accumulating at zero. It is found that for every $K\in\Lambda$, the number $N(K;z)$ of eigenvalues of $H(K)$ lying on the left of $z$, $z0$, satisfies the asymptotic relation $\lim\limits_{z\to-0}N(K;z)\bigl|\log|z|\bigr|^{-1}=\mathcal U_0$ with $0\mathcal U_0\infty$, independently on the cardinality of $\Lambda$.
Keywords: three-particle Schrödinger operator, zero-range pair attractive potentials, Birman–Schwinger principle, the Efimov effect, discrete spectrum asymptotics. 
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     author = {Mukhiddin I. Muminov and Tulkin H. Rasulov},
     title = {Universality of the discrete spectrum asymptotics of the three-particle {Schr\"odinger} operator on a lattice},
     journal = {Nanosistemy: fizika, himi\^a, matematika},
     pages = {280--293},
     publisher = {mathdoc},
     volume = {6},
     number = {2},
     year = {2015},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/NANO_2015_6_2_a14/}
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Mukhiddin I. Muminov; Tulkin H. Rasulov. Universality of the discrete spectrum asymptotics of the three-particle Schr\"odinger operator on a lattice. Nanosistemy: fizika, himiâ, matematika, Tome 6 (2015) no. 2, pp. 280-293. http://geodesic.mathdoc.fr/item/NANO_2015_6_2_a14/