Number of Eigenvalues of the Three-Particle Schrodinger Operators on Lattices

S. Albeverio, A.M. Khalkhujaev, S.N. Lakaev

2012, v.18, №3, 387-420


We consider the Hamiltonian $H_{2,v}$ of a system of three identical quantum mechanical particles on the three-dimensional lattice $Z^3$ interacting via short-range pair potentials $v$. We prove the existence of a unique positive eigenvalue $z(k)$ lying below the bottom of the essential spectrum for the two-particle discrete Schrodinger operator $H_{2,v}(k)$, $k\in t^3$, being the two-particle quasi-momentum, associated to the Hamiltonian $H_{2,v}$ under the assumption that the operator $H_{2,v}(0)$ corresponding to the zero value of $k$ has a zero energy resonance. We describe the location of the essential spectrum of the three-particle discrete Schrodinger operators $H_{3,V}(K)$, $K\in t^3$, being the three-particle quasi-momentum, via the spectra of $H_{2,v}(k)$, $k\in t^3$. We prove the existence of infinitely many eigenvalues of $H_{3,V}(0)$, in fact for the number $N[H_{3,V}(0),z]$ of eigenvalues lying below $z<0$ we find the following asymptotics \begin{equation*} \lim\limits_{z \to 0} \frac{N[H_{3,V}(0),z]}{|\log |z||} = \frac{\lambda _0}{2\pi}, \end{equation*} where $\lambda _0$ is a unique positive solution of the equation \[ \lambda = \frac{8 \sinh \pi\lambda /6}{\sqrt 3 \cosh \pi\lambda/2}. \] We prove finiteness of the number $N[H_{3,V}(K),0]$ of eigenvalues of the operator $H_{3,V}(K)$ below zero for all $K \in U_\delta^0(0)$, where $U_\delta^0(0)$ is a punctured $\delta >0$ neighborhood of the origin and find the following asymptotics \begin{equation*} \lim\limits_{|K| \to 0} \frac{N[H_{3,V}(K),0]}{|\log |K||} = \frac{\lambda _0}{\pi}. \end{equation*}

Keywords: Schrodinger operators,quantum mechanical three-particle systems,short-range potentials,eigenvalues,essential spectrum,asymptotics,zero energy resonance,excess mass phenomenon,solid state physics,Birman - Schwinger principle,lattice


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