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Asymptotic behavior of resolvent for $$N$$-body Schrödinger operators near a threshold. (English) Zbl 1049.81026
Let $$P$$ denote the $$N$$-body Schrödinger operator with the mass-center removed from the total energy operator $-\sum_{j=1}^N\frac{1}{2m_j}\Delta_{x_j}+\sum_{1\leq i<j\leq N}V_{ij}(x_i-x_j), \;\;x_j\in\mathbb{R}^3,$ where $$x_j$$ and $$m_j$$ denote the position and mass of the $$j$$-th particle, $$V_{ij}$$ is assumed to be real and relatively compact with respect to $$-\Delta$$ in $$L^2(\mathbb{R}^3)$$ and satisfies the decay $| V_{ij}(y)| \leq C_{ij}| y| ^{-\rho},$ for $$y\in\mathbb{R}^3,| y| >R$$, $$R>0$$ and $$\rho>2$$. In the paper, the spectral properties of $$P$$ near its first threshold $$E_0$$ and the asymptotic expansions of the resolvent there are studied. To do it, the author introduces an auxiliary operator $$P'$$ and study its spectral properties near $$E_0$$. Under the assumption that $$E_0$$ is not an eigenvalue of the operator $$P'$$, two Grushin problems are studied, in order to reduce the study of $$R(z)$$ to the inverse of a matrix-valued holomorphic function and establish the asymptotic expansion of $$R(z)$$ according to the spectral properties of $$E_0$$ with respect to $$P$$.

##### MSC:
 81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis 81V70 Many-body theory; quantum Hall effect 47N50 Applications of operator theory in the physical sciences 81U10 $$n$$-body potential quantum scattering theory
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