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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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