A unified approach to resolvent expansions at thresholds. (English) Zbl 1029.81067

Jensen, Arne; Nenciu, Gheorghe

Rev. Math. Phys. 13, No. 6, 717-754 (2001).

Summary: Results are obtained on resolvent expansions around zero energy for Schrödinger operators \(H = -\Delta + V(x)\) on \(L^2(\mathbb{R}^m)\), where \(V(x)\) is a sufficiently rapidly decaying real potential. The emphasis is on a unified approach, valid in all dimensions, which does not require one to distinguish between \(\int V(x) dx = 0\) and \(\int V(x) dx \neq 0\) in dimensions \(m = 1, 2\). It is based on a factorization technique and repeated decomposition of the Lippmann-Schwinger operator. Complete results are given in dimensions \(m = 1\) and \(m = 2\).

MSC:

81U05	\(2\)-body potential quantum scattering theory
81Q15	Perturbation theories for operators and differential equations in quantum theory
81Q10	Selfadjoint operator theory in quantum theory, including spectral analysis

Keywords:

Schrödinger operators; Lippmann-Schwinger operator

References:

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