Can one bind three electrons with a single proton?
Résumé
Of course not for an ideal $\mathrm{H}^{--}$ atom. But with the help of an intense homogeneous magnetic field $B$, the question deserves to be reconsidered. It is known (see e.g. \cite{BSY,BD}) that as $B\to\infty$ and in the clamped nucleus approximation, this ion is described by a one dimensional Hamiltonian \begin{equation}\label{DeltaModel} \sum_{i=1}^N-{\Delta_i\over2}-Z\delta(x_i)+\sum_{1\le i < j\le n}\delta(x_i-x_j)\quad\mbox{acting in}\quad l^2(\r^3) \end{equation} where N = 3, Z = 1 is the charge of the nucleus, and \delta stands for the well known "delta" point interaction. We present an extension of the "skeleton method", see [CDR1, CDR2], to the case of three degree of freedom . This is a tool, that we learn from [R] for the case N = 2, which reduces the spectral analysis of (1) to determining the kernel a system of linear integral operators acting on the supports of the delta interactions. As an application of this method we present numerical results which indicates that (1) has a bound state for Z = 1 and N = 3.
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