T-matrix analysis of one-dimensional weakly coupled bound states

The ground-state energy of the Hamiltonian $-\left(\frac{1}{2}\right)\frac{d^2}{d{x}^2}+\lambda V\left(x\right)\mathrm{}$ is analyzed in terms of the zeros of a perturbative expansion for the inverse of the $T$ matrix, and an expression for ${\mathrm{}(-2E)\mathrm{}}^{\frac{1}{2}}$ correct to order ${\lambda }^4$ is obtained. Modifications of the results for long-range potentials of the type $V\left(x\right)\mathrm{}\to -a_{\pm }{\mathrm{}\left|x\right|\mathrm{}}^{-n}$, $n=1,\dots ,4\mathrm{}$ at $x\to \pm \infty$ are discussed. The problem of three-body bound states is considered, leading to an expression for the ground-state energy of He-like atoms with nuclear charge $\alpha \gg 1$, in a strong magnetic field.