The exotic $H_3^{2+}$ ion in a strong magnetic field. Linear   configuration
  An accurate study of the lowest $1\sigma_g$ and the low-lying excited $1\sigma_u$, $1\pi_{u,g}$, $1\delta_{g,u}$ electronic states of the exotic molecular ion $H_3^{2+}$ in linear configuration parallel to a magnetic field is carried out. The magnetic field ranges from $10^{10}$ G up to $4.414 \times 10^{13}$ G where non-relativistic considerations are justified. The variational method is exploited and the {\it same} trial function is used for different magnetic fields. It is shown that the states of positive $z$-parity $1\sigma_g, 1\pi_u, 1\delta_{g}$ are bound states of the $H_3^{2+}$ exotic ion for all magnetic fields studied. We also demonstrate that for magnetic fields $B\gtrsim 2.35\times 10^{12}$ G the potential energy surface well corresponding to the $1\sigma_g$ state contains at least one longitudinal vibrational state. It is also shown that the negative $z$-parity states $1\sigma_u, 1\pi_g, 1\delta_{u}$, are purely repulsive in the whole range of magnetic fields studied, $B=10^{10}- 4.414 \times 10^{13}$ G. 
A. V. Turbiner, J. C. L\'opez Vieyra and N. L. Guevara