v[
        x_] := -2/(1 + x^2)

    Plot[v[x], {x, -5, 5}]

\(\(\( (*\ 
      plot\ solution\ for\ trial\ e\ *) \)\(
\)\( (*\ \(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(---\)\)\)\)\)\)\)\)\)\)\)\)\ *) \)\(
\)\(plotsol[e_] := 
      Block[{x0 = 15}, 
        sol = NDSolve[{\(-\(psi''\)[x]\) + \((v[x] - e)\)*psi[x] == 0, 
              psi[\(-x0\)] == 0.001, \(psi'\)[\(-x0\)] == 0.001}, 
            psi, {x, \(-x0\), x0}]; 

        Plot[Evaluate[psi[x] /. sol], {x, \(-x0\), x0}]]\)\)\)

plotsol[-1.08371]

\(\(\( (*\ 
      shooting\ method\ *) \)\(
\)\( (*\ \(--\(--\(--\(--\(-\(-\(--\(---\)\)\)\)\)\)\)\)\ *) \)\(
\)\(shoot[e_] := 
      Block[{x0 = 15}, 
        sol = NDSolve[{\(-\(psi''\)[x]\) + \((v[x] - e)\)*psi[x] == 0, 
              psi[\(-x0\)] == 0.001, \(psi'\)[\(-x0\)] == 0.001}, 
            psi, {x, \(-x0\), x0}]; 
psi[x0] /. sol]\)\)\)

Plot[shoot[e], {e, -2, 0}, 
      PlotRange -> {-1, 1}]

Off[
      NDSolve::ndnum]

    Off[FindRoot::frmp]

    Off[ReplaceAll::reps]

    e1 = e /. 
        FindRoot[
          shoot[e] == 0, {e, -1.1, -1}]

    plotsol[e1]

(* 
      first excited state *) 
 (* ------------------- *) 
e2 = 
      e /. FindRoot[
          shoot[e] == 0, {e, -0.15, -0.05}]

    plotsol[e2]