f[x_] = (E^(-x^2))

ρ = 10
ξ0 = 1
ϵ = -95
w = 
      NDSolve[{-ψ''[ξ] - (ρ^2) 
                f[ξ] ψ[ξ] == ϵ ψ[ξ], ψ[-ξ0] == 
            1, ψ'[-ξ0] == (Sqrt[Abs])[ϵ] ψ[-ξ0]}, ψ, {ξ, -ξ0, ξ0}]
Plot[
      Evaluate[ψ[ξ]/ψ[0]  /.  
          w], {ξ, -ξ0, ξ0}]

ρ = 10
ξ0 = 2
ϵ = -85
w = 
      NDSolve[{-ψ''[ξ] - (ρ^2) 
                f[ξ] ψ[ξ] == ϵ ψ[ξ], ψ[-ξ0] == 
            1, ψ'[-ξ0] == (Sqrt[Abs])[ϵ] ψ[-ξ0]}, ψ, {ξ, -ξ0, ξ0}]
Plot[
      Evaluate[ψ[ξ] /ψ[0] /.  
          w], {ξ, -ξ0, ξ0}, 
      PlotRange -> {-3, 3}]

ρ = 10
ξ0 = 2
ϵ = -91
w = 
      NDSolve[{-ψ''[ξ] - (ρ^2) 
                f[ξ] ψ[ξ] == ϵ ψ[ξ], ψ[-ξ0] == 
            1, ψ'[-ξ0] == (Sqrt[Abs])[ϵ] ψ[-ξ0]}, ψ, {ξ, -ξ0, ξ0}]
Plot[
      Evaluate[ψ[ξ] /ψ[0] /.  
          w], {ξ, -ξ0, ξ0}, 
      PlotRange -> {-3, 3}]

ρ = 10
ξ0 = 2
ϵ = -90.5
w = 
      NDSolve[{-ψ''[ξ] - (ρ^2) 
                f[ξ] ψ[ξ] == ϵ ψ[ξ], ψ[-ξ0] == 
            1, ψ'[-ξ0] == (Sqrt[Abs])[ϵ] ψ[-ξ0]}, ψ, {ξ, -ξ0, ξ0}]
Plot[
      Evaluate[ψ[ξ] /ψ[0] /.  
          w], {ξ, -ξ0, ξ0}, 
      PlotRange -> {-3, 3}]

ρ = 10
ξ0 = 2
ϵ = -90.38
w = 
      NDSolve[{-ψ''[ξ] - (ρ^2) 
                f[ξ] ψ[ξ] == ϵ ψ[ξ], ψ[-ξ0] == 
            1, ψ'[-ξ0] == (Sqrt[Abs])[ϵ] ψ[-ξ0]}, ψ, {ξ, -ξ0, ξ0}]
Plot[
      Evaluate[ψ[ξ] /ψ[0] /.  
          w], {ξ, -ξ0, ξ0}, 
      PlotRange -> {-3, 3}]

ρ = 10
ξ0 = 2
ϵ = -90.375
w = 
      NDSolve[{-ψ''[ξ] - (ρ^2) 
                f[ξ] ψ[ξ] == ϵ ψ[ξ], ψ[-ξ0] == 
            1, ψ'[-ξ0] == (Sqrt[Abs])[ϵ] ψ[-ξ0]}, ψ, {ξ, -ξ0, ξ0}]
Plot[
      Evaluate[ψ[ξ] /ψ[0] /.  
          w], {ξ, -ξ0, ξ0}, 
      PlotRange -> {-3, 3}]

ρ = 10
ξ0 = 2
ϵ = -75
w = 
      NDSolve[{-ψ''[ξ] - (ρ^2) 
                f[ξ] ψ[ξ] == ϵ ψ[ξ], ψ[-ξ0] == 
            1, ψ'[-ξ0] == (Sqrt[Abs])[ϵ] ψ[-ξ0]}, ψ, {ξ, -ξ0, ξ0}]
Plot[
      Evaluate[ψ[ξ] /ψ[- .5] /.  
          w], {ξ, -ξ0, ξ0}, 
      PlotRange -> {-3, 3}]

ρ = 10
ξ0 = 2
ϵ = -65
w = 
      NDSolve[{-ψ''[ξ] - (ρ^2) 
                f[ξ] ψ[ξ] == ϵ ψ[ξ], ψ[-ξ0] == 
            1, ψ'[-ξ0] == (Sqrt[Abs])[ϵ] ψ[-ξ0]}, ψ, {ξ, -ξ0, ξ0}]
Plot[
      Evaluate[ψ[ξ] /ψ[- .5] /.  
          w], {ξ, -ξ0, ξ0}, 
      PlotRange -> {-3, 3}]

ρ = 10
ξ0 = 2
ϵ = -70
w = 
      NDSolve[{-ψ''[ξ] - (ρ^2) 
                f[ξ] ψ[ξ] == ϵ ψ[ξ], ψ[-ξ0] == 
            1, ψ'[-ξ0] == (Sqrt[Abs])[ϵ] ψ[-ξ0]}, ψ, {ξ, -ξ0, ξ0}]
Plot[
      Evaluate[ψ[ξ] /ψ[- .5] /.  
          w], {ξ, -ξ0, ξ0}, 
      PlotRange -> {-3, 3}]

ρ = 10
ξ0 = 2
ϵ = -72
w = 
      NDSolve[{-ψ''[ξ] - (ρ^2) 
                f[ξ] ψ[ξ] == ϵ ψ[ξ], ψ[-ξ0] == 
            1, ψ'[-ξ0] == (Sqrt[Abs])[ϵ] ψ[-ξ0]}, ψ, {ξ, -ξ0, ξ0}]
Plot[
      Evaluate[ψ[ξ] /ψ[- .5] /.  
          w], {ξ, -ξ0, ξ0}, 
      PlotRange -> {-3, 3}]

ρ = 10
ξ0 = 2
ϵ = -71.5
w = 
      NDSolve[{-ψ''[ξ] - (ρ^2) 
                f[ξ] ψ[ξ] == ϵ ψ[ξ], ψ[-ξ0] == 
            1, ψ'[-ξ0] == (Sqrt[Abs])[ϵ] ψ[-ξ0]}, ψ, {ξ, -ξ0, ξ0}]
Plot[
      Evaluate[ψ[ξ] /ψ[- .5] /.  
          w], {ξ, -ξ0, ξ0}, 
      PlotRange -> {-3, 3}]

ρ = 10
ξ0 = 2
ϵ = -71.9
w = 
      NDSolve[{-ψ''[ξ] - (ρ^2) 
                f[ξ] ψ[ξ] == ϵ ψ[ξ], ψ[-ξ0] == 
            1, ψ'[-ξ0] == (Sqrt[Abs])[ϵ] ψ[-ξ0]}, ψ, {ξ, -ξ0, ξ0}]
Plot[
      Evaluate[ψ[ξ] /ψ[- .5] /.  
          w], {ξ, -ξ0, ξ0}, 
      PlotRange -> {-3, 3}]

ρ = 10
ξ0 = 2
ϵ = -71.95
w = 
      NDSolve[{-ψ''[ξ] - (ρ^2) 
                f[ξ] ψ[ξ] == ϵ ψ[ξ], ψ[-ξ0] == 
            1, ψ'[-ξ0] == (Sqrt[Abs])[ϵ] ψ[-ξ0]}, ψ, {ξ, -ξ0, ξ0}]
Plot[
      Evaluate[ψ[ξ] /ψ[- .5] /.  
          w], {ξ, -ξ0, ξ0}, 
      PlotRange -> {-3, 3}]

ρ = 10
ξ0 = 2
ϵ = -71.92
w = 
      NDSolve[{-ψ''[ξ] - (ρ^2) 
                f[ξ] ψ[ξ] == ϵ ψ[ξ], ψ[-ξ0] == 
            1, ψ'[-ξ0] == (Sqrt[Abs])[ϵ] ψ[-ξ0]}, ψ, {ξ, -ξ0, ξ0}]
Plot[
      Evaluate[ψ[ξ] /ψ[- .5] /.  
          w], {ξ, -ξ0, ξ0}, 
      PlotRange -> {-3, 3}]

ρ = 10
ξ0 = 2
ϵ = -71.915
w = 
      NDSolve[{-ψ''[ξ] - (ρ^2) 
                f[ξ] ψ[ξ] == ϵ ψ[ξ], ψ[-ξ0] == 
            1, ψ'[-ξ0] == (Sqrt[Abs])[ϵ] ψ[-ξ0]}, ψ, {ξ, -ξ0, ξ0}]
Plot[
      Evaluate[ψ[ξ] /ψ[- .5] /.  
          w], {ξ, -ξ0, ξ0}, 
      PlotRange -> {-3, 3}]

ρ = 1
ξ0 = 5
ϵ = - .35
w = 
      NDSolve[{-ψ''[ξ] - (ρ^2) 
                f[ξ] ψ[ξ] == ϵ ψ[ξ], ψ[-ξ0] == 
            1, ψ'[-ξ0] == (Sqrt[Abs])[ϵ] ψ[-ξ0]}, ψ, {ξ, -ξ0, ξ0}]
Plot[
      Evaluate[ψ[ξ]/ψ[0]  /.  
          w], {ξ, -ξ0, ξ0}, 
      PlotRange -> {-3, 3}]

-0.2

-0.3