4 edited body
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3 added 204 characters in body
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  sol = NDSolve[Join[eqnSubbed2 //. subs /. root, 
        {T[0] == 0, V[0] == 0, T'[0] == 1, V[1] == 0}], {T, V}, {x, 0, 1}]];
  GraphicsRow[{Plot[Evaluate[ReIm@T[x] /. sol], {x, 0, 1}], PlotLabel -> "T(x)"], 
  Plot[Evaluate[ReIm@V[x] /. sol], {x, 0, 1}], PlotLabel -> "V(x)"], 
  Plot[Evaluate[ReIm[ρ[x] /. DSolve[eqnSubbed[[2]], ρ, x][[1]] //.subs /.root /.sol]]
  , {x, 0, 1}], PlotLabel -> "ρ(x)"]}, ImageSize -> 1000]
sol = NDSolve[Join[eqnSubbed2 //. subs /. root, 
    {T[0] == 0, V[0] == 0, T'[0] == 1, V[1] == 0}], {T, V}, {x, 0, 1}]
GraphicsRow[{Plot[Evaluate[ReIm@T[x] /. sol], {x, 0, 1}], 
  Plot[Evaluate[ReIm@V[x] /. sol], {x, 0, 1}]}]
  sol = NDSolve[Join[eqnSubbed2 //. subs /. root, 
        {T[0] == 0, V[0] == 0, T'[0] == 1, V[1] == 0}], {T, V}, {x, 0, 1}];
  GraphicsRow[{Plot[Evaluate[ReIm@T[x] /. sol], {x, 0, 1}, PlotLabel -> "T(x)"], 
  Plot[Evaluate[ReIm@V[x] /. sol], {x, 0, 1}, PlotLabel -> "V(x)"], 
  Plot[Evaluate[ReIm[ρ[x] /. DSolve[eqnSubbed[[2]], ρ, x][[1]] //.subs /.root /.sol]]
  , {x, 0, 1}, PlotLabel -> "ρ(x)"]}, ImageSize -> 1000]
2 Added example eigenfunctions
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Needs["CompoundMatrixMethod`"]

subs = {γ -> 1.67, ϵ -> 0.01, d -> 0.05, H -> 1.1, gparagpar -> Function[{x}, 0.38], 
      ρ0 -> Function[{x}, Exp[-x/H]], p0 -> Function[{x}, Exp[-x/H]]};
sys = ToMatrixSystem[eqnSubbed2, {T[0] == 0, V[0] == 0, T[1] == 0, V[1] == 0}, {T, 
    V}, {x, 0, 1}, λ] //. subs
root = FindRoot[Evans[λ, sys], {λ, -1 + 3 I}]
 (* {λ -> -0.342689 + 2.65899 I} *)

Unfortunately my code doesn't currently get the eigenfunctions out as part of itimmediately at the moment. However, you can use NDSolve to get them once you find a root:

sol = NDSolve[Join[eqnSubbed2 //. subs /. root, 
    {T[0] == 0, V[0] == 0, T'[0] == 1, V[1] == 0}], {T, V}, {x, 0, 1}]
GraphicsRow[{Plot[Evaluate[ReIm@T[x] /. sol], {x, 0, 1}], 
  Plot[Evaluate[ReIm@V[x] /. sol], {x, 0, 1}]}]

enter image description here

Needs["CompoundMatrixMethod`"]

subs = {γ -> 1.67, ϵ -> 0.01, d -> 0.05, H -> 1.1, gpara -> Function[{x}, 0.38], 
      ρ0 -> Function[{x}, Exp[-x/H]], p0 -> Function[{x}, Exp[-x/H]]};
sys = ToMatrixSystem[eqnSubbed2, {T[0] == 0, V[0] == 0, T[1] == 0, V[1] == 0}, {T, 
    V}, {x, 0, 1}, λ] /. subs
FindRoot[Evans[λ, sys], {λ, -1 + 3 I}]
 (* {λ -> -0.342689 + 2.65899 I} *)

Unfortunately my code doesn't currently get the eigenfunctions out as part of it at the moment.

Needs["CompoundMatrixMethod`"]

subs = {γ -> 1.67, ϵ -> 0.01, d -> 0.05, H -> 1.1, gpar -> Function[{x}, 0.38], 
      ρ0 -> Function[{x}, Exp[-x/H]], p0 -> Function[{x}, Exp[-x/H]]};
sys = ToMatrixSystem[eqnSubbed2, {T[0] == 0, V[0] == 0, T[1] == 0, V[1] == 0}, {T, 
    V}, {x, 0, 1}, λ] //. subs
root = FindRoot[Evans[λ, sys], {λ, -1 + 3 I}]
 (* {λ -> -0.342689 + 2.65899 I} *)

Unfortunately my code doesn't currently get the eigenfunctions out immediately at the moment. However, you can use NDSolve to get them once you find a root:

sol = NDSolve[Join[eqnSubbed2 //. subs /. root, 
    {T[0] == 0, V[0] == 0, T'[0] == 1, V[1] == 0}], {T, V}, {x, 0, 1}]
GraphicsRow[{Plot[Evaluate[ReIm@T[x] /. sol], {x, 0, 1}], 
  Plot[Evaluate[ReIm@V[x] /. sol], {x, 0, 1}]}]

enter image description here

1
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