Solving infinite coupled equations

Question

I need to solve symbolically an infinite system of coupled algebraic equations that I tried to do analytically but I could not. Solutions of these equations, $V_l^m$, define the coefficients of a some function $V$ (that I'm looking for) of the form

$$ V(r,\theta ,\varphi)=\sum _{l=0}^{\infty } \sum _{m=-l}^{+l} Y_l^m (\theta ,\varphi) V_l^m(r)=Y_0^0 V_0^0+Y_1^0 V_1^0+Y_1^1 V_1^1+Y_1^{-1} V_1^{-1}+\cdots $$

where $Y_l^m (\theta ,\varphi)$ denote the spherical harmonics.

These solutions $V_l^m$ obey the general form

$$ A\, V_l^m+\frac{B}{2}\left(-\sqrt{\frac{(l+m-1) (l+m)}{(2 l-1) (2 l+1)}} V_{l-1}^{m-1}+\sqrt{\frac{(l-m) (l-m-1)}{(2 l+1) (2 l-1)}} V_{l-1}^{m+1}+\sqrt{\frac{(l-m+1) (l-m+2)}{(2 l+1) (2 l+3)}} V_{l+1}^{m-1}-\sqrt{\frac{(l+m+2) (l+m+1)}{(2 l+1) (2 l+3)}} V_{l+1}^{m+1}\right)+m R V_l^m+\left(\delta _{l,1} \delta _{m,-1}-\delta _{l,1} \delta _{m,1}\right)D\, \lambda =0$$

which involves an infinite coupled equations.

Here $ A, B, D, R, \lambda$ are real constants.

In MA input form it gives

eq[l_,m_]:=A Subscript[V,{l,m}]+B/2 (Sqrt[((l-m+1)(l-m+2))/((2l+1)(2l+3))] Subscript[V,{l+1,m-1}]+Sqrt[((l-m)(l-m-1))/((2l+1)(2l-1))] Subscript[V,{l-1,m+1}]-Sqrt[((l+m-1)(l+m))/((2l-1)(2l+1))] Subscript[V,{l-1,m-1}]-Sqrt[((l+m+2)(l+m+1))/((2l+1)(2l+3))] Subscript[V,{l+1,m+1}])+(KroneckerDelta[l,1]KroneckerDelta[m,-1]-KroneckerDelta[l,1]KroneckerDelta[m,1])D \[Lambda]+m R Subscript[V,{l,m}]

Here just for the code I used $V_{\{l,m\}}$ instead of $V_l^m$.

For your information, solution of this problem is to express $V_{\{0,0\}}$ according to the variable $\lambda$ as

$$V_{\{0,0\}}=(f)\lambda$$

where $f$ is a term that includes all other solutions (maybe in the form of an infinite continued fraction).

Any help please on this problem! Can MA do something?

Answer 1

first attempt

(apparently this isn't what the OP wanted)

I have edited the equation a bit because D is a reserved symbol and because subscripts are inconvenient:

eq[l_,m_] = d λ (KroneckerDelta[-1, m] KroneckerDelta[1, l] - KroneckerDelta[1, l] KroneckerDelta[1, m]) + a V[l, m] + 
            m R V[l, m] + 
            1/2 b (-Sqrt[(((-1 + l + m) (l + m))/((-1 + 2 l) (1 + 2 l)))] V[l-1, m-1] + 
                   Sqrt[((-1 + l - m) (l - m))/((-1 + 2 l) (1 + 2 l))] V[l-1, m+1] + 
                   Sqrt[((1 + l - m) (2 + l - m))/((1 + 2 l) (3 + 2 l))] V[l+1, m-1] - 
                   Sqrt[((1 + l + m) (2 + l + m))/((1 + 2 l) (3 + 2 l))] V[l+1, m+1]);

For every value of l, we have 2l+1 equations that can be used to determine {V[l+1,-l-1],...,V[l+1,l+1]}. There are 2l+3 unknowns, so we cannot determine all coefficients. Here I've chosen to solve for the {V[l+1,-l-1],...,V[l+1,l-1]} and leave V[l+1,l] and V[l+1,l+1] as free parameters:

S[l_] := First@Solve[Table[eq[l, m] == 0, {m, -l, l}], 
                     Table[V[l+1, m], {m, -l-1, l-1}]]

Try it out:

S[2]
(*    {V[3, -3] -> ..., 
       V[3, -2] -> ..., 
       V[3, -1] -> ..., 
       V[3, 0] -> ..., 
       V[3, 1] -> ...}    *)

We can stack these solutions in order to express the coefficients at a given value of l in terms of all smaller l-values:

Expand[S[2] /. S[1] /. S[0]]
(*    complicated expression for V[3,-3]...V[3,1] in terms of
      V[0, 0], V[1, 0], V[1, 1], V[2, 1], V[2, 2], V[3, 2], V[3, 3]}    *)

programmatically:

Clear[F];
F[l_] := Module[{f = S[l]},
  Do[f = f /. S[L], {L, l - 1, 0, -1}];
  Expand[f]]

Now calling F[2] gives the same result as the above Expand[S[2] /. S[1] /. S[0]].

second attempt

If you want to express $V_{0,0}$ in terms of $\{V_{L,-L},V_{L,-L+1},\ldots,V_{L,L}\}$ for a specific $L$, you can solve all the equations for $l<L$ for all the variables with $l<L$. For example, with $L=4$ we express $V_{0,0}$ in terms of $\{V_{4,-4},V_{4,-3},V_{4,-2},V_{4,-1},V_{4,0},V_{4,1},V_{4,2},V_{4,3},V_{4,4}\}$:

With[{L = 4},
  FullSimplify[V[0, 0] /. 
    First@Solve[
      Flatten[Table[eq[l, m] == 0, {l, 0, L - 1}, {m, -l, l}]], 
      Flatten[Table[V[l, m],       {l, 0, L - 1}, {m, -l, l}]]]]]

-((b (-10290 Sqrt[7] a^7 d λ + 42 Sqrt[15] a^2 b^3 R (35 Sqrt[7] R^2 (-V[4, -2] + V[4, 2]) + 2 b^2 (7 V[4, -4] - Sqrt[7] V[4, -2] + Sqrt[7] V[4, 2] - 7 V[4, 4])) - 294 Sqrt[15] a^4 b^3 R (7 V[4, -4] - Sqrt[7] V[4, -2] + Sqrt[7] V[4, 2] - 7 V[4, 4]) + 6 Sqrt[15] b^3 R (21 Sqrt[7] b^2 R^2 (V[4, -2] - V[4, 2]) + 49 R^4 (7 V[4, -4] + 4 Sqrt[7] V[4, -2] - 4 Sqrt[7] V[4, 2] - 7 V[4, 4]) + b^4 (-7 V[4, -4] + Sqrt[7] V[4, -2] - Sqrt[7] V[4, 2] + 7 V[4, 4])) + 49 a^5 (174 Sqrt[7] b^2 d λ + 2940 Sqrt[7] d R^2 λ + b^3 (-7 Sqrt[15] V[4, -4] + 2 Sqrt[105] V[4, -2] - 3 Sqrt[42] V[4, 0] + 2 Sqrt[105] V[4, 2] - 7 Sqrt[15] V[4, 4])) - 14 a^3 (153 Sqrt[7] b^4 d λ + 1680 Sqrt[7] b^2 d R^2 λ + 36015 Sqrt[7] d R^4 λ + 35 Sqrt[3] b^3 R^2 (7 Sqrt[5] V[4, -4] + Sqrt[35] V[4, -2] - 3 Sqrt[14] V[4, 0] + Sqrt[35] V[4, 2] + 7 Sqrt[5] V[4, 4]) - 2 b^5 (7 Sqrt[15] V[4, -4] - 2 Sqrt[105] V[4, -2] + 3 Sqrt[42] V[4, 0] - 2 Sqrt[105] V[4, 2] + 7 Sqrt[15] V[4, 4])) + a (162 Sqrt[7] b^6 d λ + 252 Sqrt[7] b^4 d R^2 λ + 50274 Sqrt[7] b^2 d R^4 λ + 370440 Sqrt[7] d R^6 λ + 14 Sqrt[3] b^5 R^2 (28 Sqrt[5] V[4, -4] + Sqrt[35] V[4, -2] - 6 Sqrt[14] V[4, 0] + Sqrt[35] V[4, 2] + 28 Sqrt[5] V[4, 4]) + 49 Sqrt[3] b^3 R^4 (77 Sqrt[5] V[4, -4] + 8 Sqrt[35] V[4, -2] - 27 Sqrt[14] V[4, 0] + 8 Sqrt[35] V[4, 2] + 77 Sqrt[5] V[4, 4]) - 3 b^7 (7 Sqrt[15] V[4, -4] - 2 Sqrt[105] V[4, -2] + 3 Sqrt[42] V[4, 0] - 2 Sqrt[105] V[4, 2] + 7 Sqrt[15] V[4, 4]))))/(3 Sqrt[ 42] (1715 a^9 - 1225 a^7 (2 b^2 + 21 R^2) + 7 a^5 (156 b^4 + 2380 b^2 R^2 + 15435 R^4) - a^3 (174 b^6 + 3045 b^4 R^2 + 31850 b^2 R^4 + 145775 R^6) + 3 a (3 b^8 + 58 b^6 R^2 + 651 b^4 R^4 + 5880 b^2 R^6 + 20580 R^8))))

You can do this for larger values of $L$, but it gets more difficult.