function construction - How to write a differential operator in Mathematica

I have a very basic question. I am trying to code the LHS of the differential equation: $$\Big[\frac{1}{\sin\theta}\frac{d}{d\theta}\big(\sin\theta\frac{d}{d\theta}\big)-\frac{m^2}{\sin^2\theta}+n(n+1)\Big]f(\theta)=0$$ into Mathematica. As you can see, in the first term, the differential operator is nested and is to properly act on the function $f(\theta)$.

I tried to use #1, but (lol) I don't know how to use it!

This is what I have:

Sin[θ]^(-1) D[Sin[θ] D[#1, θ], θ] - m^2*#1/Sin[θ]^2 + n (n + 1) #1

What next?!

Also: I don't want to just stick in f[θ] into where I have #1, because eventually I would like to be able to make change of variables!

Answer

Why not just define a Function ?

Mathematica graphics

ClearAll[m, n, x, foo]
oper = Function[{f, \[Theta]}, (1/
      Sin[\[Theta]] D[Sin[\[Theta]] D[f[\[Theta]], \[Theta]]] - 
     m^2/Sin[\[Theta]]^2 f[\[Theta]] + n (n + 1) f[\[Theta]])];
oper[foo, x]

foo[x_] := Sin[x]*Cos[x]*x^4
oper[foo, x]

Comments

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