1. The problem statement, all variables and given/known data I suppose that this is not directly a quantum mechanical problem, but this have been assigned as homework for the Quantum Mechanics course. Let be an operator L and eigenvalue equation [tex]Lf=\lambda f[/tex]. This operator, applied to a function [tex]f(x,y)[/tex], interchanges the variables i.e. [tex]Lf(x,y)=f(y,x)[/tex]. What's the general property of the eigenfunctions of this problem? Get the possible eigenvalues. 2. Relevant equations 3. The attempt at a solution Well. I think that if [tex]Lf(x,y)=f(y,x)[/tex] then if [tex]f[/tex] is an eigenfunction, obviously, [tex]\lambda f(x,y)=f(y,x)[/tex]. One possible kind of [tex]f[/tex] that fills conditions is one that is symmetric, that is [tex]f(x,y)=f(y,x)[/tex] then [tex]\lambda[/tex] for this kind of eigenfunctions will be [tex]\lambda=1[/tex]. Others are the antisymmetric ones, those for is true [tex]f(y,x)=-f(x,y)[/tex] and then the eigenvalue for this kind is [tex]\lambda=-1[/tex] But I'm sure that there are more conditions that generate other kinds of eigenfunctions, not only symmetric nor antisymmetric. My question is: there are more or the antisymmetric and symmetric ones are the only ones, and if there are more how I get them and their eigenvalues?