# solving this laplacian

given the laplacian $-y^{2}( \partial _{x}^{2} + \partial _{y}^{2} )f(x,y)=Ef(x,y)$

can we find a solution in the form

$f(x,y)= \sum_{n,m}C_{n,m} \phi (x,E) g(y,E)$

if we impose the extra boundary conditions $f( \frac{ax+b}{cx+d})=f(x)$ for the modular group

how can we calculate the energies and the eigenfunctions ?? , thanks.

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This question is about automorphic/modular forms, of course, because that "Laplacian" is the $SL(2,\mathbb R)$-invariant one on the upper half-plane, and, implicitly, the coordinate is $z=x+iy$, and the group action is on $z$, not $x$. Much more can be said in this particular case than for generic, non-compact, finite-volume Riemann surfaces.
The desired form of solution presumably dropped some subscripts... But, yes, $L^2$ solutions, and moderate-growth solutions, have expansions of the form $\sum_n c_n\cdot W_\lambda(|n|y)\cdot e^{2\pi i n x}$, where $W_\lambda$ is (up to normalization) a Bessel function depending on the eigenvalue $\lambda$. This can be seen by noting that $SL(2,\mathbb R)$ contains the subgroup $\pmatrix{1&m\cr 0&1}$, which translates the real part $x$, thereby giving a Fourier expansion. The differential operator allows separation of variables, giving a differential equation in the imaginary part, $y$, which is essentially Bessel's equation.
There is also significant continuous spectrum, spanned by "wave packets" of (non-$L^2$) Eisenstein series. For this modular curve, this is well-established and understood in the literature.