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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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up vote 4 down vote accepted

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.

Iwaniec' book Spectral methods of automorphic forms (AMS) talks about such stuff in an accessible form. My on-line modular forms class notes discuss such things, also.

It is non-trivial to prove that there is infinite point spectrum! Selberg did this c. 1956, by inventing his "trace formula". Still, even though now we know that the discrete spectrum does satisfy "Weyl's Law", the specific numerical features of the discrete spectrum seem mysterious. (Work of Phillips-Sarnak, Wolpert, and others shows that perturbations of the modular surface tend to kill off discrete spectrum.)

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.

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