I understand that for infinite-area hyperbolic surfaces, there are no $L^2$-eigenfunctions of the Laplace-Beltrami operator but there are a lot resonances.


But I am confused about the notion of resonances for finite-area hyperbolic surfaces. Are there resonances which are not eigenvalues?



Eigenvalues vs resonances
I presume by a "resonance" you mean a scattering resonance, so a pole of the determinant of the scattering matrix. On finite-area hyperbolic surfaces all scattering resonances correspond to eigenvalues, but not the other way around. The "cusp form" spectrum, which
includes all embedded eigenvalues, does not give rise to scattering resonances.


See page 20 of [Introduction to Spectral Theory on Hyperbolic Surfaces](http://www.mathcs.emory.edu/~davidb/spths.pdf) by David Borthwick.


