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I assume that you ask whether we can compute the continuous part of the spectrum!?

1.There is no continuous spectrum, if the surface is compact.

2.If the surface has finite volume, then, in general, we can not compute the contionuous contribution to the Selberg trace formula. We do not have any non-trivial bounds in full generality, can not even say whether there are any nontrivial discrete eigenvalues at all.

3.For modular surfaces, the continuous spectrum is related to the logarithmic derivative of certain $L$-functions. The computation are very complicated in full generaltiy. For $PSL_2(\mathbb{Z}) \backslash PSL_2(\mathbb{R})$ it can be realted related to the Riemann zeta function.

4.For infinite volume surfaces, there is no trace formula, but you can do still some spectral analysis, but I do not know any results here.

If you like more analysis, I can recommend "Iwaniec - Spectral theory of automorphic forms", but if you prefer some group theoretic arguments "Deitmar, Echterhoff - Principle of harmonic analysis" is pretty good, but treats also only the compact case.

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I assume that you ask whether we can compute the continuous part of the spectrum!?

1.There is no continuous spectrum, if the surface is compact.

2.If the surface has finite volume, then, in general, we can not compute the contionuous contribution to the Selberg trace formula. We do not have any non-trivial bounds in full generality, can not even say whether there are any nontrivial discrete eigenvalues at all.

3.For modular surfaces, the continuous spectrum is related to the logarithmic derivative of certain $L$-functions. The computation are were very complicated in full generaltiy. For $PSL_2(\mathbb{Z}) \backslash PSL_2(\mathbb{R})$ it can be realted to the Riemann zeta function.

4.For infinite volume surfaces, there is no trace formula, but you can do still some spectral analysis, but I do not any results here.

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I assume that you ask whether we can compute the continuous part of the spectrum!?

1.There is no smooth partcontinuous spectrum, if the surface is compact.

2.If the surface has finite volume, then, in general, we can not compute the contionuous contribution to the Selberg trace formula. We do not have any non-trivial bounds in full generality, can not even say whether there are any nontrivial discrete eigenvalues at all.

3.For modular surfaces, the continuous spectrum is related to the logarithmic derivative of certain $L$-functions. The computation are were complicated in full generaltiy. For $PSL_2(\mathbb{Z}) \backslash PSL_2(\mathbb{R})$ it can be realted to the Riemann zeta function.

4.For infinite volume surfaces, there is no trace formula, but you can do still some spectral analysis, but I do not any results here.

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