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Given a hyperbolic surface $X$, one considers the multiset of lengths of closed primitive geodesics. This multiset is called the length spectrum $\mathcal{L}(X)$.

Question: is $\mathcal{L}(X)$ a spectrum of some (differential) operator? In what context does this operator naturally arise?

I hope that there are not only folklore arguments of a philosophical nature, but specific constructions leading to such a term.

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    $\begingroup$ I have no idea about this particular case, but the English word "spectrum" is a more general term than something which just applies to operators (e.g., think of the light spectrum in physics, or even within math, think of spectra in algebraic topology). $\endgroup$
    – Sam Hopkins
    Commented Feb 6 at 18:37
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    $\begingroup$ One simple calculation that might be suggestive: in a torus ${\bf R}^d/\Lambda$ with the Euclidean metric, the length spectrum is the set of squares of magnitudes of (irreducible) elements of the lattice $\Lambda$, while the spectrum of the Laplacian is the set of squares of magnitudes of elements of the dual lattice $\Lambda^*$. (Admittedly, this is not a hyperbolic manifold, though.) $\endgroup$
    – Terry Tao
    Commented Feb 6 at 20:25
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    $\begingroup$ Another possible connection is that there is an analogy between primitive geodesics and primes (see e.g., en.wikipedia.org/wiki/Prime_geodesic ), and prime numbers are basically the spectrum of the integers (in the algebraic sense en.wikipedia.org/wiki/Spectrum_of_a_ring ). $\endgroup$
    – Terry Tao
    Commented Feb 6 at 20:29
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    $\begingroup$ Think of the spectrum of a hydrogen atom. Basically, it's a bunch of points on the real line, if you think of the real line as being frequencies of light. The length spectrum of a surface is again essentially just a bunch of points on the real line. So the comparison is pretty direct. $\endgroup$
    – Ryan Budney
    Commented Feb 7 at 4:53
  • $\begingroup$ @RyanBudney You are simply talking about a subset of the line. This subset can just as well be called a “length support” or a “length generator”, using other comparisons $\endgroup$
    – Andrey Ryabichev
    Commented Feb 7 at 8:15

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I do not know of such an operator. But there is the following lovely theorem of Huber:

Theorem: Two compact hyperbolic surfaces have the same spectrum of the Laplacian if and only if they have the same (a) area and (b) length spectrum.

For example, here are Burrin's lecture notes from a class at ETH — see Section 6.2.

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  • $\begingroup$ thanks! possibly this is the truly reason! $\endgroup$
    – Andrey Ryabichev
    Commented Feb 6 at 20:04
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    $\begingroup$ One could also mention the Selberg trace formula which directly relates the Laplace spectrum and the length spectrum for a hyperbolic surface. $\endgroup$
    – quarague
    Commented Feb 7 at 8:56

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