Timeline for Why is the length spectrum called a spectrum?
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| Feb 8 at 5:41 | comment | added | Ryan Budney | @AndreyRyabichev: true, but it's called a spectrum in physics. So it's natural to do it in math as well. Mathematics does not have any special rights when it comes to the usage of the word "spectrum". I mean, think of how much mathematicians appropriate the word "quantum". | |
| Feb 7 at 12:51 | comment | added | D. Thomine | There are also notions of Markov and Lagrange spectra, which are related to height (instead of length) of geodesics in the modular surface. However, I am not sure that the vocabulary of spectrum for these predates the length spectrum. | |
| Feb 7 at 8:51 | comment | added | Timothy Budd | I am far from an expert here, but taking the question literally, you could have a look at Bismut's work on the hypoelliptic Laplacian, which, loosely speaking, deals with a family of operators that interpolates between the (Hodge) Laplacian and the generator of geodesic flow. In this way it provides a link between the Laplace spectrum and the length spectrum (in particular, chapter 8 deals with proving the Selberg trace formula in this way). | |
| Feb 7 at 8:15 | comment | added | Andrey Ryabichev | @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 | |
| Feb 7 at 4:53 | comment | added | Ryan Budney | 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. | |
| Feb 7 at 1:54 | history | edited | RobPratt |
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| Feb 7 at 1:08 | history | became hot network question | |||
| Feb 6 at 20:29 | comment | added | Terry Tao | 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 ). | |
| Feb 6 at 20:25 | comment | added | Terry Tao | 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.) | |
| Feb 6 at 19:41 | answer | added | Sam Nead | timeline score: 12 | |
| Feb 6 at 18:37 | comment | added | Sam Hopkins | 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). | |
| Feb 6 at 17:06 | history | asked | Andrey Ryabichev | CC BY-SA 4.0 |