Talk:Isospectral

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I think the definition of isospectral is usually just taken to be the same spectrum as a set. So I think, one might just want to remove all sentences about the multiplicity of eigenvalues.

ElMaison (talk) 03:19, 31 March 2008 (UTC)Reply

Not in the applications to the spectrum of the Laplacian. (Just think about the Laplacian on a torus.) Mathsci (talk) 17:10, 6 April 2009 (UTC)Reply

Pgdoyle (talk) 16:45, 18 May 2010 (UTC) I'm asking for a citation for the statement: "In this case Selberg's trace formula shows that the spectrum of the Laplacian fully determines the length spectrum[citation needed], the set of lengths of closed geodesics in each free homotopy class, along with the twist along the geodesic in the 3-dimensional case." This statement is certainy true for orientable hyperbolic 2-manifolds (Huber's theorem), but I believe the problem is still open for hyperbolic 3-manifolds. Maclachlan and Reid do not address this issue. Instead, they simply define isospectrality of hyperbolic 3-manifolds to mean agreement of complex length spectra (Definition 12.4.1, p. 383).Reply

I'm also asking for a citation for the statement that Selberg's method "does not recapture the arithmetic examples of Milnor and Vignéras". I have heard people say that the Vignéras examples cannot be produced by Selberg's method, but I have been unable to find a reference.

The third and last paragraph in the introductory section begins as follows:

"In infinite dimensions, the spectrum need not consist solely of isolated eigenvalues. However, the case of a compact operator on a Hilbert space (or Banach space) is still tractable, since the eigenvalues are at most countable with at most a single limit point λ = 0."

In the introductory section, readers are just trying to understand what the word isospectral means. It is premature to discuss issues of what is "tractable" and what may be too difficult for mathematicians to make much progress on.

Any mention of "tractable" belongs in a later section after methods of determining which operators are isospectral has been discussed. 2601:200:C000:1A0:E9F2:EC0:68E9:DAE2 (talk) 19:16, 24 April 2021 (UTC)Reply

https://arxiv.org/pdf/2104.12885.pdf 00prometheus (talk) 22:32, 3 June 2023 (UTC)Reply