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Jacob Muller: Spectra of higher order differential operators on graphs and almost periodic functions

Time: Fri 2019-11-22 14.35 - 14.55

Location: KTH, D3

Participating: Jacob Muller, Stockholms universitet

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In this talk, we study the spectra of self-adjoint \(n\)-Laplacian operators \(\left(−d^2/dx^2 \right)^n\)  on compact finite metric graphs. We derive an effective secular equation and analyse the spectral asymptotics, exploiting the fact that the secular function is close to a trigonometric polynomial. The notion of the quasispectrum is introduced and its uniqueness is proved using the theory of almost periodic functions. For this purpose, new results concerning (classical) almost periodic functions are proved. In particular we study the roots of functions which are close to almost periodic ones and show that their roots are asymptotically close to each other. One can prove an equivalence relation of sorts between sets of roots of such functions, which has the consequence that two \(n\)-Laplacians are asymptotically isospectral if and only if they have the same quasispectrum. The results on almost periodic functions have wider applications outside the theory of differential operators. (Joint work with Pavel Kurasov, Stockholm University)