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- W4214963584 abstract "According to Courant's theorem, an eigenfunction as-sociated with the $n$-th eigenvalue $lambda_n$ has at most $n$ nodal domains. A footnote in the book of Courant and Hilbert, states that the same assertion is true for any linear combination of eigenfunctions associated with eigenvalues less than or equal to $lambda_n$. We call this assertion the emph{Extended Courant Property}.smallskipIn this paper, we propose simple and explicit examples for which the extended Courant property is false: convex domains in $R^n$ (hypercube and equilateral triangle), domains with cracks in $mathbb{R}^2$, on the round sphere $mathbb{S}^2$, and on a flat torus $mathbb{T}^2$." @default.
- W4214963584 created "2022-03-05" @default.
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- W4214963584 date "2017-05-10" @default.
- W4214963584 modified "2023-09-23" @default.
- W4214963584 title "On Courant's nodal domain property for linear combinations of eigenfunctions, Part I" @default.
- W4214963584 doi "https://doi.org/10.48550/arxiv.1705.03731" @default.
- W4214963584 hasPublicationYear "2017" @default.
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