Timeline for Discrete and continuous representation in Hilbert space
Current License: CC BY-SA 4.0
5 events
| when toggle format | what | by | license | comment | |
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| Aug 10 at 10:05 | comment | added | terceira | The situation where the operator has no classical eigenvectors (i.e., not in the original Hilbert space) but does have so-called generalised ones (in this case exponential functions or, in the the FT mirror, delta distributions) was studied in great detail 50 years ago (key words: generalised eigenvectors, Gelfand triples, distribution theory). The literature is vast but you could start with the monographs of K. Maurin, easy to find with a search machine. | |
| Aug 10 at 1:32 | comment | added | Nate Eldredge | The Laplacian $\Delta$ indeed doesn't have any eigenfunctions in $L^2(\mathbb{R}^n)$. One way to to see this is to apply the Fourier transform (an isometric isomorphism), under which $\Delta$ becomes multiplication by $|x|^2$ which clearly doesn't have any eigenfunctions. | |
| Aug 10 at 0:30 | comment | added | Nate Eldredge | You can use LaTeX style formatting on this site, which would be more readable than what you have. See math.stackexchange.com/help/notation for info. | |
| Aug 9 at 10:55 | history | edited | YCor | CC BY-SA 4.0 |
fixed typos
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| Aug 9 at 10:50 | history | asked | Alucard-o Ming | CC BY-SA 4.0 |