Timeline for Rigged Hilbert spaces and the spectral theory in quantum mechanics
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 25, 2022 at 9:49 | comment | added | Giovanni Ghezzi | It seems the only quasi-rigorous treatment of Rigged Hilbert Spaces is explained in the book "General eigenfunction expansions and unitary representations of topological groups" by Krzysztof Maurin | |
| Feb 15, 2021 at 3:07 | answer | added | user1504 | timeline score: 13 | |
| Feb 14, 2021 at 21:57 | history | became hot network question | |||
| Feb 14, 2021 at 21:30 | comment | added | Jochen Glueck | [continuation] Solvability of the Schrödinger equation is not a problem in this setting: it follows from Stone's theorem and the theory of $C_0$-groups that, for every initial value $u_0 \in L^2(\mathbb{R}^d)$, the equation $\dot \psi = i\Delta \psi$ has a unique mild solution in $L^2(\mathbb{R}^d)$ | |
| Feb 14, 2021 at 21:30 | comment | added | Jochen Glueck | @gmvh: I'm not sure I follow your comment. The common point of view is to consider the Laplace operator $\Delta$ as an unbounded but closed operator on $L^2(\mathbb{R}^d)$. In other words, $\Delta$ maps from the domain $D(\Delta) := W^{2,2}(\mathbb{R}^d) \subseteq L^2(\mathbb{R}^d)$ to $L^2(\mathbb{R}^d)$, and its graph is closed as a subset of $L^2(\mathbb{R}^d) \times L^2(\mathbb{R}^d)$. This is also the setting in which it makes sense to consider $\Delta$ as a self-adjoint operator. [to be continued]. | |
| Feb 14, 2021 at 16:28 | answer | added | Carlo Beenakker | timeline score: 14 | |
| Feb 14, 2021 at 16:19 | answer | added | Siddharth Bhat | timeline score: 7 | |
| Feb 14, 2021 at 15:51 | answer | added | Nik Weaver | timeline score: 19 | |
| Feb 14, 2021 at 15:48 | comment | added | jjcale | You should learn first John von Neumann's foundations auf quantum mechanics, especially unbounded operators. | |
| Feb 14, 2021 at 14:47 | comment | added | MathMath | @gmvh indeed. But most books use $L^{2}(\mathbb{R}^{d})$. It is a really difficult process if you are studying it for the first time, as I am, and most of the discussions are not "quite there". | |
| Feb 14, 2021 at 14:25 | comment | added | gmvh | Taking the Hilbert space to be $L^2(\mathbb{R}^d)$ is of course already a bit problematic in view of the Laplacian occurring in the Schrödinger equation; the more appropriate setting would appear to be a Sobolev space $W^{k,2}(\mathbb{R}^d)$ with $k\ge 1$ in order to at least allow a weak solution. | |
| Feb 14, 2021 at 14:21 | history | edited | gmvh | CC BY-SA 4.0 |
Fixed spelling, improved formatting
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| Feb 14, 2021 at 13:57 | history | asked | MathMath | CC BY-SA 4.0 |