Timeline for Make mathematical sense of the Dirac well Potential Equation
Current License: CC BY-SA 3.0
11 events
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| Jul 21, 2015 at 21:56 | comment | added | paul garrett | @ChristianRemling, yes, indeed, thanks... and to give an example-objection: one way to interpret the (mathematical) difficulty in interpreting $\delta'$ as a "potential" in analogous fashion is that it is not inside $H^{-1}$, and it itself cannot be "legally" applied to solutions to $(\Delta-\lambda)u=\delta'$, since these solutions will only be in $H^{1/2-\epsilon}$, not $H^{3/2+\epsilon}$, etc. | |
| Jul 21, 2015 at 21:29 | comment | added | Christian Remling | Perhaps it's useful to point out that there has been an ongoing controversy (and a lot of ink has been wasted) on what $-d^2/dx^2 +\delta'$ "should" mean. I guess what I'm trying to say in this context is that this kind of thing isn't rocket science; you make some rigorous definition and try to convince others that this is a good interpretation of what you're trying to make sense of (that said, there is agreement on how to make sense of $\delta$ here). | |
| Jul 21, 2015 at 13:26 | history | edited | paul garrett | CC BY-SA 3.0 |
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| Jul 21, 2015 at 13:11 | history | edited | paul garrett | CC BY-SA 3.0 |
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| Jul 21, 2015 at 12:45 | comment | added | paul garrett | @JeanDuchon, yes, what I wrote was careless. But/and I think the issue is not truly about literal pointwise multiplication, for a reason I will try to clarify in the edit I will do just now... | |
| Jul 21, 2015 at 8:34 | comment | added | Jean Duchon | The product of a continuous function $y$ and Dirac $\delta$ "function" is $y(0)\delta$, a measure, not a locally integrable function. It's Carlo Beenakker's answer that is right, isn't it? | |
| Jul 16, 2015 at 18:26 | history | edited | paul garrett | CC BY-SA 3.0 |
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| Jul 16, 2015 at 18:26 | comment | added | paul garrett | Ooops, yes, I'll change that sign on the epsilon. The general, basic things about Sobolev spaces are treated in most or many books on PDEs, especially linear ones. Folland's book (or Tata lectures), Brezis' book, are two that do this sort of thing. The question of multiplication is an immediate corollary, then, since $H^{+s}$ and $H^{-s}$ (in various contexts) are in duality, so "pair" to $L^1$, at least. The Sobolev imbedding business is treated in those sources, certainly. Googling "Sobolev space" should give lots of useful results, too. | |
| Jul 16, 2015 at 17:42 | comment | added | Gateau au fromage | Thank you. On the first line, you mean $-1/2-\epsilon$ in the exponent, right? A reference discussing these specific types of issues would be great. | |
| Jul 16, 2015 at 17:39 | vote | accept | Gateau au fromage | ||
| Jul 16, 2015 at 16:24 | history | answered | paul garrett | CC BY-SA 3.0 |