Timeline for Existence of non-trivial solutions to Dirichlet problem with a potential lying between eigenvalues.
Current License: CC BY-SA 2.5
8 events
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| Sep 7, 2010 at 14:09 | comment | added | Dorian | This is really fantastic Pietro! The whole minmax formulation of the eigenvalues was something that I had just skipped in my studies as it didn't seem important but wow what a nice application! I'm at least put at ease that what I was thinking was indeed correct. | |
| Sep 7, 2010 at 5:04 | comment | added | Victor Protsak | Pietro, this is so cool! I haven't seen a comparison principle used in good 15 years, and the min-max principle for eigenvalues (except $\lambda_1$) for comparable time. | |
| Sep 6, 2010 at 22:45 | comment | added | Pietro Majer | Note also that if you make the stronger assumption $V(x)+\epsilon\leq -\lambda_k$ with $\epsilon>0$, since $\lambda_k(-\Delta+V+\epsilon)=\lambda_k(-\Delta+V)+\epsilon$, the above weak form of monotonicity is sufficient to conclude $\lambda_k(-\Delta+V)\leq-\epsilon$; analogously for the other inequality. | |
| Sep 6, 2010 at 21:42 | comment | added | Pietro Majer | At least the weak inequality is immediate: for a fixed $u\in H^1_0$ the Rayleigh quotient is increasing wrto $V$. Since the above formula express $\lambda_k$ as a certain min-max over the same sets, it follows that $\lambda_k(-\Delta+V)\leq \lambda_k(-\Delta+W)$ whenever $V\leq W$. For the strict inequality it's jus a bit more delicate, but the idea is the same. | |
| Sep 6, 2010 at 21:19 | comment | added | Dorian | Is there an easy way to see why we should have monotonicity with respect to $V$? | |
| Sep 6, 2010 at 21:18 | vote | accept | Dorian | ||
| Sep 6, 2010 at 20:49 | history | edited | Pietro Majer | CC BY-SA 2.5 |
added 29 characters in body
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| Sep 6, 2010 at 20:44 | history | answered | Pietro Majer | CC BY-SA 2.5 |