Maximum of a series of integrals of Hermite functions - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T01:59:54Z http://mathoverflow.net/feeds/question/66147 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/66147/maximum-of-a-series-of-integrals-of-hermite-functions Maximum of a series of integrals of Hermite functions Mateus Araújo 2011-05-27T02:31:54Z 2011-05-27T11:42:04Z <p>Given the function $$f(A) := \sum_{n=1}^{\infty}\left( \int_A \varphi_0\varphi_n \right)^2,$$ where $A$ is any measurable subset of $\mathbb{R}$, and $\varphi_n$ is the $n$th Hermite function, I want to know for which sets $f$ attains its maximum.</p> <p>I've already proven that $$f(A)\le \frac{1}{4}$$ for all $A$, and that $f(\mathbb{R}^+) = 1/4$. But it is crucial for my problem to find other sets that would also maximise $f$ or prove that none exists, and for doing that I am at a loss. Even finding local maxima would be interesting to me. </p> <p>The proof I already obtained is somewhat long and very indirect, so I won't include it here; also, I'm very interested to see how a mathematician would approach this problem.</p> http://mathoverflow.net/questions/66147/maximum-of-a-series-of-integrals-of-hermite-functions/66171#66171 Answer by Piotr Migdal for Maximum of a series of integrals of Hermite functions Piotr Migdal 2011-05-27T10:09:48Z 2011-05-27T11:42:04Z <p>Write your quantity as: $$f(A)=\hbox{Tr}\left[ P_A|0\rangle\langle0|P_A(\mathbb{1}-|0\rangle\langle0|) \right],$$ where $P_A$ is the projection on A, and $|0\rangle\langle0|$ is the projection on $\varphi_0$. Note that then you need only to investigate properties of the $\varphi_0$, not every Hermite function (as they form an orthonormal basis).</p> <p>With the properties of Tr and projection operators you get $$f(A)= \hbox{Tr}[|0\rangle\langle0|P_A|0\rangle\langle0|]-\hbox{Tr}[(|0\rangle\langle0|P_A|0\rangle\langle0|)^2]$$ $$=\lambda-\lambda^2.$$</p> <p>So:</p> <ul> <li>indeed, $\max f(A) = \frac{1}{4}$,</li> <li>$f(A)=\frac{1}{4}$ iff $\int_{A} \varphi_0^2(x)dx=\frac{1}{2}$. </li> </ul> <p>Depending what you need the formula for, but if it is about the filtering of the higher-order modes, <a href="http://www.springerlink.com/content/n57345p035262823/" rel="nofollow">here (sec. 6.)</a> is a numerical remark. </p>