Smooth dependence of the spectrum on the operator - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T08:56:18Z http://mathoverflow.net/feeds/question/76152 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/76152/smooth-dependence-of-the-spectrum-on-the-operator Smooth dependence of the spectrum on the operator Kofi 2011-09-22T19:31:20Z 2011-11-30T09:04:38Z <p>I would like to know if there are theorems that state under which circumstances spectra of operator families depend smoothly on the parameter.</p> <p>To clarify, suppose I have a 1-parameter family $T_h$ of self-adjoint operators in $L(H)$, $h \in I$ open and suppose that every $T_h$ has a discrete and well-ordered spectrum. Under which circumstances are the maps $\lambda_n: L(H) \longrightarrow \mathbb{R}$ smooth?</p> <p>Think of the operator $h\Delta + \mathrm{id} \in H^2(S^1)$ for example. The spectrum is $h^2n^2$, $n \in \mathbb{Z}$, so the maps $\lambda_n: L(H^2)\longrightarrow \mathbb{R}$ that give the $n$-th Eigenvalue depend smoothly on $h$ except at $h=0$, where they are not even defined anymore. So in case of differential operators, I could imagine that it has to do something with invertibility of the symbol? </p> <p>I am interested in reading suggestions :-)</p> http://mathoverflow.net/questions/76152/smooth-dependence-of-the-spectrum-on-the-operator/76157#76157 Answer by Michael Renardy for Smooth dependence of the spectrum on the operator Michael Renardy 2011-09-22T19:43:01Z 2011-09-22T19:43:01Z <p>Look at Kato's book on Perturbation Theory for Linear Operators.</p> http://mathoverflow.net/questions/76152/smooth-dependence-of-the-spectrum-on-the-operator/82258#82258 Answer by rtwmartin for Smooth dependence of the spectrum on the operator rtwmartin 2011-11-30T09:04:38Z 2011-11-30T09:04:38Z <p>Another good book for such results is Volume 1 of Reed and Simon's Methods of Modern Mathematical Physics - Functional Analysis. See the section on convergence of unbounded operators, in particular norm resolvent / strong resolvent convergence. </p>