Question

I would like to know if there are theorems that state under which circumstances spectra of operator families depend smoothly on the parameter.

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?

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?

I am interested in reading suggestions :-)

Answer 1

Look at Kato's book on Perturbation Theory for Linear Operators.

Answer 2

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.