Continuity of the spectrum with respect to the metric

Question

The following question is quite natural, but I am not aware of a reference dealing with it: let $M$ be a compact (smooth) manifold (posssibly with boundary) and $E$ a vector bundle on $M$ with an Hermitian or Euclidean metric, and let $g_u,\: u\in[0,1]$ be a smooth family of Riemannian metrics on $M$. For each $p=0,\ldots,{\rm dim}\: M$ we get a map ${\rm sp}(\Delta^p):\: [0,1] \to {\mathbb R}^{\mathbb N}$ which associates tu $u$ the ordered spectrum (with multiplicities) of the Laplace operator on $p$-forms on $(M,g_u)$ with coefficients in $E$. The question is: is this map continuous (with respect to the product topology on ${\mathbb R}^{\mathbb N}$)?

I believe that this is true, and that you can prove it using the min-max characterization of the spectrum, but as it seems a rather basic question I would think that somebody already answered some version of it.

Answer 1

However, for continuity of individual eigenvalues (with appropriate provisos in the presence of multiplicity), the min-max characterization works fine. More refined statements (e.g. about the behavior of the spectrum for smooth or analytic one-parameter families of metrics) are contained in Kato's classic book `Perturbation theory for Linear Operators'.

Answer 2

Maybe you find this this helpful. It answers the analogous question for the Dirac operator.