Consider a one-parameter family $A_t$ of unbounded positive self-adjoint operators with discrete spectrum (for example, one can consider a one-parameter family of Laplacians on a compact Riemannian manifold). Kato's real-analytic perturbation theory says that the eigenvalues $\lambda_t$ of $A_t$ vary real-analytically, and we can also select a complete orthonormal basis of eigenfunctions $\varphi_k(t)$ that vary real-analytically in $t$. My question is, let's say that the perturbation is smooth instead of real-analytic. How do the eigenvalues vary then? Also, can we still find an orthonormal basis of eigenfunctions that vary at a certain scale of regularity? I am particularly interested in the second question, as I am curious if something nasty happens when eigenvalues have multiplicities. Thanks!

Consider a one-parameter family $A_t$ of unbounded positive self-adjoint operators with discrete spectrum (for example, one can consider a one-parameter family of Laplacians on a compact Riemannian manifold). Kato's real-analytic perturbation theory says that the eigenvalues $\lambda_t$ of $A_t$ vary real-analytically, and we can also select a complete orthonormal basis of eigenfunctions $\varphi_k(t)$ that vary real-analytically in $t$. My question is, let's say that the perturbation is smooth instead of real-analytic. How do the eigenvalues vary then? Also, can we still find an orthonormal basis of eigenfunctions that vary at a certain scale of regularity (in particular, are the eigenfunctions differentiable with respect to $t$)? I am particularly interested in the second question, as I am curious if something nasty happens when eigenvalues have multiplicities. Thanks!