Approximate eigenvectors for a set of non-commuting self-adjoint operators This problem is motivated by finding the right mathematical setting for expressing the compatibility of classical physics with quantum mechanics. 
Let $\mathcal H$ be a Hilbert space and $S$ a bounded self-adjoint operator. Then define $x \in \mathcal H$ to be an $\epsilon$-approximate eignevector if $\|Sx-\lambda x\| < \epsilon.\|x\|$ where $\lambda = \langle Sx,x\rangle /\|x\|^2$. Given a finite set of bounded self-adjoint operators $S_1,\cdots,S_n$ and bounds $\vec\epsilon = (\epsilon_1,\cdots,\epsilon_n)$, we say $S_1,\cdots,S_n$ are $\vec\epsilon$-classical if $\mathcal H$ is spanned by elements $x \in \mathcal H$ that are $\epsilon_i$-approximate eigenvectors of $S_i$ for all $i$. 
A simple example is $\mathcal H = L^2(\mathbb R)$, $T_1 =$ mult by $x$, $T_2 = i\cdot d/dx$, $S_1 = \tau_{c_1}(T_1), S_2 = \tau_{C_2}(T_2)$ where $\tau_C(x) = \max(\min(x,C),-C)$ is a cutoff function (n.b. no measuring instrument can register unbounded values). Then $x_{a,b} = e^{iax}\cdot e^{(x-b)^2/2\sigma^2}$ are the best candidates for approximate eigenvectors of $S_1$ and $S_2$. 
The question is to find natural bounds on the commutators $[S_i,S_j]$ that imply $S_1,\cdots,S_n$ are $\vec\epsilon$-classical with $\vec\epsilon$ a function of $\|S_i\|$ and the bounds on $[S_i,S_j]$. I have such results using the Hilbert-Schmidt norms of the commutators but the resulting $\vec\epsilon$ seem much too big. I seek ideas or references for work in this direction.
 A: I can give you a few papers, and these have references to others.  I discuss joint approximate eigenvectors in the context of approximate joint measurement in [1].  You need to know that often it is $K$-theory that tells you if such a basis can be found or not.  In [2] is a section called ``joint Wannnier spread'' that seems to be what you are after.  It dicusses the distinction between Hilbert Schmidt errors and operator norm errors. I learned of this line of reasoning from Hastings, who discusses the case of 2 observables.  All this work is on finite dimensional Hilbert space, but the joint pseudospectrum I discuss in [1] can work in infinite dimensions.
[1] Loring, Terry A. "K-theory and pseudospectra for topological insulators." Annals of Physics 356 (2015): 383-416.
[2] Loring, Terry A., and Adam P. W. Sørensen. "Almost commuting unitary matrices related to time reversal." Communications in Mathematical Physics 323.3 (2013): 859-887.
[3] Hastings, M. B. "Topology and phases in fermionic systems." Journal of Statistical Mechanics: Theory and Experiment 2008.01 (2008): L01001.
