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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.

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    $\begingroup$ Welcome to Mathoverflow $\endgroup$
    – Peter Michor
    Commented Jun 7, 2018 at 15:36
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    $\begingroup$ Very good question. Are you sure, though, that the assumption on your $\epsilon$-approximate eigenvalue (if I may call it like that) $\lambda$ to be of the form $\lambda = \langle Sx,x\rangle / \|x\|^2$ isn't a bit too restrictive?. For instance, the lowest spectral value of $S$ can never be of this form unless it is an eigenvalue of $S$. As every spectral value of a self-adjoint operator is an approximate eigenvalue, this shows that the set of $\epsilon$-approximate eigenvalues does not contain the set of approximate eigenvalues, in general. $\endgroup$
    – Jochen Glueck
    Commented Jun 7, 2018 at 16:29
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    $\begingroup$ Did you consider coherent states of irreducible SU(2)-representations ? $\endgroup$
    – jjcale
    Commented Jun 7, 2018 at 18:32
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    $\begingroup$ "if $H$ is spanned by elements $x\in H$ that are $\varepsilon_i$-approximate eigenvectors of $S_i$ for all $i$." I'm not sure you want just that little: if $x$ is a common approximate eigenvector, then a whole neighborhood of $x$ consists of approximate eigenvectors, so the spanning property becomes pretty much void. $\endgroup$
    – fedja
    Commented Jun 7, 2018 at 20:03
  • $\begingroup$ Good point. I want to say that, if you start in any state $x$, then by a classical measurement, you project ("collapse" in Copenhagen language) that state into an approximate eigenstate of all $S_i$. For one $S$, the state is an integral of its projections into eigenstates and its norm squared is the integral of the squared norms of the projections. I want something like that but yes, this needs to be thought through. $\endgroup$
    – David Mumford
    Commented Jun 8, 2018 at 0:01

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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.

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