I'm wondering if there are any mathematical physicists/analysts out there that can help me with some notation I've seen in Reed and Simon's books on analysis. Unfortunately I don't have time to read all four volumes at the moment, and I've carefully combed through volume three (scattering) where I thought it would be defined.
What does the symbol $(L^\infty)_\epsilon$ mean? Presumably this is a modification on $L^\infty(\mathbb{R}^3)$ but I haven't been able to find a definition or defining property of functions in this space.
To give some context, given a Schrodinger operator $ - \Delta + V$, they'll usually define $V \in R + (L^\infty)_\epsilon$ where $R$ is a Rollnik class and proceed in a theorem from there. Unfortunately this level of analysis is very new to me and I only need a cursory understanding of this material for a project I'm working on.
Thanks for your help!
I found this notation in Simon's paper "Hamiltonians defined as quadratic forms", Commun. Math. Phys. 21 (1971), 192-210. (PDF available on his web page).
A footnote at the bottom of the first page reads:
$$X + (L^\infty)_\varepsilon = \{ f \mid (\forall \varepsilon) f = x_\varepsilon + g_\varepsilon \text{ with } x_\varepsilon \in X; \|g_\varepsilon\|_\infty < \varepsilon\}.$$ So a function is in $X + (L^\infty)_\varepsilon$ iff it looks like a function from $X$ plus an arbitrarily small bounded perturbation. As an example, Simon's footnote adds that the function $r^{-1}$ is in $L^2 + (L^\infty)_\varepsilon$ on $\mathbb{R}^3$. (It isn't in $L^2$, but for any $\varepsilon$ you can write it as the sum of an $L^2$ function plus a bounded function of sup norm less than $\epsilon$; just cut it off outside a large ball.)