`Special' deformations

The WNA dramatically fails (see Fig. 3.5) for dilation, translations and rotations (see Table 3.2 for their definitions in 2D). It is not surprising that the WNA is `bad' for these deformations because their $D({\mathbf s})$ are slowly-changing delocalized functions of $s$. However, what is remarkable is that $\tilde{C}_{{\mbox{\tiny E}}}(\omega)$ for this type of deformations vanishes in the limit $\omega\rightarrow0$. Such deformations we would like to call `special' [14]. Generally, we would like to define a a deformation as `special' if the associated fluctuation intensity is $\nu_{{\mbox{\tiny E}}}=0$.

A special result that follows from the considerations of Appendix E is that a linear combination of special deformation is also special. Therefore the special deformations constitute a linear space of functions. We believe that this linear space is spanned by the following basis functions: one dilation, $d$ translations, and $d(d-1)/2$ rotations. However we are not able to give a rigorous mathematical argument that excludes the possibility of having a larger linear space. This is discussed in Appendix D (and the generalization to arbitrary potential $U{({\mathbf r})}$ is presented). In other words, we believe that any special deformation can be written as a linear combination of dilation, translations and rotations.