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 are slowly-changing delocalized functions of . However, what is remarkable is that for this type of deformations vanishes in the limit . Such deformations we would like to call `special' . Generally, we would like to define a a deformation as `special' if the associated fluctuation intensity is .
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, translations, and 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 is presented). In other words, we believe that any special deformation can be written as a linear combination of dilation, translations and rotations.