Transient and Recurrent Spectrum

spectral theory loses sight of the fact that operators of interest are typically acting on a concrete space, normally L’(R“. d“x) for some I’. Physics is often connected with the local structure in R” and much recent progress (reviewed, e.g., in 191) uses the geometry in R“. Of particular interest is G,“(t) = ilF(I.ul < R)e-““o?:,l. where F(A) is multiplication by the characteristic function of A. For example. the celebrated RAGE theorem (see, e.g., 126)) says that y E Kd,, :g K,< if and only if, for all R. 1 .7 2*.I., G,“(t) dt --t 0 as Tco under the sole condition that F(lxl < R)(H + i) ’ is compact. something true for virtually all Schr6dinger operators. Let us introduce four subspaces (since G” tw < G” + G”, they are subspaces): (1) The transport vectors, denoted TRANS, being any u, for which there is a > 0 so that G,“,(t) = O(t‘“‘) for all N. (2) The flight vectors, denoted FLIGHT. being any cp for which G;(t) = O(t--,‘) for each fixed R and N. (3) The weak flight vectors, denoted WEFL, being any q for which )-= G;(t) dt < co -02 for each fixed R.