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The spectra of parallel flows (that is, flows governed by first-order differential operators parallel to one direction) are investigated, on both $L^2$ spaces and weighted-$L^2$ spaces. As a consequence, an example of a flow admitting a purely singular continuous spectrum is provided. For flows admitting more regular spectra the density of states is analyzed, and spaces on which it is uniformly bounded are identified. As an application, an ergodic theorem with uniform convergence is proved., Comment: 18 pages, no figures

Long-time behavior and regularity are studied for solutions of the Stark equationut=i(−Δ−x1+V(x))u,u(0,x)∈L2(Rn). It is shown that for a class of short-range potentialsV(x) the gain of local smoothness and the decay as |t|→∞ are close to those of the corresponding Schrödinger equationut=i(−Δ+V(x))u.

Let u = u(x, t) be a solution to the IVP for the Schrodinger equation iu 1 = (−Δ + V(x))u ≡ Hu , u(x, 0) = u 0 (x) ϵ P ac L 2 (R n ) (Pac is the projection on the absolutely continuous subspace of H). Assume that for some e > 0 the multiplication operator (1+|x|) 1+i V(x):H 1-e (R n ) → L 2 (R n ) is bounded. Then u(x, t) = u1(x, t) + u2(x, t) where, for every s > 1 2 , ∫ R ∫ R n (1 + |x| 2 ) −s |(1+H) 1 4 u 1 (x,t)| 2 dxdt ⩽ C ‖u 0 ‖ L 2 (R n ) 2 , and for every integer j, sup‖(I + H) j u 2 (·, t)‖ L 2 (R n ) ⩽ C j ‖ U 0 ‖ L 2 (R n ) tϵR

The initial value problem for the Schrodinger type equation ∂u ∂t = iP(D)u , u (0, x ) = u 0 ( x ), x ∈ R n , is considered. Here P ( ξ ) is a principal type polynomial of order m ( i.e ., ¦▽P(ξ)¦ ⩾ C(1 + ¦ξ¦) m − 1 , ¦ξ¦ ⩾ R) , or a certain type of other real symbol such as P(ξ) = ¦ξ¦ m , m > 1 . The following results are proved: (a) If u 0 ∈ L 2 ( R n ), then for any fixed R > 0 write u 0 = u 1 + u 2 , where u 1 and u 2 are square integrable and the Fourier transform of u 1 , vanishes for ¦ξ¦ ⩾ R and that of u 2 vanishes for ¦ξ¦ ⩽ R . If u 1 ( t , x ) = e itP ( D ) u 1 ( x ), u 2 ( t , x ) = e itP ( D ) u 2 ( x ), then u 1 is real analytic in R × R n and satisfies for every multi-index α, sup t, x ¦D α u 1 (t, x)¦ ⩽ C α ∥u 0 ∥ L 2 , while (1+|x| 2 ) − 1 2 u 2 (t,x)∈L 2 (R t ;H (m−1) 2 (R x n )) . (b) If u 0 ∈ L 2, S ( R n ), S >0 then for every t ≠0, u ( t ,·)∈ H loc ( m −1) S ( R n ). (c) if u 0 ∈H S (R n ), S> 1 2 then for a.e. x ∈ R n , u ( t , x )→ u 0 ( x ) as t →0.

In this paper we consider factorizations of the form I − K = ( I + K − )( D + F )( I + K + ), where K − , K + , and D are lower, upper, and diagonal operators relative to a maximal chain P of orthoprojections in a separable Hilbert space. In the case when K , K − , and K + are Hilbert-Schmidt, we determine the minimal rank of the operator F which occurs in the middle term of the factorization.