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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

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.

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.