The generic behavior of solutions to some evolution equations: asymptotics and Sobolev norms

We study generic behavior of solutions to a large class of evolution equations. The methods are applied to Schrodinger evolution on the circle.

In this paper, we develop methods to control the generic behavior of solutions to various evolution equations. The word "generic" will refer to the "coupling constant" that appears in front of the diagonal operator (a differential operator in most applications) which is perturbed by a potential. We were motivated by recent results where the behavior of Sobolev norms was studied for various evolution equations [3,9,11]. The general situation of fixed coupling constant was studied in many papers (e.g., [8] and references therein).
The structure of the paper is as follows. In the first section, we prove simple results for the "typical" behavior of solutions to 2×2 system of differential equations which preserve ℓ 2 norm of the initial value in Cauchy problem. The section 2 deals with "integrable" case when the transport equation is considered on the circle. In section 3, we prove results similar to those in section 1 but for the general N × N systems. The section 4 contains the applications of these results to evolution in Hilbert spaces, e.g., the non-stationary Schrödinger equation. In the last section, we consider the most difficult case when the so-called "gap condition" deteriorates in time. We handle only the short-range case, the general situation is far more difficult and will be considered elsewhere. The paper is concluded with Appendix which contains some well-known results we use in the main text. The first two sections and Appendix have mostly pedagogical value, the main results are in sections 3 and 5.
In the text, we will use the following notations: for f 1 (2) ≥ 0, f 1 f 2 means there is a constant C so that f 1 ≤ Cf 2 . The symbols e j will denote the standard basis vectors in C N . 1. The model case of 2 × 2 system We start this section with the following model system of two ODEs.
where Λ = λ 1 0 0 λ 2 , λ 1(2) ∈ R and Hermitian V (t) in some way to be specified below. Notice that X(t) is unitary in this setting. The standard problem to address is, of course, the long-time asymptotics of X(t). In the case of strong decay, i.e. when V (t) ∈ L 1 (R + ), X(t) has a limit as can be easily seen from iteration of the corresponding integral equations. Take That reduces the problem to with and it decay as fast as v 12 . Does Y (t) have a limit as t → ∞? Clearly, for q(t) ∈ L 1 (R + ) the answer is yes but already for q(t) ∈ L p (R + ), p > 1 this is not the case in general. Indeed, take q(t) = iǫ on t ∈ [0, ǫ −1 ]. Then, on that interval, Y (t) = cos(ǫt) − sin(ǫt) sin(ǫt) cos(ǫt) a rotation by ǫt. Clearly, q L 1 [0,ǫ −1 ] = 1 but q L p [0,ǫ −1 ] = ǫ 1−p −1 . Therefore, by taking q(t) = ǫ n , ǫ n → 0 on consecutive intervals, we can arrange for q to be in L p (R + ) but the solution has no limit. In the meantime, we will see that it makes sense to talk about the following problem: let λ 1 = 0, λ 2 = k, v 11 = v 22 = 0 and v 12 = q(t). Thus, and we will study the asymptotics for generic frequency k. As the discrete version of (4) one might consider the following recursion X n+1 (z) = ρ −1/2 n 1 γ n −γ n 1 · z 0 0 1 X n (z) X 0 (z) = I 2×2 , |z| = 1, ρ n = 1 + |γ n | 2 This is different from the recursion for polynomials orthogonal on the circle [10] only by sign "−" in front of γ n and by slightly different formula for ρ n . Remark 1. The simple calculation shows that X(t, k) = x 11 (t, k) x 12 (t, k) x 21 (t, k) x 22 (t, k) = x 11 (t, k) x 12 (t, k) −e ikt x 12 (t, k) e ikt x 11 (t, k) for real k.
Open problem. Is it true that for q(t) ∈ L 2 (R + ) and Lebesgue a.e. k ∈ R, the limit lim t→∞ X(t, k) exists? If so, how do the points of convergence correspond to frequencies k for which [M q](k) < ∞, where M q is the Carleson-Hunt maximal function?
The similar problem is known for the Krein systems [4] Z t = ikq(t) q(t) 0 Z, Z(0) = I, k ∈ R In this case, though, the solution Z is J-unitary, with We will consider (4) and will prove a simple result which has an analog in the theory of Krein systems (the so-called Szegő case) which gives a somewhat weaker type of convergence. That will be a warm-up for a later discussion of the general case. Consider the first column (x 11 (t, k), x 21 (t, k)) t of X(t, k). The functions x 11(21) (t, k) are holomorphic in k and have the following properties for any fixed t. Lemma 1.1. For any locally integrable q(t), we have (a) |x 11 (t, k)| 2 + |x 21 (t, k)| 2 = 1 − 2 Im k t 0 |x 21 (τ, k)| 2 dτ, k ∈ C (b) |x 11(21) Proof. This is a direct corollary of the differential equations.
Assuming, in addition, that q(t) is square summable, we get The function x 11 (∞, k) is nonzero function in the unit ball in H ∞ (C + ) and x 11 (k, t) → x 11 (∞, k) in the weak- * sense on R (d) The following estimates hold (e) We have Proof. If x 21 = e ikt y, then we have the following integral equations Therefore, Due to the estimate |x 11 (t, k)| ≤ 1 and Young's inequality, the L 1 norm of the integrand in τ 1 is smaller than q 2 2 (Im k) −1 . That implies the uniform convergence on compacts of C + and the bound from below for x 11 . The representation gives uniform convergence to zero for x 21 (t, k) and the estimate on its L 2 norm. The function x 11 (∞, k) is obviously in the unit ball of H ∞ (C + ) and is nonzero due to the estimate from below. Since x 11 (t, k)(k + i) −1 , x 11 (∞, k)(k + i) −1 are both uniformly bounded in H 2 (C + ) and as t → ∞ uniformly on the compacts in C + , we have Let us prove estimates in (d) now. Iterate (8) once and take the first term (the second allows stronger estimate) by Plancherel, whereq t (ω) is the Fourier transform of q(τ ) · χ [0,t] (τ ). Therefore, we have as Im k → +∞. The function ln |x 11 (t, k)| is subharmonic in C + so Since ln |x 11 (t, s)| ≤ 0, we can take k = iy, y → ∞ and compare the first terms in asymptotics of l.h.s. and r.h.s. to get (5). The estimates for x 11 (∞, k) are deduced similarly. Let us prove (e). Writing x 11 (∞, k) = 1 + h(k) yields 2 Re h ≤ −|h| 2 since |x 11 | ≤ 1 and Re h(iy) = y π Re h(t) t 2 + y 2 dt Taking y → +∞, we get (this we might call the "trace formula" for our evolution). Therefore, we have In a similar way, one can show that x 12 (t, k) has a limit in C + which we will denote by x 12 (∞, k). This, of course, implies that x 12 (t, k) → x 12 (∞, k) in the weak- * sense on the real line. In the next lemma, we will improve the convergence result.
Proof. Take arbitrary t 1 < t 2 and notice that we have the following semigroup property Hence, So, x 12 (t 1 , t 2 , k) 2 → 0 by (6). Thus, x 1j (t, k) is Cauchy in L 2 (R) and must have the limit which will coincide with weak- * limit x 1j (∞, k).

The transport equation on the circle
Another model very important for us is the transport equation. Consider with the simplest initial condition u(0, x) = 1. We can either say that x ∈ T or x ∈ R but all functions are 2π-periodic in x.
Notice that on the Fourier side, this equation iŝ where D is diagonal: (Dg) n = ng(n) in ℓ 2 (Z). We will work under the assumption There is no any loss of generality since we can always satisfy this condition by subtractingq(t, 0)I which corresponds to unimodular factor for u(t, x). Remark 3. Let P {0,1} be the Fourier projection onto the zeroth and the first modes, P c {0,1} = I − P {0,1} . Then, u Notice that if one drops the third term in the r.h.s., then the equation becomes equivalent to the model considered in the previous section.
Of course, the solution to (9) can be written explicitly. After periodization, we have If, e.g., q, q x ∈ C(R + , T), then this is the classical solution. The meaningful question is to study the asymptotics of u(t, x − kt) or, rather, On the Fourier side (with respect to x coordinate), φ(t, n, k) = t 0 e inτ kq (τ, n)dτ andφ (t, 0, k) = 0, t > 0 due to assumption (10). We have the following Lemma 2.1. If q(t, x) ∈ L 2 (R + , T) and T q(t, x)dx = 0, then by Plancherel. If φ(∞, x, k) is defined as function on T × R having Fourier coefficients ∞ 0 e inτ kq (τ, n)dτ then we easily have the first statement of the lemma. For the second part, it is sufficient to show that for a.e. k we have since then the convergence will follow from the Cauchy criterion. Let us introduce Here we used the standard Carleson estimate for the maximal function [6]. Since h m (t, k) is monotonic, we have h m (t, k) → 0 for a.e. k.
We get the following simple corollary Corollary 2.1. If q(t, x) ∈ L 2 (R + , T) and T q(t, x)dx = 0, then for a.e. k we have u(t, x − kt, k) → ν(x, k) in the following sense We know for a.e. k the limit φ(∞, x, k) exists as a function in H 1/2 (T). All other statements follow from Lemmas 6.1, 6.2 in Appendix.

Remark 4. Similarly, one can show
provided that q ∈ iR. It follows from lemma 6.2, estimates on the maximal function, and dominated convergence theorem. Moreover, for a.e. k, all Fourier coefficients of u(t, x − kt, k) converge as t → ∞ regardless of whether q is purely imaginary or not.
Remark 5. We considered the simplest case of initial data, i.e. u(0, x, k) = 1. The general case u(0, x, k) = f (x) is almost identical due to multiplicative structure of the problem. If the potential is square summable and purely imaginary, then we have the full measure set of k (that depends only on q) for which the equation is globally well-posed for f in the Krein algebra L ∞ (T) ∩ H 1/2 (T) [2]. We also have the stability and the asymptotics at infinity.
There is an instructive case q(t, x) = 2q(t) cos x with q(t) -purely imaginary square summable on R. In this situation, Notice that for a.e. k the function ν(x, k) is infinitely smooth. Moreover, expanding into the Taylor series, Notice that since the Bessel function J 0 (z) has positive zeroes ([1], Chapter 9), it is possible to choose q such that on arbitrary interval k ∈ I which means there is no hope to get Consider the case when the transport equation is given on the cylinder of large size 2πh u t = ku x + q(t, x)u, u(0, x) = 1, and u is h-periodic in x, q is purely imaginary. Scaling in x gives where ψ(t, θ, k) = u(t, hθ, k),q(t, θ) = q(t, hθ). For the new differential operator, ih −1 ∂ x , the gaps in the spectrum are of the size h −1 but, nevertheless, we have by scaling. The r.h.s. measures the L 2 norm in time of the space averages of q. If it is bounded, then H 1/2 norm of u(T, x, k), when averaged over (0, h), is bounded for most k. We expect this phenomenon for general situation when the gap condition deteriorates.
The calculations presented in this section can be easily carried out for the case when q is more regular, e.g. q ∈ L 2 (R + , H 1/2 (T)). That will lead to better regularity of the solution.

The model case of N × N system
In this section, we consider the following evolution Sometimes we will allow the eigenvalues to degenerate, that will require more careful analysis. Denote δ j = λ j+1 − λ j , j = 1, . . . , N − 1. We will also assume that V (t) is locally integrable on R + and that V jj (t) = 0 for all j. The last assumption can be made without loss of generalization. It is obvious that X(t, k) = {x mn (t, k), 1 ≤ m, n ≤ N } is unitary for real k. For general k, the following lemma holds true.
Proof. The proof is a trivial corollary from the differential equation itself and monotonicity of the square root. and Thus u(τ, k), V (τ )e 1 ∈ L 1 (R + ) by Cauchy-Schwarz and that proves convergence of u(t, k), e 1 to some π f (k) and and we have and ψ(t, k) → 0 uniformly on compacts in C + . That proves (a) through (b).
The properties of π f (k) stated in (c) follow from the mean-value inequality for subharmonic function ln |π f (k)| and (19). If f = e 1 , then (18) and (20) yield The proof of (d) repeats the arguments in lemma 1.2.
As a simple corollary of (c), we get existence of the weak- * limits for x 1j (t, k) on the real line (j = 2, . . . , N ). Denote them by x 1j (∞, k). The next lemma gives a stronger convergence result and is an analog of lemma 1.3 Proof. For any t 1 < t 2 the semigroup property yields The Cauchy-Schwarz and unitarity of X give is Cauchy in L 2 (R) and must have a limit equal to the the weak- * limit x 1j (∞, k) .
This result is somewhat surprising since we do not assume anything about P c 1 V P c 1 except local integrability. Now, we are going to prove results on convergence of all elements of the matrix X and need to assume more on V . Let V ∈ L 2 loc (R + ) and T is a fixed positive constant. We start with the following simple observation. Fix 1 < j < N and consider vector u(t) (it will be different for different j but we suppress this dependence for shorthand) which solves and satisfies the following boundary conditions. Let a(t) denote the vector containing the first j −1 components of u, b(t) is the j-th component of u, and c(t) contains the j + 1, . . . , N components of u. Then, we require that c(0) = 0, b(0) = 1, and a(T ) = 0. This solution does not have to exist, but for Im k large enough or small V it does, it is unique, and it allows two different representations. One of them is through X. Let X j = P 1≤k≤j XP 1≤k≤j , where P 1≤k≤j is the projection onto the first j coordinates. Then, assuming that u exists, By the Laplace theorem for determinants, we have where ∆ j = det X j . Provided that u l exists for any l = 1, . . . , j, iteration yields The b 1 (t, k) can be identified with x 11 (t, k). The existence of u(t, k) for large Im k and its analytical properties follow from the standard asymptotical method for systems of ODEs close to diagonal. We can where Q nl are the corresponding blocks of iV and Let U 1 and U 2 be solutions to the following Cauchy problems Since Q 11 and Q 33 are antisymmetric, we have the following obvious estimates The integral equations for the boundary conditions specified are Then, forũ, we have the operator equatioñ where f (t) = e j and D is the corresponding integral operator. If y = Im k >> 1, then D 2 is contraction in the ball of radius 1 in the space B, where and L is the space with the norm · ∞ + · 2 . In fact, by (25), (26), Cauchy-Schwarz and Young inequalities, (1)). This is a well-know result in the asymptotical theory of ODE but it is valid for either large positive Im k or fixed Im k > 0 and small V 2 . It does not require any information on Q 11 and Q 33 . Notice that b(T, k) = exp(iλ j kT )(1 + O((Im k) −1 )). Then, (24) yields invertibility of each X j for large Im k. Also, since ∆ j is entire in k, the formula b(T, k) = ∆ j ∆ −1 j−1 allows to define b for any k as a meromorphic function.
For k = iy, y >> 1 we have the following asymptotical expansioñ and . Substituting the Duhamel expansions for Ψ 1(2) into the formula above, one gets The similar calculation can be done in the general case when Im k → +∞.
We are ready to prove the following Theorem 3.1. If Proof. Consider g(t, k) for any t > 0. It is entire in k and |g(t, k)| ≤ 1 for real k since X j is a contraction for Im k ≥ 0. Moreover, we know its asymptotics for large Im k which implies that g(t, k) is in the unit ball in H ∞ (C + ) and Arguing like in the proof of lemma 1.2, we write g(t, k) = 1 + h(t, k). Then Re h(t, k) ≤ 0 and Write X(t, k) in the block form Now, that all necessary uniform bounds are obtained, we can prove the convergence result. For any t 1 < t 2 , the semigroup property in the block form yields the identity is Cauchy in H 2 (C + ) and we have convergence g(t, k) → g(∞, k) uniformly over the compacts in C + . Since each g(t, k) is analytic contraction, the limit g(∞, k) as an analytic contraction as well. The bounds (29) can be obtained through the argument identical to the one used to handle g(t, k).
Remark 6. Notice that the theorem was proved under the assumption that all eigenvalues of Λ are non-degenerate. That was used in the proof of the asymptotics for b. In the meantime, due to cancelation in (28), the statement of theorem 3.1 as well as (31) holds under the assumption that λ j < λ j+1 and the other eigenvalues can degenerate.
The estimates for determinants and (31) can be obtained for W j−1 as well and that yields the following important result.
Proof. The estimate (31), applied to X j and W j−1 , yields Expanding in the last raw, we have where {A lm } are cofactors of X j .
Lemma 3.4. If Z is j × j contraction then the adjoint C = adj Z is contraction as well. In particular, Proof. Take any α = (α 1 , . . . α j ) with α 2 = 1. Replace the first raw of Z by α and denote the resulting matrix by Z α . By Laplace theorem, On the other hand, Hadamard's estimate gives where h l is the ℓ 2 -length of the l-th raw of Z α . Since α is arbitrary, we get (33). This implies Ce 1 ≤ 1. Take any unitary U . We have U CU −1 = adj (U ZU −1 ). Therefore, CU −1 e 1 ≤ 1 Since U is arbitrary, Cx ≤ x for any x.

By lemma, we have
and Using the semigroup property one can show that x jj (t, k) exp(−iktλ j )−1 is Cauchy in L 2 (R) which implies existence of the limit.
Proof. The proof is a standard application of the semigroup property and the previous results.
We are going to consider now a somewhat special case when the frequencies degenerate in different ways. The first situation is a model for Schrödinger evolution on 1-d torus.
Assume that λ j−1 < λ j = λ j+1 < λ j+2 for some j : 1 < j < N . We will try to understand how the P {j,j+1} X(t, k)P {j,j+1} part of X(t, k) behaves for large t. Consider the following evolutions Obviously, W is k-independent and is unitary since V j,j+1 =V j+1,j . For Ψ, we have Ψ = exp(ikλ j t)W . Notice that for real or purely imaginary V j,j+1 the matrix that diagonalizes the perturbation is t-independent and we can assume that V j,j+1 (t) = 0 without loss of generality. We will consider the general case. The proof of the following result repeats the argument given above with minor changes which we will explain. Theorem 3.3. Assume that λ j−1 < λ j = λ j+1 < λ j+2 and Proof. We repeat the proofs of theorems 3.1 and 3.2 with the following modifications. Instead of a single vector u satisfying b(0, k) = 1, we consider its N × 2 matrix version. Let us denote the matrix containing first j − 1 rows of u by a, b is formed by j, j + 1 rows and is therefore 2 × 2 matrix, and c is built of j + 2, . . . , N rows of u. Then, the boundary conditions would be The analogs of (22) and (23) are Multiplying from the left with the adjoint of X j+1 (T, k) and taking the 2 × 2 blocks in the "southeastern corner", we have ∆ j+1 (T, k) · I 2×2 = A(T, k) · b(T, k), By Remark 6, we already know that Combining these estimates and using the Laplace theorem for determinants, we have The analog of (27) is The similar perturbation argument gives b(T, iy) = Ψ(0, T, iy) Since we know asymptotical expansion for ∆ j+1 as Im k → +∞, (35) yields Notice that 0 ≤ Γ + I j+1 (V ) · I 2×2 I ′′ (V ) By lemma 3.4, A(T, k) is contraction for Im k ≥ 0 and so is C(T, k) for real k. C(T, k) is also entire in k and we know its asymptotics for large Im k which implies that C(T, k) is contraction for k ∈ C + . Write C(T, k) = I 2×2 + H(T, k). Then, 2 Re H(T, k) + |H(T, k)| 2 ≤ 0 (40) Since (Re H(T, k)ξ, ξ) is harmonic for any ξ ∈ C 2 , comparison of asymptotics for y → ∞ gives Next, notice that and R r(T, k) 2 dk I ′′ (V ) due to (36). On the other hand, we know the asymptotics of ∆ j−1 (T, k) which together with (39) and (41) give Denote the matrix under the norm in (42) by µ. We have Y µ ≤ µ since Y is a contraction and therefore By (37), we have (34). The rest is standard.
The method can be carried over to the case when the multiplicity of frequencies are higher than 2. Using these results we can obtain the "asymptotics" of solution for any λ 1 ≤ . . . ≤ λ N provided that V ∈ L 2 (R + ). However, the constants in our estimates will blow up when some δ j = λ j+1 − λ j ∼ 0.
Assume we are in the situation when λ 1 = λ 2 = . . . = λ m < λ m+1 . The simple matrix version of lemma 3.2 gives for large m assuming only λ 1 < λ 2 < . . . and some off-diagonal decay for V ? The conjecture might be that for suitable L 2 condition on V . That could lead to better understanding of Schrödinger evolution on the circle.
The calculations below will be extensively used in later sections to handle special evolutions equations. We will empasize the dependence of I ′ (V ) on j by writing where q j (t) = q −j (t) and Assume also that |λ l − λ m | |l − m|. Then for any j we have (remember that q 0 (t) = 0) (b) Take the same V but assume that λ j ∼ j m . Then, In the second case, the condition on V can be relaxed. If λ 2j = λ 2j+1 ∼ j m , the estimate for I ′′ j (V ) is similar.

The case of Hilbert spaces and applications to Schrödinger evolution on the circle
In this section, some results from the previous section are generalized to the case N = ∞. We will also give various applications to the Schrödinger evolution on the circle.
Consider the selfadjoint operator Λ on H = ℓ 2 (Z) with discrete spectrum {λ n } where λ n is nondecreasing sequence. Let Q(t) be operator-valued function with norm Q(t) bounded for a.e. t and The weak solution to is the solution to which follows from the Duhamel formula. Here X 0 (τ, t, k) denotes solution to the unperturbed evolution. We will write u(t, k) = X(0, t, k)ψ.
There are some general results that prove the weak solution is in fact a "strong solution" provided that the initial value and potential Q are "regular enough". We will study the behavior of weak solution. The condition (45) is sufficient for the iterations of (46) to converge in the space L ∞ ([0, T ], H) for any T > 0 with the obvious estimate The following stability result will allow us to use the standard approximation technique. Let Π n = P {−n,...,n} , a projection in ℓ 2 (Z). Proof. Each term in the corresponding series is a multilinear operator and use linearity for the second term, etc. Then, can be written as a sum of m terms and each of them converges to zero. Indeed, It is now easy to prove , then X(τ, t, k) is unitary and it satisfies the semigroup property Proof. The semigroup property and preservation of the norm follow from the Approximation lemma and the corresponding results for finite systems of ODE's. We also have X(0, t, k) · X(t, 0, k) = X(t, 0, k) · X(0, t, k) = I which implies that X is unitary.
Next, we prove an analog of theorem 3.2. Denote the matrix elements of V (t) and X(t, k) by V mn (t) and x mn (t, k), respectively. For simplicity, we again make an assumption that V nn (t) = 0 for any n. Theorem 4.1. Assume that V ∈ L 1 loc (R + ), λ −1 < λ 0 = 0 < λ 1 , and Proof. For truncated potentials V (n) = Π n V Π n , the theorem 3.2 is applicable and the resulting estimates are uniform in n. Since for each fixed k we have convergence we can go to the limit as n → ∞ to get R |x 00 (t, k) − 1| 2 dk I ′ (V ), The rest is the standard application of semigroup property and unitarity of X.
Most results from the previous section can be adjusted similarly including the case when the frequencies have multiplicity. In particular, we can consider Schrödinger evolution on, say, one-dimensional circle u t = −iku θθ + iV (t, θ)u, u(0, k) = ψ(θ) ∈ L 2 (T).
We are interested in the weak solution and assume that V is real-valued and V (t, θ) ∞ ∈ L 1 loc (R + ). In this case, on the Fourier side, equation takes form where Λ is diagonal λ 0 = 0, λ n = n 2 and all eigenvalues but the principal one (i.e., We also always assume without loss of generality that T V (t, θ)dθ = 0, t > 0 Consider the following k-independent evolution Notice that ifV (2n, t) = 0 for all t, then Ψ n (t) are trivial. Also, ifV (2n, t) is real or purely imaginary, we can write the explicit formula for Ψ n (t). That can be satisfied, e.g., if V is even or odd. Take where X 0 (0, t, k) is the free Schrödinger evolution.
and V (t, θ) ∈ L 2 (R + × T). Then for any ψ ∈ L 2 (T) the weak solution u(t, k) satisfies for any I ⊂ R, |I| < ∞. The operator H is defined as bounded operator from L 2 (T) to the space of functions h(θ, k) satisfying h(θ, k) 2 L 2 (T) ∈ L 1 loc (R).
Proof. Denote the matrix elements ofX(t, k) = W −1 (0, t, k)X(t, k) byx mn (t, k). Fix n and take, say, ψ(θ) = e inθ . Letû be the corresponding solution, i.e. the α(n)-th column ofX. The choice ψ(θ) = e −inθ gives the α + 1-th column. From the Approximation lemma, theorem 4.1 (adapted by the theorem 3.3), and (44) we know Moreover, (31) gives the following uniform estimates which implies for any γ < 1. Therefore, By linearity, we can prove existence of the limit for any trigonometric polynomial ψ = T (θ). Denote the corresponding limit by [HT ](θ, k). That gives a linear operator H defined on the set dense in L 2 (T). We have for any k and t and so for any Therefore, H can be extended to a bounded operator on L 2 (T) such that If ψ is fixed, the last identity implies that Hψ = ψ for a.e. k.
Notice that the condition (47) was used only to guarantee the global existence of the weak solution for any k and can probably be dropped. The solution corresponding to the initial value ψ = 1 is special in a way that we always have R ln |x 11 (t, k)|dk − V 2 2 and that means |x 11 (t, k)| > 0 for a.e. k (compare with (13)).
The analogous bound can be proved for any sufficiently smooth function ψ. Assuming that V is only bounded on the strip R + × T this estimate shows that k-averaged H γ norm is finite and grows not faster than √ t.
Remark 7. Now, assume that Due to (31), we have |M | < Cσ −1 T and by taking σ large we have that at least a half of the first T columns are strongly localized for many k. In this argument, M can depend on T , in principle.
In the case of transport equation, our method allows us to reproduce that the solutions are in H 1/2 . Indeed, we have where x mn (t, k) are the matrix elements of the evolution operator in the Fourier representation. In the meantime, we have x mn (t, k) = exp(ikmt)x n−m,0 (t, k) which yields . Evolution with deteriorating gap condition: the short-range interactions.
This section contains the main results of the paper. Unfortunately, they handle only the short-range potentials and even in this case are far from optimal.
Consider, e.g., the following model V is real and u(1, θ) = 1 Similar evolution equation appears as the WKB correction in the three-dimensional Schrödinger dynamics [5]. We assume that V is real-valued and satisfies where 0 ≤ γ ≤ 1 is to be specified later.
On the Fourier side, the equation can be written aŝ and Λ is diagonal with elements n 2 , n ≥ 0. The multiplicity of each eigenvalue is two as long as n > 0, the principal eigenvalue is non-degenerate. Clearly, this case can not be handled by the methods considered in the previous section since the distance between eigenvalues decays like t −2 which might lead to significant growth of the Sobolev norms even for "typical" k. Instead, as results of the previous section suggest, we should introduce the scaled Sobolev norms This conjecture is supported, e.g., by calculations (14) made for transport equation or by the Remark 7. If true, it implies |n|>Ct |û(t, n)| 2 → 0 for any C and since u 2 = 1, we have |n|<Ct |û(t, n)| 2 → 1 so the most of L 2 (T) norm is concentrated on, roughly, t first harmonics. We will call this phenomenon the concentration of L 2 norm. It does not seem to be possible to obtain any asymptotical result similar to the case when the gap condition does not deteriorate and, perhaps, the "scattering" for this model should be defined in terms of the boundedness of scaled Sobolev norms.
The simple substitution τ = t −1 reduces the problem to equation Thus, (50) can be reduced to studying the standard problem on the circle where the potential grows in the controlled way. We will study the growth of the standard Sobolev norm. Assume for a second that we could prove (which we can not! but compare to (43)) Then, for the original problem that would mean and so Notice also that by the standard time-scaling it would be sufficient to prove (52) only for t = 1.
We will start with rather simple apriori estimates. Consider the simplified version of (50) We start with well-known estimate Lemma 5.1. Assume that V (t, θ) is real trigonometric polynomial of degree smaller than T α for any t ∈ [0, T ] and |V (t, θ)| T −γ . Then, for any k ∈ R Proof. The proof is elementary. Differentiating (54) in angle, multiplying byū θ and integrating, we get Clearly, we have concentration of L 2 norm for all k as long as α < γ. This argument holds for transport equation as well and can be easily modified to control the higher Sobolev norms. On the other hand, for the transport equation, the L 2 norm can really smear over first T 1−γ harmonics as can be easily seen from van der Corput lemma applied to (12).
In the case just considered, the potential had an extra smoothness in θ. The other extreme case is when V oscillates.
Clearly, by taking α + 2γ > 5/2, we have localization of almost all of the L 2norm on the first harmonic for most k but this argument does not say much about the Sobolev norms.
We also can improve this result to get real analyticity for a.e. k. Proof. We will work on the Fourier side. Summing (58) Multiply (58) by N − 3 and sum from N = 4 to ∞. (60) gives Taking, say, N ∼ 4l, we have sup t≥0 |x N (t, k)| 2 ≤ Cµ(k) σ α 1 l l which shows that the solution is real analytic for a.e. k.
In theorem 5.1, the integration is restricted to an interval (a, b) which must be finite, not containing 0. Below we show that this condition can be dropped. Proof. Notice that the function t α v(t) ∈ L ν (R + ) for some ν(γ) < 2 and therefore M (k) ∈ L ζ (R) with ζ dual to ν. Multiply (57) by N and sum from N = 2 to infinity. We have where ǫ > 0. By Young's inequality, we have Taking ǫ = (2 − ν)/ν, we get The Gronwall lemma yields which implies (61).
The similar argument can handle the higher Sobolev norms.
The next theorem studies the L p (R, dk) norms of Theorem 5.3. Assume that conditions of the theorem 5.1 hold. Then, for any 2 ≤ p ≤ ∞, N > 1, we have Sum these inequalities in m from N/2 to N . We get N S N (T, k) The argument similar to the one employed in the proof of lemma 5.1 gives Interpolation between (63) and (64) gives the statement of the theorem.

Appendix
In this section, we collect rather standard results that we used in the main text. The following lemma is well-known