Time-dependent delta-interactions for 1D Schrödinger Hamiltonians

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Time-dependent delta-interactions for 1D Schrödinger Hamiltonians

Taoufik Hmidi, Andrea Mantile, Francis Nier

To cite this version:

Taoufik Hmidi, Andrea Mantile, Francis Nier. Time-dependent delta-interactions for 1D Schrödinger

Hamiltonians. 2009. �hal-00371803�

Time-dependent delta-interactions for 1D Schr¨ odinger Hamiltonians.

T. Hmidi

, A. Mantile

, F. Nier

Abstract

The non autonomous Cauchy problemi∂_tu=−∂_xx² u+α(t)δ0uwithut=0=u0is considered inL²(R) . The regularity assumptions forαare accurately analyzed and show that the general results for non autonomous linear evolution equations in Banach spaces are far from being optimal. In the mean time, this article shows an unexpected application of paraproduct techniques, initiated by J.M. Bony for nonlinear partial differential equations, to a classical linear problem.

MSC (2000): 37B55, 35B65, 35B30, 35Q45,

keywords: Point interactions, solvable models in Quantum Mechanics, non autonomous Cauchy problems.

1 Introduction

This work is concerned with the dynamics generated by the particular class of non-autonomous quantum Hamiltonians: Hα(t)=−_dx^d22+α(t)δ, defining the time dependent delta shaped perurba- tions of the 1D Laplacian. Quantum hamiltonians with point interactions were first introduced by physicists as a computational tool to study the scattering of quantum particles with small range forces. Since then, the subject has been widely developed both in the theoretical framework as well as in the applications (we refer to [2] for an extensive presentation). For real values of the coupling parameterα, the rigorous definition of: Hα=−dx^d22+αδarises from the Krein’s theory of selfadjoint extentions. In particular,Hαidentifies with the selfadjoint extension of the symmetric operator: H0=−_dx^d22,D(H0) =C₀^∞(R\ {0}) defined through the boundary conditions

ψ^′(0⁺)−ψ^′(0⁻) =αψ(0)

ψ(0⁺)−ψ(0⁻) = 0 (1.1)

ψ(0^±) denoting the right and left limit values ofψ(x) asx→0 [2]. Explicitely, one has D(Hα) =

ψ∈H²(R\ {0})∩H¹(R)ψ^′(0⁺)−ψ^′(0⁻) =αψ(0) (1.2) Hαψ=− d²

dx²ψ in R\ {0}. (1.3)

Whenα(t) is assigned as a real valued function of time, the domainD(Hα(t)) changes in time with the boundary condition (1.1), while the form domain is given by H¹(R). The quantum evolution

∗IRMAR, UMR-CNRS 6625, Universit´e Rennes 1, Campus de Beaulieu, 35042 Rennes Cedex, France, andrea.mantile@univ-rennes1.fr

†IRMAR, UMR-CNRS 6625, Universit´e Rennes 1, Campus de Beaulieu, 35042 Rennes Cedex, France, andrea.mantile@univ-rennes1.fr

‡IRMAR, UMR-CNRS 6625, Universit´e Rennes 1, Campus de Beaulieu, 35042 Rennes Cedex, France, Francis.Nier@univ-rennes1.fr

associated to the family of operators

Hα(t) is defined by the solutions to the equation i_dt^du=Hα(t)u

u_|t=0=u0. (1.4)

The mild solutions are the solutions to the associated integral equation u(t) =e^it∆u0−i

Z t 0

e^i(t−s)∆q(s)δ0 ds (1.5) withq(s) =α(s)u(s,0) . The questions are about:

• the regularity assumptions ont→α(t) for which (1.4) defines a unitary strongly continuous dynamical system onL²(R)

• the meaning of the differential equation (1.4), according to the regularity oft→α(t) . General conditions for the solution of this class of problems have been long time investigated.

In the framework of evolution equations in Banach spaces, Kato was the first who obtained a result

for the Cauchy problem _d

dtu=A(t)u

ut=0=u0 , (1.6)

when t →A(t) is an unbounded operator valued function [6]. This result, which applies to the quantum dynamical case for A(t) = −iH(t), requires the strong differentiability of the function t → A(t) and the time independence of the domain D(A(t)). Afterwards, a huge literature was devoted to this problem in the main attempt of relaxing the above conditions (e.g. in [11] and [9]; a rather large bibliography and an extensive presentation of the subject are also given in [5]).

In particular, the time dependent domain case was explicitely treated in [8] by Kisy´nski using coercivity assumptions and C_loc² -regularity of t → A(t). The regularity conditions in time were substantially relaxed into a later work of Kato [7] who proves weak and strong existence results, for the solutions to (1.6), when{A(t)}forms a stable (a notion defined in [7]-Definition 2.1 expressing uniform bounds for the norms of resolvents) family of generators of contraction semigroups leaving invariant a dense set Y of the Banach space X, and the map t → A(t) is norm-continuous in L(Y, X) .

Due to the particular structure, point interaction Hamiltonians allows rather explicitely energy and resolvent estimates, so that most of the techniques employed in the analysis of non-autonomous Hamiltonians can be used to deal with the equation (1.4), provided thatα(t) is regular enough. At this concern, the Yafaev’s works [12], [13] and [10] (with M. Sayapova) on the scattering problems for time dependent delta interactions in the 3D setting are to be recalled: There the condition α∈ C_loc² (t0,+∞) is used to ensure the existence of a strongly differentiable time propagator for the quantum evolution. Such a condition, however, could be considerably relaxed. In our case, for instance, a first appraoch consist in adapting the strategy of [8] by constructing a family of unitary maps Vt,t0 such that: Vt,t0Hα(t)V_t,t^∗₀ has a constant domain; then, it is possible to solve the evolution problem for the deformed operator by using results from [7]. To fix the idea, let yt

be the time dependent vector field defined by

y˙t=g(yt, t)

yt0 =x (1.7)

with g(·, t) ∈ C⁰(0, T;C₀^∞(R)), supp g(·, t) ⊂ (−1,1), and g(0, t) = 0 for each t. Under these assumptions, (1.7) allows an unique solution depending continuously from time and Cauchy data {t0, x}. Using the notation: yt=F(t, t0, x), one has

∂xF(t, t0, x) =e

Rt

t0∂1g(ys,s)ds

>0 ∀x∈R, (1.8)

∂1g(·, s) denoting the derivative w.r.t. the first variable. This condition allows to consider the map ofx→F(t, t0, x) as a time-dependent local dilation and one can construct the family of time dependent unitary transformation associated to it

( (Vt0,tu) (x) = (∂xF(t, t0, x))¹²u(F(t, t0, x)) Vt,t0 =V_t⁻_0,t¹

Under the action ofVt,t0, the equation (1.4) reads as

( i_dt^dv=Vt,t0Hα(t)Vt0,tv−i ¹₂[∂yg](y, t) +g(y, t)∂y v vt=0=u0

(1.9) with

Vt,t0Hα(t)Vt0,t=−∂yb²∂y+a²−(∂yab) +b0α(t)δ b(y, t) =e

Rt

t0∂1g(F(s,t,y),s)ds

; b0=b(0, t) a(y, t) = 1

2 Z t

t0

∂₁²g(F(s, t, y), s)e

Rt

t0∂1g(^{F(s′,t,y),s′})^ds′ds and Vt,t0u=v. Set: A(t) = iVt,t0Hα(t)Vt0,t+ ¹₂∂yg(y, t) +g(y, t)∂y

, the domain D(A(t)) is the subspace ofH²(R\ {0})∩H¹(R) identified by the boundary condition

b(0, t)

u^′(0⁺)−u^′(0⁻)

=α(t)u(0)

For α ∈ W^1,1(0, T) and sign(α(t)) = const., one can determine (infinitely many) g(y, t) such that: _k¹b(0, t) =α(t) for a fixed constantk. With this choice, the operator’s domain is constant, D(A(t)) =Y. Moreover, under the same conditions onα, one can shows thatA(t) defines a stable family of skew-adjoint operators t-continuous in L(H¹, H⁻¹)-operator norm. Thus, one can use the Theorem 5.2 and Remark 5.3 in [7] to get strongly solutions for the related evolution problem.

This is summarized in the following Proposition.

Proposition 1.1 Let α ∈W^1,1(0, T), sign(α(t)) =const. andu0 ∈ D(Hα(0)). There exists an unique solution utof the problem (1.4), with: ut∈D(Hα(t))for eacht andu^′t∈C⁰(0, T;L²(R)).

In spite of those improved results with already known general tools, our aim is to prove that they are far from being optimal. An additional structure can lead to the same conclusions with weaker regularity assumptions. This is the question that we propose to explore with one dimensional δ-interactions which allow direct and explicit computations. The main result of this paper is the following.

Theorem 1.2 1) Assume α ∈ H_loc¹⁴ (R) , then for any u0 ∈ H¹(R) the integral equation (1.5) admits a unique solution u∈C(R;H¹(R)) withi∂tu−α(t)u(t,0)δ0∈C(R;H⁻¹(R))and (1.4) is weakly well-posed.

2)With the same assumption, (1.5) defines a unitary strongly continuous dynamical systemU(t, s) on L²(R).

3) If additionally α∈H_loc^3/4(R), then for anyu0∈D(Hα(0)) the solution uof (1.5) belongs to the spaceC¹(R;L²(R)), withu(t)∈D(Hα(t)) for everyt∈R.

Remark. The problem of defining the quantum evolution for 1D time dependent delta interactions has also been considered in a nonlinear setting [1] whereαis assigned as a function of the particle’s state, in our notation: α(t) = γ|ut(0)|^2σ, with γ ∈ R, σ ∈R₊. In this framework, the authors prove that solutions to the nonlinear evolution problem exist locally in time foru0∈H^ρwithρ >¹₂

and: U(t, s)u0(0) ∈ H

ρ 2+¹₄

loc as a function of time. This corresponds to the condition: α ∈ H_loc^ν with ν > ¹₂. Although a linear problem is considered here, the result of Theorem 1.2 improves significantly the regularity condition: α∈H_loc^1/4(R) is weaker than a continuity assumption.

In what follows,Dxdenotes ¹_i∂x=F⁻¹◦(ξ×)◦F, whereF is the Fourier transform in position (or in time) normalized according to

Fϕ(ξ) = Z

R

e^−iξxϕ(x)dx .

The Sobolev spaces are denoted by H^s(R), s∈ R, their local version by H_loc^s (R). The notation u∈H^s+0(R) means that there exists ε >0 such thatu∈H^s+ε(R) (inductive limit) and its local version u∈ H_loc^s+0(R) allows εR >0 to depend on R >0 while considering the interval [−R, R].

More generally the Besov spaces are defined through dyadic decomposition: For (p, r)∈[1,+∞]² ands∈R,the spaceB^s_p,r is the set of tempered distributionsusuch that

kuk^Bsp,r:=

2^qsk∆quk^Lp

ℓ^r <+∞,

where ∆q =ϕ(2^−qD),q∈N, is a cut-off in the Fourier variable supported inC⁻¹2^q≤ |ξ| ≤C2^q. Details are given in Appendix A. Finally, the notation ’.’, appearing in many of the following proofs, denotes the inequality: ’≤C’, beingCa suitable positive constant.

2 Proof of Theorem 1.2

This theorem is a consequence of simple remarks, explicit calculations and standard applications of paraproduct estimates in 1D Sobolev spaces. Let us start with some elementary rewriting of the Cauchy problem (1.4).

2.1 Preliminary remarks

• First of all equation (1.4) or its integral version (1.5) are local problems in time so thatt0= 0, t∈[−T, T] for some T >0 and even suppα⊂[−T /2, T /2] can be assumed after replacing αwithαT(s) =α(s)χ(_T^s) for some fixedχ ∈C₀^∞((−1/2,1/2)) and χ≡1 nears= 0. The dependence ofH^s-norms ofαT with respect toT will be discussed when necessary.

• The equation (1.4) or its integral version (1.5) makes sense in S^′(R_x) as soon as u(t,0) is well defined for almost allt∈[−T, T] and q(t) =αT(t)u(t,0) is locally integrable. Then it can be written after applying the Fourier transform as a local problem inξ∈R

i∂tu(t, ξ) =b |ξ|²u(t, ξ) +b q(t) b

u(0, ξ) =ub0(ξ) (2.1)

with q(t) =αT(t)u(t,0) =αT(t) Z

Ru(t, ξ)b dξ . (2.2) This is equivalent to the integral form

u(t, ξ) =b e^−it|ξ|2ub0(ξ)−i Z t

0

e^{−i(t−s)|ξ|2}q(s)ds (2.3) with q(t) =q0(t)−iαT(t)

Z t 0

Z

R

e^{−i(t−s)|ξ|2}q(s)dsdξ (2.4) by setting q0(t) = αT(t)[e^it∆u0](0) . The assumption u0 ∈ H^s(R), s > 1/2, (resp. u0 ∈ L¹(R)) ensures that [e^it∆u0](0) ∈ C⁰([−T, T]) (resp. t^1/2[e^it∆u0](0) ∈ C⁰([−T, T])). Such an assumption as well as looking foru(t)∈H¹(R) ensures that the quantitiesq0(t) andq(t) make sense for almost allt∈[−T, T].

• With the support assumption suppαT ⊂[−T /2, T /2], the convolution equation (2.3) can be written

q(t) =q0(t)−iαT(t) Z t

0

Z

R

1[−T,T](t−s)e^{−i(t−s)|ξ|2}1[−T,T](s)q(s)dsdξ in D^′(R) :=q0(t)−iαT(t)Lq(t) :=q0(t) +Lαq(t). (2.5)

• Onceqis known after solving (2.5), equation (2.3) witht∈Rreads simply b

u(t, ξ) =e^−it|ξ|2bu0(ξ)−ie^−it|ξ|2F q1[0,t]

(−|ξ|²). (2.6)

• Whenu0 andqare regular enough the time-derivative of the quantity (2.3) gives i∂t(∂tu)(t, ξ) =b |ξ|²∂tbu(t, ξ) +q^′(t).

By Duhamel formula, this implies

∂tbu(t, ξ) =e^−it|ξ|2∂tu(0, ξ)b −i Z t

0

e^{−i(t−s)|ξ|2}q^′(s)ds , while (2.1) says fort= 0

∂tu(0, ξ) =b −i|ξ|²ub0(ξ)−iq(0). Therefore we obtain fort∈R

i∂tu(t, ξ) =b e^−it|ξ|2

|ξ|²bu0(ξ) +q(0)

+e^−it|ξ|2F

(∂sq)1[0,t]

(−|ξ|²). (2.7)

2.2 Reduced scalar equation for q

Let us now study the equation (2.5) written:

q=q0+L^αq with

Lαq:=−iαT(t)Lq=−iαT(t) Z t

0

Z

R

1[−T,T](t−s)e^{−i(t−s)|ξ|2}1[−T,T](s)q(s)dsdξ . Solving this fixed point equation relies on the next result.

Proposition 2.1 The estimate

kLqkH^s .T¹²k1[−1,1](Dt)qkL²+T^12−θ

kq1[0,T]kH^s−θ+kq1[−T,0]kH^s−θ

holds for every s∈Randθ∈[0,¹₂]. Proof: Owing toR

Re^±iλ|ξ|2 _2π^dξ = ^{√e±i π}_4πλ⁴ forλ >0,Lqwrites as

√1πLq(t) =e^−iπ4 Z

R

1[0,T](s)q(s)1[0,T](t−s)

(t−s)¹² ds+e^iπ4 Z

R

1[−T,0](s)q(s)1[−T,0](t−s) (s−t)¹² ds Passing to the Fourier transform, we get

√1

πLcq(τ) =e^−iπ4 Z T

0

e^−itτ

√t dt

!

F(1[0,T]q)(τ) +e^iπ4 Z T

0

e^itτ

√t dt

!

F(1[−T,0]q)(τ)

One easily checks Z T

0

e^±itτ

√t dt ≤2√

T and

Z T 0

e^±itτ

√t dt

.|τ|⁻¹². This yields for everyθ∈[0,¹₂], τ ∈R

Z T 0

e^±itτ

√t dt

.T¹²1[−1,1](τ) +T^12−θ|τ|^−θ1_{^R_\[−1,1]}(τ).

Thus we get for every s∈R, θ∈[0,¹₂],

kLqkH^s .T¹²k1[−1,1](Dt)qkL²+T^12−θ

k1[0,T]qkH^s−θ+k1[−T,0]qkH^s−θ

.

Proposition 2.2 1. Letu0 ∈H^s(R_x) withs >1/2 and let α∈H_loc¹⁴ (R_t). Then the equation (2.5) has a unique solutionq∈H_loc¹⁴ (R_t). Moreover, for a fixed u0 ∈H^s(R_x)with s >1/2, the map α7→q is locally Lipschitzian from H¹⁴(R_t) toH¹⁴(R_t).

2. Let u0 ∈ H¹(R_x) and let α ∈ H_loc³⁴ (R_t). Then the equation (2.5) has a unique solution q∈H_loc³⁴ (R_t). Moreover, for a fixedu0∈H¹(R_x)the mapα7→qis locally Lipschitzian from H³⁴(R_t)toH³⁴(R_t).

3. Let u0 ∈H^1+2ε(R_x) and letα∈H_loc^34+ε(R_t), for some ε >0. Then the equation (2.5) has a unique solution q∈H_loc^34+ε(R_t).

Proof: 1)Let us first prove that q0∈H¹⁴ whenu0∈H^12+ε. Write first (e^it∆u0)(0) =

Z 1

−1

e^−it|ξ|2cu0(ξ) dξ 2π+

Z +∞ 1

e^−itτ c u0(√

τ) +cu0(−√

τ) dτ 2π√

τ

=I(t) +II(t).

The first term I(t) defines aC^∞function with

kIk^Bs_∞,∞.ku0kL²,

for everys∈R. On the other hand, the Fourier transform of II equals cII(−τ) = 1[1,∞[(τ) uc0(√

τ) +cu0(−√ τ) 1

√τ . The Sobolev regularity of the second term, II, is given by:

kIIk²H^ν . Z +∞

1

τ^2ν cu0(√

√ττ)

2

dτ

.ku0k²H^2ν−1/2 , (2.8)

for any ν∈R. Now, write

q0(t) =αT(t)I(t) +αT(t)II(t).

Lemma A.2-b) applied to the first term, implies

kαTIk_H¹₄ .kαTk_H¹₄kIk_B∞,∞^s+ε .kαTk_H¹₄ku0kL².

For the second term, use Lemma A.2-a), Sobolev embeddings and (2.8) kαTIIk_H¹₄ .kαTk_H¹₄kIIk

B

1 2 2,∞∩L^∞

.kαTk_H¹₄kIIk_H¹₂+ε 2

.kαTk_H¹₄ku0k_H¹₂^+ε. By combining these estimates, we get

kq0k_H¹₄ .kαTk_H¹₄ku0k_H¹₂^+ε.

It remains to estimate αT. Letχe∈ D(R) withχe≡1 in [−1,1] and setα(t) =e χ(t)α(t) . By usinge again Lemma A.2-a), we get for 0≤T ≤1

kαTk_H¹₄ .kαek_H¹₄χ(T⁻¹·)

B

1 2 2,∞∩L^∞. A change of variable in the Fourier transformF

χ(T⁻¹.)

(τ) =Tχ(T τ) leads tob χ(T⁻¹.)

H^µ ≤T^12−µkχkH^µ and χ(T⁻¹.)

B_2,∞^µ .T^12−µkχkB_2,∞^µ , (2.9) forµ≥0 andT ≤1. Hence we get

kαTk_H¹₄ .keαk_H¹₄. and

kq0k_H¹₄ .kαek_H¹₄ku0k_H¹₂+ε (2.10) In order to estimate the the operatorL, use Lemma A.2-a)

kL^αqk_H¹₄ .kαTk_H¹₄ kLqk

B

1 2 2,∞∩L^∞

.kαek_H¹₄ kLqk_H¹₂^+ε , while Proposition 2.1 says

kLqk_H¹₂^+ε.T¹²kqkL²+T^14−ε

k1[0,T]qk_H¹₄ +k1[−T,0]qk_H¹₄ . Hence we get for 0≤T ≤1 and by Lemma A.2-a)

kLqk_H¹₂^+ε.T^14−εkqk_H¹₄ This yields

kL^αqk_H¹₄ .kαek_H¹₄T^14−εkqk_H¹₄ . (2.11) This proves thatLis a contracting map in H¹⁴ for sufficiently small timeT. The time T depends only on keαk_H¹₄ and then we can construct globally a unique solutionq ∈ H_loc¹⁴ (R) for the linear problem (2.5).

It remains to prove the continuity dependence ofqwith respect toα. Letα,α¯∈H_loc¹⁴ andq,q¯the corresponding solutions then we have

q(t)−q(t) =¯ αT(t)Lq(t)−α¯T(t)Lq(t),¯ with α¯T(t) = ¯α(t)χ(t/T).

SinceL is linear onqthen

q(t)−q(t) =¯ αT(t)L(q−q)(t) + (α¯ T −α¯T)(t)Lq(t)¯

=Lα(q−q)(t) +¯ Lα−α¯q(t).¯ To estimate the terms of the r.h.s we use (2.11)

kLα(q−q)¯k_H¹₄ .keαk_H¹₄T^14−εkq−q¯k_H¹₄ , kL^α−α¯q¯k_H¹₄ .

eα−αe¯

H¹⁴T^14−εkq¯k_H¹₄ . With the choice ofT done above we get

kq−q¯k_H¹₄ . eα−eα¯

H¹⁴kq¯k_H¹₄. This achieves the proof of the continuity.

2)Write againq0(t) =αT(t)I(t) +αT(t)II(t). Lemma A.2-b) implies kαTIk_H³₄ .kαTk_H³₄kIk

B

3 4+ε

∞,∞

.kαTk_H³₄ku0kL². SinceH³⁴ is an algebra the inequality

kαTk_H³₄ .kχ(T⁻¹·)k_H³₄keαk_H³₄ .T⁻¹⁴keαk_H³₄

holds forT ∈[0,1], owing to (2.9).

It follows

kαTIk_H³₄ .T⁻¹⁴kαek_H³₄ku0kL². The second term is estimated with (2.8):

kαTIIk_H³₄ .kαTk_H³₄kIIk_H³₄ .T⁻¹⁴keαk_H³₄ku0kH¹. Finally we get forT ∈[0,1]

kq0k_H³₄ .T⁻¹⁴kαek_H³₄ku0kH¹.

Using Lemma A.2-a)-d), Sobolev embeddings and Proposition 2.1 (with θ= ₁₂⁵ and θ= ¹₆), gives kLαqk_H³₄ .kαTkL^∞kLqk_H³₄ +kαTk_H³₄kLqkL^∞

.kαekL^∞T¹²¹ kq1[0,T]k_H¹₃ +kq1[−T,0]k_H¹₃

+T⁻¹⁴keαk_H³₄kLqk_H¹₂^+ε

.kαek_H³₄T¹²¹kqk_H¹₃ +T⁻¹⁴keαk_H³₄T¹³kqk_H¹₃^+ε. (2.12) Thus we get for 0≤T ≤1,

kLαqk_H³₄ .kαek_H³₄T¹²¹kqk_H³₄.

This proves thatLis a contracting map in H³⁴ for sufficiently small timeT. The time T depends only on keαk_H³₄ and then we can construct globally a unique solutionq ∈ H_loc³⁴ (R) for the linear problem.

For the locally Lipschitz dependence with respect to α, the proof is left to the reader: it can be done easily done like for the caseα∈H¹⁴.

3)Like in the proof of the second point 2) we get

kq0k_H³₄^+ε.kαTk_H³₄^+ε ku0kL²+kIIk_H³₄^+ε .T^−14−εku0kH^1+2ε.

Reproducing the same computation as (2.12) leads to kLαqk_H³₄^+ε.kαTkL^∞kLqk_H³₄^+ε+kαTk_H³₄^+εkLqkL^∞

.kαek^L∞T¹²¹ kq1[0,T]k_H¹₃^+ε+kq1[−T,0]k_H¹₃^+ε

+T^−14−εkαek_H³₄^+εkLqk_H¹₂^+ε .kαekL^∞T¹²¹kqk_H¹₃+ε+T^−14−εkαek_H³₄+εT^13+εkqk_H¹₃+2ε

.T¹²¹kαek_H³₄^+εkqk_H³₄^+ε.

With the fixed point argument we can conclude the proof.

2.3 Regularity of u

We start with the following result.

Lemma 2.3 Fors∈R, let H_T^s be the closed subset ofH^s(R_t) H_T^s ={u∈H^s(R_t), suppu⊂[−T, T]},

endowed with the norm k kH^s. For anyT >0 and any s∈R, there is a constant CT,s such that

∀f ∈H_T^2s−14 , F⁻¹

Ff(−|ξ|²)

H^s ≤CT,skfk_H^2s−1₄ . Proof: It suffices to compute

Z

R

(1 +|ξ|²)^s bf(−|ξ|²)

2

dξ= Z +∞

0

(1 +τ)^s 2τ^1/2

bf(−τ)

2

dτ

≤ max

τ∈[0,1]

bf(−τ)

2

+ Z _∞

1

(1 +τ)^s−1/2 bf(−τ)

2

dτ

≤ max

τ∈[0,1]

bf(−τ)

2

+kfk²_H^s₂−1 4 ,

where f(τ) =b he^{iτ x}χ(x)e , fiwithχe∈ D(R) with value 1 in [−T, T].By duality we have forν∈R sup

0≤τ≤1|fb(τ)| ≤ kfkH^ν sup

0≤τ≤1ke^iτ·χ(e·)kH^−ν

≤C_T,ν¹ kfkH^ν.

The main result of this section is the following.

Proposition 2.4 1. Let u0 ∈ H¹(R_x), α ∈ H_loc¹⁴ (R_t), then the equation (1.5) has a unique solutionu∈C(R;H¹(R)).

2. Let u0 ∈D(Hα(0)),α∈H_loc³⁴ (R_t),then the equation (1.5) has a unique solution ubelonging to the spaceC¹(R;L²(R)), with u(t)∈D(Hα(t))for allt∈R.

Proof: 1)The solution of (1.5) is obtained via the equation (2.6) b

u(t, ξ) =e^−it|ξ|2ub0(ξ)−ie^−it|ξ|2F q1[0,t]

(−|ξ|²).

Let us check that we have the required regularity for u. Applying Lemma 2.3 to (2.6) implies ku(t)kH¹ ≤ ku0kH¹+Ckq1[0,t]k_H¹₄. (2.13) Lemma A.2-a) and Lemma B.1 yield

kq1[0,t]k_H¹₄ .kqk_H¹₄k1[0,t]k

B

1 2,∞2 ∩L^∞

≤Ctkqk_H¹₄.

This proves thatu∈L^∞_loc(R;H¹).It remains to prove the continuity in time ofu. First notice that we need for this purpose to prove only the continuity in time ofv(t) :=u(t)−e^it∆u0.This will be done in two steps. In the first one we deal with the caseα∈H_loc³⁴ . In the second one, we go back to the caseα∈H_loc¹⁴ .

• Case α∈H_loc³⁴ . Remark that according to Proposition 2.2-2) we can construct a unique solution q∈H_loc³⁴ for the problem (2.5). An easy computation gives fort, t^′∈R

kv(t)−v(t^′)k²H¹ . Z

R

sin² (t−t^′2/2)

(1 +|ξ|²)F q1[0,t]

(−|ξ|²)²dξ +kq(1[0,t]−1[0,t^′])k²_H¹₄.

Using the fact sin²x≤ |x|^ε,∀ε∈[0,1],and Lemma 2.3 gives Z

R

sin² (t−t^′2/2)

(1 +|ξ|²)F q1[0,t]

(−|ξ|²)²dξ.|t−t^′εkq1[0,t]k_H¹₄^+ε₂.

It suffices now to use Lemma A.2-a) Z

R

sin² (t−t^′2/2)

(1 +|ξ|²)F q1[0,t]

(−|ξ|²)²dξ .|t−t^′εkqk_H¹₄+ε 2. For the second term we use again Lemma A.2-c) combined with the proof of Lemma B.1

kq(1[0,t]−1[0,t^′])k_H¹₄ .kqk_H¹₄^+εk1[0,t]−1[0,t^′]k_H¹₂−ε

.kqk_H¹₄^+ε|t−t^′ε This concludes the proof of the time continuity ofuwhenα∈H_loc³⁴ .

•Case α∈H_loc¹⁴ . We smooth out the functionαleading to a sequence of smooth functionsαn that converges strongly to αin H_loc¹⁴ . To eachαn we associate the unique solutions qn andun. From the first step un belongs toC(R;H¹). Similarly to (2.13) we get forn, m∈N

kun−umkL^∞_{[−T ,T]}H¹ ≤CTkqn−qmk_H¹₄.

By Proposition 2.2-a), {qn} is a Cauchy sequence inH¹⁴ and thus {un} converges uniformly tou in L^∞_T H¹. This gives thatu∈C([−T, T], H¹), for everyT >0.

2) Recall from (2.7) that

i∂tu(t, ξ) =b e^−it|ξ|2F(Hα(0)u0)(ξ) +e^−it|ξ|2F

(∂sq)1[0,t]

(−|ξ|²).

Since u0 ∈D(Hα(0)) then the first term of the r.h.s belongs to C(R;L²). On the other hand we haveD(Hα(0)⊂H^32−d for anyd >0. It follows from Proposition 2.4-1)that we can construct a unique solution q∈H³⁴. Now, letw(t) :=i∂tu−e^it∆Hα(0)u0.Then Lemma 2.3 yields

kw(t)kL² .kq^′1[0,t]k_H−1 4. Lemma A.2-a) and Lemma B.1 imply

kq^′1[0,t]k_H−1

4 .kq^′k_H−1 4k1[0,t]k

B

1 2,∞2 ∩L^∞

≤CTkqk_H³₄. Thus we get for every t∈[−T, T]

kw(t)kL² ≤CTkqk_H³₄. (2.14) It follows thatw∈L^∞_loc(R;L²). To prove the continuity in time ofwwe use the same argument as for the first point of this proposition. We start with a smooth functionα, that is α∈H

3 4+ε loc . This gives according to Proposition 2.2-3) a unique solutionq∈H^34+ε. We have the following estimate, obtained similarly to caseα∈H

3 4

loc discussed above, kw(t)−w(t^′)kL² .|t−t^′|^εkq^′1[0,t]k_H−1

4+ε

2 +kq^′(1[0,t]−1[0,t^′])k_H−1 4. Using Lemma A.2-a) with s=−¹₄+^ε₂ gives

kq^′1[0,t]k_H−1 4+ε

2 .kqk_H³₄^+ε.

For the second term of the r.h.s we use Lemma A.2-c- withs=−¹₄+ε, s^′= ¹₂−εand Lemma B.1 kq^′(1[0,t]−1[0,t^′])k_H−1

4 .|t−t^′|^εkqk_H³₄^+ε.

This achieves the proof of the continuity of w in time for α ∈ H_loc^34+ε. Now for α ∈ H_loc³⁴ we do like the first point of the proposition: we smooth out αand we use the continuity dependence of q with respect to α stated in Proposition 2.2-2) combined with the estimate (2.14). By writing i∂tu=Hα(t)u(t) we get that for everyt∈R, u(t)∈D(Hα(t)).

A Paraproducts and product laws

The aim of this section is to prove some product laws used in the proof of the main results. For this purpose we first recall some basic ingredients of the paradifferential calculus. Start with the dyadic partition of the unity: there exists two radial positive functionsχ∈ D(R) andϕ∈ D(R\{0}) such that

χ(ξ) +X

q≥0

ϕ(2^−qξ) = 1, ∀ξ∈R. For every tempered distributionv∈ S^′, set

∆₋1v=χ(D)v ;∀q∈N, ∆qv=ϕ(2^−qD)v andSq =

q−1

X

j=−1

∆j.

For more details see for instance [4][3]. Then Bony’s decomposition of the product uvis given by uv=Tuv+Tvu+R(u, v),

with

Tuv=X

q

Sq−1u∆qv and R(u, v) = X

|q^′−q|≤1

∆qu∆q^′v .

Let us now recall the definition of Besov spaces through dyadic decomposition. For (p, r)∈[1,+∞]² ands∈R, the spaceB^s_p,r is the set of tempered distributionusuch that

kukB_p,r^s :=

2^qsk∆qukL^p

ℓ^r <+∞.

This definition does not depend on the choice of the dyadic decomposition. One can further remark that the Sobolev space H^s coincides with B^s_2,2. Below is the Bernstein lemma that will be used for the proof of product laws and which is a straightforward application of convolution estimates and Fourier localization.

Lemma A.1 There exists a constant C such that forq, k∈N, 1≤a≤b and for f ∈L^a(R), sup

|α|=kk∂^αSqfkL^b≤C^k2^q(k+a1−1b)kSqfkL^a, C^−k2^qkk∆qfkL^a≤ sup

|α|=kk∂^α∆qfkL^a≤C^k2^qkk∆qfkL^a.

The following product laws have been used intensively in the proof of our main result.

Lemma A.2 In dimensiond= 1 the product (u, v)7→uvis bilinear continuous a) fromH^s×(B_2,¹²_∞∩L^∞)toH^s as soon as|s|<¹₂;

b) from H^s×B_∞^s+ε_,∞ toH^s as soon ass≥0 andε >0.

c) from H^s×H^s′ toH^s+s′−12 as soon as s, s^′<¹₂ ands+s^′>0.

d) Fors≥0,H^s∩L^∞ is an algebra. Fors > ¹₂,H^s is an algebra.

Proof: a)Using the definition and Bernstein lemma we obtain kTuvk²H^s .X

q

2^2qskSq−1uk²L^∞k∆qvk²L²

.kvk²

B

1 2 2,∞

X

q

2^2q(s−12)kSq−1uk²L^∞

.kvk²

B

1 2,∞2

X

q

X

p≤q−1

2^q(s−12)2^p2k∆pukL²

2

.kvk²

B

12 2,∞

X

q

X

p≤q−1

2^{(q−p)(s−12)}(2^psk∆pukL²)2

.kvk²

B

1 2,∞2

kuk²H^s.

We have used in the last line the convolution lawℓ¹⋆ ℓ²→ℓ².

For the second termTvuwe use the fact that Sq−1mapsL^∞to itself uniformly with respect toq.

kTvuk²H^s.X

q

2^2qskSq−1vk²L^∞k∆quk²L²

.kvk²L^∞kuk²H^s.

To estimate the remainder term we use the fact

∆qR(u, v) = X

j≥q−4

|j−j′ |≤1

∆q(∆ju∆j^′v).

According to Bernstein lemma one gets

2^qsk∆q(R(u, v))kL² .2^q(s+12) X

j≥q−4

|j−j′ |≤1

k∆jukL²k∆j^′vkL²

. X

j≥q−4

|j−j′ |≤1

2^{(q−j)(s+12)}2^jsk∆jukL²2^j′12k∆j^′vkL²

.kvk

B

1 2,∞2

X

j≥q−4

2^{(q−j)(s+12)}2^jsk∆jukL². It suffices now to apply the convolution inequalities.

b)First remark that the cases= 0 is obvious: L²×L^∞→L²andB^ε_∞,∞֒→L^∞.Hereafter we consider s >0 . To estimate the first paraproduct, use the embeddingB^s+ε_∞,∞֒→B_∞^s_,2,forε >0.

kTuvk²H^s.X

q

2^2qskSq−1uk²L²k∆qvk²L^∞

.kukL²kvk^B_∞,2^s .kukL²kvkB_∞,∞^s+ǫ

For the second term we use the result obtained in the parta):

kTvuk²H^s.X

q

2^2qskSq−1vk²L^∞k∆quk²L²

.kvk²L^∞kuk²H^s

.kvk²_B^s+ǫ_∞,∞kuk²H^s

To estimate the remainder term we write

2^qsk∆q(R(u, v))kL².2^qs X

j≥q−4

|j−j′ |≤1

k∆jukL²k∆j^′vk^L∞

.kvk^L∞ X

j≥q−4

|j−j′ |≤1

2^(q−j)s2^jsk∆jukL².

Sinces >0 then we obtain by using the convolution inequalities kR(u, v)k^Hs ≤ kvk^L∞kuk^Hs. c), d)These results are standard, see for example [4].

B Sobolev and Besov regularity of cut-offs

Lemma B.1 1. For any ν <1/2,t 7→ 1[0,t](s) belongs toC(R₊;H^ν(R)). More precisely we have for|t−t^′| ≤1

k1[0,t]−1[0,t^′]k^Hν .|t−t^′|^12−ν.