DIRECT AND INVERSE SCATTERING FOR THE NONLINEAR SCHRÖDINGER EQUATION WITH

A POTENTIAL^{∗}

Ricardo Weder^{†}

Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas Universidad Nacional Autónoma de México Apartado Postal 20-726, México DF 01000 weder@servidor.unam.mx

Resumen

En este artículo se presenta una reseña de resultados recientes en la teoría de dispersión para la ecuación de Schrödinger no lineal en una dimensión y con un potencial. En particular, en la construcción del operador de dispersión para energías pequeñas y en la resolución del problema inverso. Específicamente, damos condiciones en el potencial y en la nolinealidad tales que el operador de dispersión para energías pequeñas determina unívocamente el potencial y la nolinealidad, y damos un método para la reconstrucción de ambos. Estos resultados están basados en la estimación L^{1} − L^{∞} que demostramos en [10]. Palabras claves: dispersión inversa, ecuación de Schrödinger no lineal.

Abstract

In this paper we review recent results on the scattering theory for the nonlinear Schrödinger equation with a potential on the line. In particular, on the construction of the low-energy scattering operator and on the solution of the inverse scattering problem. Namely, we give conditions on the potential and on the nonlinearity such that the low-energy scattering operator determines uniquely the potential and the nonlinearity, and we give a method for the reconstruction of both. These results are based on the L^{1} − L^{∞} estimate that we proved in [10]. Key words: inverse scattering, nonlinear Schrödinger equation. AMS Subject Classification: 35Q40, 35R30, 81U40.

∗MAT - Serie A, 3 (2001), 13-20.

†Fellow Sistema Nacional de Investigadores. Research partially supported by Universidad Nacional Autónoma de México under Proyecto PAPIIT-DGAPA IN 105799.

1 Introduction

In this paper I wish to discuss some recent results in the scattering theory of nonlinear evolution equations of dispersive type. We will consider in particular the nonlinear Schrödinger equation. This is one of the main equations of mathematical physics and it is a typical case that shows many of the main features of other equations. The aim of direct scattering theory is to study the behavior for large times of the solutions, and in particular to prove that -under appropriate conditions- the solutions are asymptotic as t → ±∞ to solutions of a simpler, linear equation with constant coefficients. Since the solutions of the later equation are obtained by Fourier transform, direct scattering theory allows us to describe the asymptotic behavior of the solutions to our nonlinear evolution equation in a simple way. Of course, this situation excludes the case of nonlinear boundstates, that are solutions periodic in time. This requires either that the initial data is small or that the interaction is repulsive in a appropriate sense. The operator that to the inital data of the asymptotic solution as t → −∞ asigns the inital data of the asymptotic solution as t → ∞ is the scattering operator.The purpose of inverse scattering theory is to obtain information on the potential and the nonlinearity, from the scattering operator. In other words, we wish to obtain as much information as possible on the potential and the nonlinearity, from the asymptotic behavior of the solutions; more precisely, from the relation of the asymptotic behaviors as t → ±∞. Some of the main problems are uniqueness: does the scattering operator uniquely determines the potential and the nonlinearity?, and reconstruction: to obtain formulae that allow to reconstruct the potential and the nonlinearity from the scattering operator.

We will discuss the following nonlinear Schrödinger equation with a potential,

i\frac{∂}{∂t}u(t, x) = −\frac{d^{2}}{dx^{2}} u(t, x) + V_{0}(x)u(t, x) + F (x, u), u(0, x) = ϕ(x),  (1.1)

where t, x ∈ R, the potential, V_{0}, is a real-valued function and F (x, u) is a complex-valued function. In the case where the potential is zero, V_{0} ≡ 0, there is a very large literature on the direct scattering problem . See for example [1], [2], [3],[4] [5], [7] and [8]. When V_{0} ≡ 0 the linearized equation is the free Schrödinger equation,

i\frac{∂}{∂t}u(t, x) = −\frac{d^{2}}{dx^{2}} u(t, x), u(0, x) = ϕ(x).  (1.2)

Let us denote by H_{0} the self-adjoint realization of −\frac{d^{2}}{dx^{2}} in L^{2} with domain the Sobolev space W_{2,2} and let us define e^{−itH_{0}} by functional calculus. The solution to (1.2) is given by, e^{−itH_{0}}ϕ. It follows immediately from the explicit formula for the kernel of e^{−itH_{0}} that the following L^{1} − L^{∞} estimate holds,

∥e^{−itH_{0}}ϕ∥_{L^{∞}} ≤ C \frac{1}{\sqrt{|t|}} ∥ϕ∥_{L^{1}} .  (1.3)

The estimate (1.3) expresses in a quantitative way a important property of the solutions to (1.2). Namely, that as t → ±∞ the solutions not only propagate to spacial infinity, but they also spread uniformly in space. This is known in the physics literature as wave packet spreading. This spreading is essential in the study of the nonlinear Schrödinger equation. This can be seen as follows. For simplicity, suppose that ϕ ∈ L^{1} ∩ W_{1,2}. Then, we have that,

∥e^{−itH_{0}}ϕ∥_{L^{∞}} ≤ C \frac{1}{\sqrt{1 + |t|}} (∥ϕ∥_{L^{1}} + ∥ϕ∥_{W_{1,2}}) .

Suppose moreover, that F (x, u) = λ|u|^{(p−1)}u, for some constant λ and some large positive p. Then, if ϕ is small enough; initially F will be small and the solution to (1.1) with V_{0} ≡ 0 will propagate for small times essentially like a solution to the free Schrödinger equation (1.2). Moreover, because of the spreading the solution will be even smaller as t increases, and this will give the nonlinear term no chance to become very big and to make the solution blow-up in a finite time. If the initial data is not small, but the nonlinearity is repulsive -in a proper sense- a similar phenomenon takes place. In both cases there is a balance within the linear and the nonlinear terms that makes possible the existence of solutions global in time, and that permits the analysis of the large time asymptotics of the solutions in terms of the solutions to the free Schrödinger equation (1.2), i.e., scattering takes place. But what happens when the the potential V_{0} is not identically zero? In this case the linearized equation is the following Schrödinger equation with a potential,

i\frac{∂}{∂t}u(t, x) = −\frac{d^{2}}{dx^{2}} u(t, x) + V_{0}(x)u(t, x), u(0, x) = ϕ(x),  (1.4)

and we would need that an estimate as (1.3) holds with H_{0} replaced with the perturbed Hamiltonian, H := H_{0}+V_{0}. The problem is that such an estimate is at worst not true and at best quite hard to come by. Suppose for example that H has an eigenvalue, E, with eigenvector ϕ, i.e., that Hϕ = Eϕ. Then, e^{−itH}ϕ = e^{−itE}ϕ. This solution is periodic in time and it does not spreads at all; ∥e^{−itH}ϕ∥_{L^{∞}} = ∥ϕ∥_{L^{∞}}. However, we can hope that an estimate as (1.3) holds for initial data in H_{c}, where H_{c} is the subspace of continuity for H, that is to say, the orthogonal complement of the subspace spanned by all eigenvectors of H. In one dimension this is rather delicate estimate-due to the singularity at low energythat has been proved only recently in [10]. We give this result below, but first we state some standard notations and definitions. For any γ ∈ R, let us denote by L_{γ}^{1} the Banach space of all complex-valued measurable functions, ϕ, defined on R and such that

∥ϕ∥_{L_{γ}^{1}} := ∫ |ϕ(x)| (1 + |x|)^{γ} dx < ∞.

We say that V_{0} ∈ L_{1}^{1} is generic if the Jost solutions to the stationary Schrödinger equation at zero energy are linearly independent, and we say that V_{0} is exceptional if they are linearly dependent. See [10] for details. We denote by P_{c} the orthogonal projector in L^{2} onto H_{c}.

THEOREM 1.1. (The L^{1} − L^{∞} estimate [10] ). Suppose that V ∈ L_{γ}^{1} where in the generic case γ > 3/2 and in the exceptional case γ > 5/2. Then for some constant C,

∥e^{−itH}P_{c}∥_{B(L^{1},L^{∞})} ≤ \frac{C}{\sqrt{|t|}}.  (1.5)

COROLLARY 1.2. (The L^{p} − Lṕ estimate [10]). Suppose that the conditions of Theorem 1.1 are satisfied. Then for 1 ≤ p ≤ 2 and \frac{1}{p} + \frac{1}{ṕ} = 1,

∥e^{−itH}P_{c}∥_{B(L^{p},Lṕ)} ≤ \frac{C}{|t|^{( \frac{1}{p} − \frac{1}{2} )}} .  (1.6)

The L^{1} − L^{∞} estimates have many applications. See for example [1]. In fact, in page 27 of [1] the problem of proving estimate (1.5) was posed as an interesting open problem of independent importance, but actually [10] already existed in preprint form when [1] was published. Corollary 1.2 follows from Theorem 1.1 using the fact that e^{−itH} is unitary in L^{2} and by interpolation. The proof of (1.5) given in [10] is based in a careful analysis of the low energy behavior of the Jost solutions that uses the fact that the Jost solutions are obtained as solutions of integral equations of Volterra type and techniques of ordinary differential equations. The L^{1} − L^{∞} estimate (1.5) is the key to the solution of the lowenergy direct scatering problem and of the inverse scattering problem in [10] and [13]. We state the results below after we introduce some more standard notations. As usual, we say that F (x, u) is a C^{k} function of u in the real sense if for each x ∈ R, ℜF and ℑF are C^{k} functions with respect to the real and imaginary parts of u.We assume that F is C^{2} in the real sense and that (\frac{∂}{∂x}F ) (x, u) is C^{1} in the real sense. If F = F_{1} + iF_{2} with F_{1}, F_{2} real-valued, and u = r + is, r, s ∈ R we denote,

F^{(2)}(x, u) := ∑_{j=1}^{2} [ |\frac{∂^{2}}{∂r^{2}} F_{j}(x, u)| + |\frac{∂^{2}}{∂r∂s}F_{j}(x, u)| + |\frac{∂^{2}}{∂s^{2}} F_{j}(x, u)| ] ,

(\frac{∂}{∂x}F )^{(1)} (x, u) := ∑_{j=1}^{2} [ |\frac{∂}{∂r} (\frac{∂}{∂x}F_{j}) (x, u)| + |\frac{∂}{∂s} (\frac{∂}{∂x}F_{j}) (x, u)| ]

Let us define,

M := {u ∈ C(R, W_{1,p+1}) : sup_{t∈R}(1 + |t|)^{d}∥u∥_{W_{1,p+1}} < ∞} ,

with norm

∥u∥_{M} := sup_{t∈R}(1 + |t|)^{d}∥u∥_{W_{1,p+1}},

where p ≥ 1, and d := \frac{1}{2} \frac{p−1}{p+1}. For functions u(t, x) defined in R^{2} we denote u(t), for u(t, ·).

THEOREM 1.3. (The low-energy scattering operator [13]) Suppose that V_{0} ∈ L_{γ}^{1}, where in the generic case γ > 3/2 and in the exceptional case γ > 5/2, that H has no eigenvalues, and that

N(V_{0}) := sup_{x∈R} ∫_{x}^{x+1} |V_{0}(y)|^{2} dy < ∞.

Furthermore, assume that F is C^{2} in the real sense, that F (x, 0) = 0, and that for each fixed x ∈ R all the first order derivatives, in the real sense, of F vanish at zero. Moreover, suppose that \frac{∂}{∂x}F is C^{(1)} in the real sense. We assume that the following estimates hold:

F^{(2)}(x, u) = O (|u|^{p−2}) , (\frac{∂}{∂x}F )^{(1)} (x, u) = O (|u|^{p−1}) , u → 0, uniformly for x ∈ R,

for some ρ < p < ∞, and where ρ is the positive root of \frac{1}{2} \frac{ρ−1}{ρ+1} = \frac{1}{ρ}. Then, there is a δ > 0 such that for all ϕ_{−} ∈ W_{2,2} ∩ W_{1,1+ \frac{1}{p}} with ∥ϕ_{−}∥_{W_{2,2}} + ∥ϕ_{−}∥_{W_{1,1+ \frac{1}{p}}} ≤ δ, there is a unique solution, u, to (1.1) such that u ∈ C(R, W_{1,2}) ∩ M and,

lim_{t→−∞} ∥u(t) − e^{−itH}ϕ_{−}∥_{W_{1,2}} = 0.

Moreover, there is a unique ϕ_{+} ∈ W_{1,2} such that

lim_{t→∞} ∥u(t) − e^{−itH}ϕ_{+}∥_{W_{1,2}} = 0.

Furthermore, e^{−itH}ϕ_{±} ∈ M and

∥u − e^{−itH}ϕ_{±}∥_{M} ≤ C ∥e^{−itH}ϕ_{±}∥_{M}^{p} ,

∥ϕ_{+} − ϕ_{−}∥_{W_{1,2}} ≤ C [∥ϕ_{−}∥_{W_{2,2}} + ∥ϕ_{−}∥_{W_{1,1+ \frac{1}{p}}} ]^{p} .

The scattering operator, S_{V_{0}} : ϕ_{−} ↪ ϕ_{+} is injective on W_{1,1+ \frac{1}{p}} ∩ W_{2,2}.

Note that, ρ ≈ 3.56.

The wave operators for the linear scattering problem corresponding to equation (1.1) with F ≡ 0 and equation (1.2) are given by:

W_{±} := s − lim_{t→±∞} e^{itH} e^{−itH_{0}}.

It is proven in [6] that the limits above exit in the strong topology in L^{2} and that RangeW_{±} = H_{c}. The corresponding scattering operator is given by,

S_{L} := W_{+}^{∗} W_{−}.

The scattering operator below relates asymptotic states that are solutions to the linear Schrödinger equation and it is the appropriate one for the reconstruction of V_{0}.

S := W_{+}^{∗} S_{V_{0}} W_{−}.  (1.7)

In the following theorem we reconstruct S_{L} from S.

THEOREM 1.4. ([13]) Suppose that the assumptions of Theorem 1.3 are satisfied. Then for every ϕ_{−} ∈ W_{2,2} ∩ W_{1,1+ \frac{1}{p}}

\frac{d}{dϵ} S(ϵϕ)|_{ϵ=0} = S_{L}ϕ,  (1.8)

where the derivative in the left-hand side of (1.8) exists in the strong convergence in W_{1,2}.

COROLLARY 1.5. ([13]) Under the conditions of Theorem 1.3 the scattering operator, S, determines uniquely the potential V_{0}.

In the case where F (x, u) = ∑_{j=1}^{∞} V_{j}(x)|u|^{2(j_{0}+j)}u we can also reconstruct the V_{j}, j = 1, 2, · · ·.

LEMMA 1.6. ([13]) Suppose that the conditions of Theorem 1.3 are satisfied, and moreover, that F (x, u) = ∑_{j=1}^{∞} V_{j}(x)|u|^{2(j_{0}+j)}u, where j_{0} is an integer such that, j_{0} ≥ (p−3)/2, for |u| ≤ η, for some η > 0, and where V_{j} ∈ W_{1,∞} with ∥V_{j}∥_{W_{1,∞}} ≤ C^{j}, j = 1, 2, ⋯, for some constant C. Then, for any ϕ ∈ W_{2,2} ∩ W_{1,1+ \frac{1}{p}} there is an ϵ_{0} > 0 such that for all 0 < ϵ < ϵ_{0}:

i ((S_{V} − I)(ϵϕ), ϕ)_{L^{2}} = ∑_{j=1}^{∞} ϵ^{2(j_{0}+j)+1} [∫ ∫ dt dx V_{j}(x) |e^{−itH}ϕ|^{2(j_{0}+j+1)} + Q_{j}] , (1.9)

where Q_{1} = 0 and Q_{j}, j > 1, depends only on ϕ and on V_{k} with k < j. Moreover, for any x́ ∈ R, and any λ > 0, we denote, ϕ_{λ}(x) := ϕ(λ(x − x́)). Then, if ϕ ≠ 0:

V_{j}(x́) = \frac{lim_{λ→∞} λ^{3} ∫ ∫ dt dx V_{j}(x) |e^{−itH}ϕ_{λ}|^{2(j_{0}+j+1)}}{∫ ∫ dt dx |e^{−itH_{0}}ϕ|^{2(j_{0}+j+1)}} .  (1.10)

COROLLARY 1.7. ([13]) Under the conditions of Lemma 1.6 the scattering operator, S, determines uniquely the potentials V_{j}, j = 0, 1, · · ·.

The method to reconstruct the potentials V_{j}, j = 0, 1, ⋯, is as follows. First S_{L} is obtained from S using (1.8). By any standard method for inverse scatering for the linear Schrödinger equation on the line (recall that H has no eigenvalues) we reconstruct V_{0}. We then reconstruct S_{V_{0}} from S using (1.7). Finally (1.9) and (1.10) give us, recursively, V_{j}, j = 1, 2, · · ·.

For the proof of these results, the extension to the multidimensional case and to the nonlinear Klein-Gordon equation see [9], [10], [11], [12], [13], [14] and [15]. In these papers also a discussion of the literature is given.

References

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[3] Ginibre J., Introduction aux Équations de Schrödinger non Linéaires, Onze Édition, Paris (1998).

[4] Kato T., Nonlinear Schrödinger equations, in Schrödinger Operators, Holden H.& Jensen A. (Eds.), Lecture Notes in Physics V. 345, Springer–Verlag, Berlin (1989), pp. 218-263.

[5] Racke R., Lectures in Nonlinear Evolution Equations. Initial Value Problems, Aspects of Mathematics V. E 19, F. Vieweg & Son, Braunschweig/Wiesbaden (1992).

[6] Schechter M., Operator Methods in Quantum Mechanics, North Holland, New York (1981).

[7] Strauss W.A., Non linear scattering theory, in Scattering Theory in Mathematical Physics, Lavita J.A & Marchand J. P. (eds.), D. Reidel, Dordrecht-Holland/Boston (1974), pp. 53-78.

[8] W. A. Strauss W.A., Nonlinear Wave Equations, CBMS–RCSM V. 73, American Mathematical Society, Providence (1989).

[9] Weder R., Inverse scattering for the nonlinear Schrödinger equation, Commun. Part. Diff. Equations 22 (1997), pp. 2089-2103.

[10] Weder R., L^{p} − Lṕ estimates for the Schrödinger equation on the line and inverse scattering for the nonlinear Schrödinger equation with a potential, J. Funct. Analysis 170 (2000), pp. 37-68.

[11] Weder R., The W_{k,p}-continuity of the Schrödinger wave operators on the line, Comm. Math. Phys. 208 (1999), pp. 507-520.

[12] Weder R., Inverse scattering on the Line for the nonlinear Klein-Gordon equation with a potential, J. Math. Anal. Appl. 252 (2000), pp. 102-123.

[13] Weder R., Inverse scattering for the nonlinear Schrödinger equation. Reconstruction of the potential and the nonlinearity. Preprint 2000. To appear in Mathematical Methods in the Applied Sciences.

[14] R. Weder, Inverse scattering for the nonlinear Schrödinger equation II. Reconstrution of the potential and the nonlinearity in the multidimensional case. Preprint 2000. To appear in Proceedings of the American Mathematical Society.

[15] R. Weder, Multidimensional inverse scattering for the Klein-Gordon equation with a potential. Preprint,2000.