We show that for a one-dimensional Schrödinger operator with a potential, whose moment is integrable, the -th derivative of the scattering matrix is in the Wiener algebra of functions with integrable Fourier transforms. We use this result to improve the known dispersive estimates with integrable time decay for the one-dimensional Schrödinger equation in the resonant case. -th
This paper is concerned with the one-dimensional Schrödinger equation
with a real-valued potential contained in one of the spaces , associated with the norms ,
We recall (e.g. Reference 15 or Reference 16, §9.7) that for the operator has a purely absolutely continuous spectrum on and a finite number of eigenvalues in Associated with the absolutely continuous spectrum is the scattering matrix .
which maps incoming to outgoing states at a given energy Here . is the transmission coefficient and are the reflection coefficients with respect to right and left incident. At the edge of the continuous spectrum the scattering matrix generically looks like
More precisely, this happens when the zero energy is non-resonant, that is, if the equation has no bounded (distributional) solution. In the resonant situation the behavior of the scattering matrix is more delicate. For it is already non-trivial to establish continuity of the scattering matrix at (for de l’Hospital’s rule suffices). This question arose around 1985 in an attempt to clarify whether the low-energy asymptotics of the scattering matrix, obtained for in Reference 6, are valid for The problem was solved independently by Guseĭnov .Reference 12 and Klaus Reference 13 (for a refined version see Reference 1 and Reference 9, Theorem 2.1). It also plays an important role in semiclassical analysis; see Reference 5 and the references therein. Furthermore, if with then, away from , one can take derivatives of the scattering matrix up to order , and in the non-resonant case it is again easy to see that they are continuous at , This clearly raises the question about continuity at . of the derivative in the resonant case. We will establish this as one of our main results in Theorem -th2.2. This result is new even for the first derivative. We remark that if decays exponentially, then is analytic in a neighborhood of and the full Taylor expansion can be obtained Reference 2Reference 3.
It is important to emphasize that this question is not only of interest in scattering theory but also plays a role in the solution of the Korteweg–de Vries equation via the inverse scattering transform (see e.g. Reference 11 where continuity of higher derivatives is needed) and in deriving dispersive estimates for Equation 1.1. The latter case has attracted considerable interest (e.g. Reference 9Reference 10 and the references therein) due to its importance for proving asymptotic stability of solitons for the associated non-linear evolution equations (see e.g. Reference 4Reference 14).
As an application of our results we will establish the following dispersive decay estimate with integrable time decay in the resonant case:
This theorem is an improvement of an earlier result by Goldberg Reference 10 who established it for If there is no resonance (i.e. no bounded solution) this result (with . was shown for ) in Reference 9, Theorem 1.2. For extensions to discrete one-dimensional Schrödinger equations (Jacobi operators) see Reference 7Reference 8.
2. Low energy scattering
In this section we establish some properties of the scattering data for our operator with , To this end we introduce the Banach algebra . of Fourier transforms of integrable functions,
with the norm plus the corresponding unital Banach algebra ,,
with the norm We also use the fact, which is known as Wiener’s lemma .Reference 17, that if and for all then ,.
Since we conclude that
Here and throughout the rest of this paper a prime will always denote a derivative with respect to the spatial variable As an immediate consequence of the estimates .Equation 2.4 and Equation 2.5 we have the following strengthening of Equation 2.7.
The fact that also solve for leads to the scattering relations
where the transmission coefficient and the reflection coefficients can be expressed in terms of Wronskians. To this end we denote by
the usual Wronskian and set
The transmission and reflection coefficients are elements of the Wiener algebra, which was established in Reference 9, Theorem 2.1. Here we extend this result to the derivatives of the scattering data.