Dispersive estimates for matrix Schr\"{o}dinger operators in dimension two

We consider the non-selfadjoint operator [\cH = [{array}{cc} -\Delta + \mu-V_1&-V_2 V_2&\Delta - \mu + V_1 {array}]] where $\mu>0$ and $V_1,V_2$ are real-valued decaying potentials. Such operators arise when linearizing a focusing NLS equation around a standing wave. Under natural spectral assumptions we obtain $L^1(\R^2)\times L^1(\R^2)\to L^\infty(\R^2)\times L^\infty(\R^2)$ dispersive decay estimates for the evolution $e^{it\cH}P_{ac}$. We also obtain the following weighted estimate $$ \|w^{-1} e^{it\cH}P_{ac}f\|_{L^\infty(\R^2)\times L^\infty(\R^2)}\les \f1{|t|\log^2(|t|)} \|w f\|_{L^1(\R^2)\times L^1(\R^2)},\,\,\,\,\,\,\,\, |t|>2, $$ with $w(x)=\log^2(2+|x|)$.


Introduction
The free Schrödinger evolution on R d , (1) e −it∆ f (x) = C d 1 t d/2 R d e −i|x−y| 2 /4t f (y)dy, satisfies the dispersive estimate In recent years many authors (see, e.g., [30,39,23,41,24,49,20,9,15,25,5], and the survey article [43]) worked on the problem of extending this bound to the perturbed Schrödinger operator H = −∆ + V , where V is a real-valued potential with sufficient decay at infinity (some smoothness is required for d > 3). Since the perturbed operator may have negative point spectrum one needs to consider e itH P ac (H), where P ac (H) is the orthogonal projection onto the absolutely continuous subspace of L 2 (R d ). Another common assumption is that zero is a regular point of the spectrum of H.
Date: May 5, 2014. 1 Although the L 1 → L ∞ estimates are very well studied in the three dimensional case, there are not many results in dimension two. In [41], Schlag proved that e itH P ac L 1 (R 2 )→L ∞ (R 2 ) |t| −1 (2) under the decay assumption |V | x −3− and the assumption that zero is a regular point of the spectrum. For the case when zero is not regular, see [16]. Yajima, [48], established that the wave operators are bounded on L p (R 2 ) for 1 < p < ∞ if zero is regular. The hypotheses on the potential V were relaxed slightly in [29].
Note that the decay rate in (1) is not integrable at infinity for d = 1, 2. However, in dimensions d = 1 and d = 2, zero is not a regular point of the spectrum of the Laplacian (the constant function is a resonance). Therefore, for the perturbed operator −∆ + V , one may expect to have a faster dispersive decay at infinity if zero is regular. Indeed, in [35,Theorem 7.6], Murata proved that if zero is a regular point of the spectrum, then for |t| > 2 Here w 1 and w 2 are weight functions growing at a polynomial rate at infinity. It is also assumed that the potential decays at a polynomial rate at infinity (for d = 2, it suffices to assume that w 2 (x) = x −3− and |V (x)| x −6− where x := (1 + |x| 2 ) 1 2 ). This type of estimates are very useful in the study of nonlinear asymptotic stability of (multi) solitons in lower dimensions since the dispersive decay rate in time is integrable at infinity (see, e.g., [42,31]). Also see [45,8,36,47] for other applications of weighted dispersive estimates to nonlinear PDEs.
In [43], Schlag extended Murata's result for d = 1 to the L 1 → L ∞ setting (also see [22] for an improved result). In [17], the authors obtained an analogous estimate for d = 2: If zero is a regular point of the spectrum of H, then with w(x) = log 2 (2 + |x|) provided |V (x)| x −3− .
In this paper we extend Schlag's result (2) and our result (3) for the 2d scalar Schrödinger operator to the 2d non self-adjoint matrix Schrödinger operator Such operators appear naturally as linearizations of a nonlinear Schrödinger equation around a standing wave. Dispersive estimates in the context of such linearizations were obtained in [11,40,44,19,13,32,25]. Note that, by Weyl's criterion and the decay assumption on V 1 and V 2 below, the essential spectrum of H is (−∞, −µ] ∪ [µ, ∞). Recall the Pauli spin matrix As in [19], we make the following assumptions: A5) The threshold points ±µ are regular points of the spectrum of H, see Definition 4.3 below.
As it was noted in [19], the first three assumptions are known to hold in the case of the linearized nonlinear Schrödinger equation (NLS) when the linearization is performed about a positive ground state standing wave. Let, for some µ > 0, ψ(t, x) = e itµ φ(x) be a standing wave solution of the NLS for some γ > 0. Here φ is a ground state: It was proven, see for example [46,6], that the ground state solutions exist and further are positive, smooth, radial, exponentially decaying functions, see [19] for further discussion.
Linearizing about this ground state yields the matrix Schrödinger equation with potentials V 1 = (γ + 1)φ 2γ and V 2 = γφ 2γ . Note that V 1 > 0 and V 1 > |V 2 |, which is the same as Assumption A1). Assumption A2) holds because of the exponential decay of φ. Also note The assumption A4) seems to hold for this example in the three-dimensional case as evidenced in the numerical studies [14,33].
The assumption A5) is also standard, since the behavior of the resolvent near the thresholds, ±µ, determine the decay rate (see [43,16] for the scalar case). We do not consider the case when the thresholds ±µ are not regular in this paper.
In an attempt for brevity of this paper, we will try to use the lemmas from the scalar results [43,17] as much as possible. The most important step in the proof of Theorem 1.1 is the analysis of the resolvent around the thresholds ±µ. Once we obtain these expansions, it will be possible to relate and/or reduce the proof to the scalar case for most of the terms.
In addition to being of mathematical interest, we wish to note that such estimates above are of use in the study of non-linear PDEs, particularly the NLS. Much work studying the NLS linearizes the equation about groundstate or standing wave solutions. We note, in particular, [36,21,47,11,31,34,12,13,44,4] and the survey paper [42].
For the spectral theory of the matrix Schrödinger operator, we refer the reader to [19].
Since most of the proofs presented in [19] are independent of dimension, we cite the results without proof. Further spectral theory for the three dimensional case can be found in [3,10]. and Ran(H − z j ) is closed. The zero eigenvalue has finite algebraic multiplicity, i.e., the generalized eigenspace ∪ ∞ k=1 ker(H k ) has finite dimension. In fact, there is a finite m ≥ 1 so that ker(H k ) = ker(H k+1 ) for all k ≥ m.
As in the scalar case, see [23,16] etc., the proofs will hinge on the limiting absorption principle of Agmon [2]. We now state such a result from [19] for (H − z) −1 for |z| > µ.
Define the space It is clear that X * σ = X −σ . The limiting absorption principle of Agmon is formulated below.
We also need the following spectral representation of the solution operator, see [19,Lemma 12].
and the sum is over the discrete spectrum {λ j } j and P λ j is the Riesz projection corresponding to the eigenvalue λ j .
This representation is to be understood in the weak sense. That is for ψ, φ in W 2,2 × In light of this representation, the first claim of Theorem 1.1 follows from the following theorem. Let χ be a smooth cutoff for the interval [−1, 1]. Theorem 2.4. Under the assumptions A1) -A5), we have, for any t ∈ R, The second claim of Theorem 1.1 follows from the following theorem and Theorem 2.4 by a simple interpolation (see [17]) Theorem 2.5. Under the assumptions A1) -A5), we have, for |t| > 2, where w(x) = log 2 (2 + |x|) and 0 < α < β−3 2 .

Properties of the Free Resolvent
For z ∈ (−∞, −µ] ∪ [µ, ∞), the free resolvent is an integral operator where R 0 denoting the scalar free resolvent operators, We first recall some properties of R 0 (z).
To simplify the formulas, we use the notation .. If the derivative bounds hold only for the first k derivatives we write f = O k (g).
Below, using the properties of R 0 listed above, we provide an expansion for the matrix free resolvent, R 0 , around λ = 0 (i.e. z = µ). In the next section, we will obtain analogous expansions for the perturbed resolvent. Similar lemmas were proved in [28,41,17] in the scalar case. The following operators and the function arise naturally in the resolvent expansion (see (18)) Note that Further, for notational convenience we define the matrices We will use the notation K(x, y)M 11 or KM 11 to denote the operator with the convolution We also use the following notation, for a matrix operator M if we write with f a scalar-valued function, we mean that all entries of the matrix M satisfy the bound.
Lemma 3.1. We have the following expansion for the kernel of the free resolvent Here G 0 (x, y) is the kernel of the operator in (26), g ± (λ) is as in (25), and the component functions of E ± 0 satisfy the bounds Proof. The expansion of the scalar free resolvent was derived in [17, Lemma 3.1]. For the free resolvent evaluated at the imaginary argument, the proof easily follows from the properties of the Hankel function listed above.

Resolvent Expansion Around the Threshold µ
It is convenient to write the potential matrix as By assumption A3), we have We employ the symmetric resolvent identity where The key issue in the resolvent expansion around the threshold µ is the invertibility of the operator M ± (λ) for small λ. Using Lemma 3.1 in (29), we can write M ± (λ) as where T is the transfer operator on L 2 × L 2 with the kernel Consider the contribution of the term with g ± (λ) in (31). Recalling the formulas for v 1 and v 2 , we obtain This gives us the following expansion: For λ > 0 with M ± (λ), P and T as above. Then Further, the error term, Here · HS is the Hilbert Schmidt operator norm on Proof. The expansion is proven above. The bounds for E ± 1 = v 2 E ± 0 v 1 follow from the bounds for E ± 0 in Lemma 3.1 and in Corollary 3.2 since We make the following definitions.
Definition 4.2. We say the operator T : Note that Hilbert-Schmidt operators and finite rank operators are absolutely bounded.
Definition 4.3. Let Q = I − P be the projection orthogonal to the span of (a, b) T . We say µ is a regular point of the spectrum H provided that QT Q is invertible on Q(L 2 × L 2 ). We Note that by the resolvent identity Since Q is a projection, it is absolutely bounded. By assumption A3), (26), (24), and (19), Hilbert-Schmidt operator. Therefore, QD 0 Q is a sum of an absolutely bounded operator and an Hilbert-Schmidt operator, which is absolutely bounded.
We also note the following orthogonality property of Q: In the scalar case, see e.g. [28,16], the invertibility of QT Q is related to the absence of distributional L ∞ solutions of Hψ = 0. It is possible to prove a similar relationship for the matrix case. Define S 1 to be the Riesz projection onto the kernel of QT Q as an operator on Q(L 2 × L 2 ).
Proof. Since φ ∈ S 1 (L 2 × L 2 ), we have Qφ = φ. Also using Q = I − P , we obtain Noting that (a, b) T = v 2 (1, 0) T , and that P project onto the span of (a, b) T , we have in the sense of distributions. It thus follows that Thus (H − µI)ψ = 0. Now we prove that ψ ∈ L ∞ × L ∞ . The first bound in (23) and the fact that the entries of φ are in L 2 and the entries of v 2 are in L ∞ ∩ L 2 imply that the second entry of ψ is bounded. We note that the first entry of ψ is (y), b(y))φ(y)dy.
Since P φ = 0, we can rewrite this as The boundedness of this integral follows immediately from the bound We refer the reader to Lemma 5.1 of [16] for more details.
It is also possible to prove a converse statement relating certain L ∞ × L ∞ solutions of (H − µI)ψ = 0 to the non-invertibility of QT Q as in Lemma 5.1 and Lemma 5.2 of [16] (also see [28]). We don't include these statements and proofs since they can be obtained from the scalar case as above.
The regularity assumption A5) allows us to invert the operators M ± (λ) for small λ as follows: Lemma 4.5. Let 0 < α < 1. Suppose that µ is a regular point of the spectrum of H. Then for sufficiently small λ 1 > 0, the operators M ± (λ) are invertible for all 0 < λ < λ 1 as bounded operators on L 2 × L 2 . Further, one has is a finite-rank operator with real-valued kernel. Further, the error term satisfies the bounds Proof. We give a proof for the operator M + (λ), the expansion for M − (λ) is similar. We drop the subscript '+' from the formulas. Using Lemma 4.1 with respect to the decomposition Denote the matrix component of the above equation by A(λ) = {a ij (λ)} 2 i,j=1 . Since QT Q is invertible by assumption, by the Fehsbach formula invertibility of A(λ) hinges upon the existence of d = (a 11 − a 12 a −1 with h(λ) = g(λ) + T r(P T P − P T QD 0 QT P ) = g(λ) + c, where c ∈ R as the kernels of T , QD 0 Q and v 1 , v 2 are real-valued. The invertibility of this operator on P L 2 for small λ follows from (25). Thus, by the Fehsbach formula, Note that S has finite rank. This and the absolute boundedness of QD 0 Q imply that A −1 is absolutely bounded. To avoid confusion, we will write S as a sum of four components rather than in a matrix form.
Finally, we write Therefore, by a Neumann series expansion, we have The error bounds follow in light of the bounds for E 1 (λ) in Lemma 4.1 and the fact that, as an absolutely bound operator on L 2 , |A −1 (λ)| 1, |∂ λ A −1 (λ)| λ −1 , and (for 0 < λ < η < λ 1 ) In the Lipschitz estimate, the factor λ − 1 2 −α arises from the case when the derivative hits A −1 (λ). We finish this section by noting that, using Lemma 4.5 in (29), one gets

Proof of Theorem 2.5 for energies close to µ
Let χ be a smooth cut-off for [0, λ 1 ], where λ 1 is sufficiently small so that the expansions in the previous section are valid. We have In the proof of this theorem we need the following Lemmas, which are standard and their proofs can be found in [17].
We start with the contribution of the free resolvent in (39) to (40). Recall (22): Therefore, the following proposition follows from the corresponding bound for the scalar free resolvent, Proposition 4.3 in [17]. The proof uses Lemma 5.2 with E(λ) = i 2 J 0 (λ|x − y|). . Now consider the contribution of the term involving (h ± ) −1 S in (39) to (40). Using Lemma 3.1 we have Using the orthogonality property (34) and the definition (36) of S, we obtain Also recall that h ± (λ) = − a 2 + b 2 L 1 g ± (λ) + c, c ∈ R, and (from (25)) that g + (λ) = − 1 2π log λ + z with g − (λ) = g + (λ) and z − z = i 2 . Therefore we can write where a i , b i , c i are real. Using this the following proposition will follow from the bounds obtained in [17].
Proof. First consider the contribution of the first term in (43): where the equality follows from Lemma 5.2.
The contribution of the second summand in (43) can be handled using the bound which is essentially Lemma 4.5 in [17] and it is proved by using Lemma 5.2.
The contribution of the third (similarly the fourth) summand in (43) can also be handled using (44) along with the bound The last inequality follows from the absolute boundedness of S, the bound where k(x, x 1 ) = 1 + log − |x − x 1 | + log + |x 1 |, and We now consider the error term, E ± 2 (λ). Note that Using this, the absolute boundedness of S, the decay bounds |a(x)| + |b(x)| x − 3 2 −α− , the bound (46), and the bounds in Lemma 3.1 and Corollary 3.2 as in the proof of (45), we obtain (for 0 < λ < η λ < λ 1 ) Therefore the contribution of the error term is controlled by using Lemma 5.3 as in Lemma 4.6 of [17].
Now we consider the contribution of the term QD 0 Q in (39) to (40).
Proof. Using Lemma 3.1 and (34) we have Since QD 0 Q is absolutely bounded, E 3 satisfies the same bounds that we obtained for the error term E 2 above.
Finally the contribution of E ± (λ) in (39) to (40) can be handled exactly as in Proposition 4.9 of [17]: This finishes the proof of Theorem 5.1.

Proof of Theorem 2.5 for energies separated from the thresholds
In this section we complete the proof of Theorem 2.5 by proving Theorem 6.1. Under the assumptions of Theorem 2.5, we have for t > 2 We employ the resolvent expansion We first note that the contribution of the term m = 0 can be bounded by x integrating by parts twice (there are no boundary terms because of the cutoff). We approach the energies separated from zero differently from the small energies. In particular, we won't use Lemma 3.1, but instead employ a component-wise approach. Recall that For the case m > 0 we won't make use of any cancelation between '±' terms. Thus, we will only consider R − 0 , and drop the '±' signs. Using (16), (17), (18), and (19) we write As such we can write It is easy to see that Therefore, we can use the right hand side of (50) for each component of R 0 . The argument for the high energy now proceeds as in Section 5 of [17]. We provide a sketch of the details for the convenience of the reader.
We first control the contribution of the finite born series in (49) for m > 0. Note that the contribution of the mth term of (49) to the integral in (48) can be written as a finite sum of integrals of the form .., m, m + 1}, and Here, with a slight abuse of notation, W (x) denotes either ±V 1 (x) or ±V 2 (x) (since we only use the decay assumption and do not rely on cancelations, this shouldn't create any confusion).
Using the representation (51), and the discussion following it, we note the following bounds hold on λ > λ 1 > 0, Thus, for σ > 1 2 + k, Once again, we estimate the '±' terms separately and omit the '±' signs.
We write the contribution of the remainder term in (49) to (48) as where Using (10), (9), and (60) (provided that M ≥ 2) we see that This requires that |V (x)| x −3− . One can see that the requirement on the decay rate of the potential arises when, for instance, both λ derivatives act on one resolvent, this twice differentiated resolvent operator maps X 5 2 + → X − 5 2 − by (10), or is in X − 5 2 − by (60). The potential then needs to map X − 5 2 − → X 1 2 + for the next application of the limiting absorption principle. This is satisfied if |V (x)| x −3− .
The required bound now follows by integrating by parts twice:

Proof of Theorem 2.4 for energies close to µ
In this section we will prove the following As in Section 5, we will use lemmas from the proof for the scalar case given in [41].
Using (39), we write First note that the contribution of the free resolvent terms in (65) to (64) immediately boils down to the scalar case because of (22).
Note that using (20), with Consider the contribution of '+' terms in (65) with QD 0 Q: The bound for the first term is in [41,Lemma 16], since M 11 v 1 QD 0 Qv 2 M 11 have the same cancellation (compare (34) above with (44) in [41]), and mapping properties as vQD 0 Qv in [41], provided that |a(x)| + |b(x)| x −3/2− . The last term is killed by the '+' and '-' cancellation. For the second and third terms, we note that the '+' and '-' cancellation says we need only consider The following propositions finishes the proof of Theorem 7.1 for the contribution of QD 0 Q terms in (65).
The same bound holds for the contribution of The following variation of stationary phase will be useful in the proof. See Lemma 2 in [41].
Proof of Proposition 7.2. Recall that from (23) we have Also recall that The contribution of ρ is: After an integration by parts, we can bound the λ integral above by The last equality follows from the bounds on R 2 and ∂ λ R 2 , and by noting that This bound suffices for the contribution of ρ since QD 0 Q is absolutely bounded and v 2 (y 1 )(1 + log − |y − y 1 |) L 2 y 1 1.
For the remaining terms in (66), we only consider the case of ω − and t > 0 (the bound for ω + follows from an integration by parts since the phase has no critical point). The It suffices to prove that this integral is The phase, φ = λ 2 − λ|x − x 1 |/t, has a critical point at λ 0 = |x − x 1 |/2t. Let By Lemma 7.3 we estimate the λ integral by Proof of Proposition 7.4. It suffices to prove that the λ integral is as in the proof of Proposition 7.2.
Noting that , and the bounds (23) on R 2 and its derivative, it suffices to prove that This follows by a single integration by parts.
Proof of Proposition 7.5. Using (21) we have Noting the bounds we see that the proof for the contribution of the term containing J 0 follows from the proof of Proposition 7.2, since this term satisfies the same bounds that J 0 does.
The bound for the contribution of the error term, E ± , in (65) to (64) follows from [41, Lemma 18] since E ± satisfies the bounds that the lemma requires and also R 0 satisfies the same bounds that R 0 satisfies.

Proof of Theorem 2.4 for energies separated from the thresholds
We note [41, Lemma 3], which we modify slightly to match the notation we have employed throughout this paper. We define (1 + log − |x − y|) 2 |W (y)| dy. ∞ 0 λe i(tλ 2 ±λ j∈J |x j+1 −x j |) χ(λ)χ(λ/L) In the proof of Theorem 2.5 for energies separated from the threshold, we encountered this integral in (52). By the discussion in that proof the finite terms of the Born series in (49) can be written as a finite sum of terms in this form where W is ±V 1 or ±V 2 . We note that by the decay assumptions on V 1 and V 2 , we always have W K < ∞. Therefore Lemma 8.1 suffices to handle the contribution of the finite terms of the Born series, (49).
It remains to consider the contribution of the tail of the series (49), see (61) and (62).
Note that for λ|x − x 1 | > 1, the scalar free resolvent R 0 (λ 2 )(x, x 1 ) has the oscillatory term e ±iλ|x−x 1 | . If a λ derivative hits one of the free resolvents at the edges the oscillatory term produces |x − x 1 | which can not be bounded uniformly in x. This was not an issue in the weighted case since we are able to allow some growth in x and y.
Note that oscillatory part changes the phase in the integral and G ±,x (λ) and its derivatives does not grow in x since differentiating G ±,x (λ) in λ produces |x−x 1 |−|x| = O(|x 1 |) (which can be killed by the decay assumption on the potential). In [41,Proposition 4], this implies the required bound by an application of stationary phase and by using limiting absorption principle.