SCATTERING THEORY FOR SEMILINEAR SCHR¨ODINGER EQUATIONS WITH AN INVERSE-SQUARE POTENTIAL VIA ENERGY METHODS

. We solve the scattering problems for nonlinear Schr¨odinger equations with an inverse-square potential by applying the energy methods. The methods are optimized to the abstract semilinear Schr¨odinger evolution equations with nonautonomous terms.


1.
Introduction. In this article we consider the scattering problems for following nonlinear Schrödinger equations with an inverse-square potential (1.1) g(u) is a nonlinear term, for example, g(u) = |u| p−1 u (pure power nonlinearity; local type) and g(u) = u (|x| −γ * |u| 2 ) (usual Hartree nonlinearity; nonlocal type). The contraction methods are the simple and useful methods for showing the global existence of nonlinear Schrödinger equations. But Okazawa-Suzuki-Yokota [15] showed the global unique existence with unsatisfactory conditions for (NLS) a with local nonlinearities.
Now we study the properties of global solutions to (NLS) a . Especially, we consider the scattering problems. These are mainly concerned with two properties: (sp1) Asymptotic behavior of global solution; (sp2) The existence of wave operators.
These are the problems for the relation between the global solution u to (NLS) a and the limit lim t→±∞ exp(itP a )u(t) = u ± .

Then (1.4) is converted into
, v(0) = v + := i −N/2 exp(−i∆)e i|x| 2 /4 exp(−i∆)u + on Σ 1 (R N ). (1.6) This is nothing but a nonautonomous Cauchy problem. Hayashi-Tsutsumi [10] solved the final-valued problems by practicing the contraction methods to the associated integral equations of (1.6). Later, Hayashi-Ozawa [9] solved the problems (1.4) directly via the contraction methods. Latterly, Suzuki [23] studied the scattering problems for (NLS) a with the usual Hartree type for a > a(N ). Especially in L 2 -supercritical case (2 < γ < min{4, N }), it occurs on an unsatisfactory restriction of a. If we solve the scattering problems (sp1) and (sp2) for (NLS) a in weighted energy class (Σ 1 (R N ) or X 1 (R N ) ∩ D(|x|)) with general a, we need to apply the pseudo-conformal transform (1.5) and solve the nonautonomous Cauchy problems. The transform (1.5) still works, however, the contraction methods are not suitable way to solve. Thus we need another method like the energy methods which established by Okazawa-Suzuki-Yokota [16] for the self-excited systems and which applied Propositions 1.2 and 1.3. Our goals are to establish new methods and to apply the scattering problems in this paper. Recently, the scattering problems for (NLS) a with the pure power type are partially solved in H 1 (R N ), not in Σ 1 (R N ). Zhang-Zheng [26] proved the Morawetz inequality and constructed the wave operators in a ≥ 0 if N = 3, and in a > a(N ) On the one hand, Lu-Miao-Murphy [11] considered the profile decomposition and constructed the wave operators in a > a(N ) and 7/3 < p ≤ 3 if N = 3, and in Note that both of studies applied the contraction methods without the pseudo-conformal transform (1.5) and excluded the case a = a(N ) and near of a(N ) for higher dimension N . Our approach in this paper guarantees that the topology of limits is stronger (not H 1 (R N ) = D((1 + P a ) 1/2 ) (a > a(N )) but D((1 + P a ) 1/2 ) ∩ D(|x|)) but the restriction of a and p is weaker, and more suitable for scattering problems (not as in [26,11] but a ≥ a(N ) and 1 + 4/N ≤ p < 1 + 4/(N − 2)). This paper is divided into four sections. In Section 2 we establish new methods (the energy methods) for nonautonomous semilinear Schrödinger equations. In Section 3 we consider the scattering problems for (NLS) a ; especially, nonlocal nonlinearity. In Section 4 we have some comments as concluding remarks.

2.
Energy methods for abstract nonautonomous semilinear Schrödinger equations. In this section we consider following Cauchy problems for abstract nonautonomous semilinear Schrödinger equations   where S is nonnegative and selfadjoint in the (complex) Hilbert space X. X S := D((1 + S) 1/2 ) and X * S := D((1 + S) −1/2 ) is the dual space of X S . Here we see the triplet X S ⊂ X = X * ⊂ X * S . We give a typical example. Let S = −∆ in X = L 2 (R N ). Then we see that X S = H 1 (R N ) and X * S = H −1 (R N ). Now we present new methods. First we give assumption for the nonlinearity g : S . For the simple notation we use B M := {u ∈ X S ; u X S ≤ M }. (A1) Existence of energy functional: there exists G ∈ C([−T, T ] × X S ; R) whose real Fréchet differential d X S G(t, u) is exactly g(t, u), that is, given u ∈ X S and t ∈ [−T, T ], for every ε > 0 there exists δ = δ(u, ε) > 0 such that (A4) Hölder-like continuity of G: given M > 0, for all δ > 0 there exists a constant C 1,δ (M ) > 0 such that (A5) Hölder type continuity of G t : G(t, u) is partially differentiable in t for every u ∈ X S . Moreover, there exists ϕ ∈ L 1 (−T, T ), and for any M > 0 and δ > 0 there exists a constant C 2,δ (M ) > 0 such that (A6) Gauge condition: Re g(t, u), i u X * S ,X S = 0 ∀ t ∈ [−T, T ], ∀ u ∈ X S ; (A7) Closedness condition: let I ⊂ (−T, T ) be an open interval. Assume that {w n } n is any bounded sequence in L ∞ (−T, T ; X S ) such that w n (t) → w(t) (n → ∞) weakly in X S a.a. t ∈ I, g(t, w n ) → f (n → ∞) weakly * in L ∞ (I; X * S ). (2.1) Then Here Theorem 2.1. Assume that (A1)-(A7) and u 0 ∈ X S . Then there exist T 0 ∈ (0, T ] (dependent on u 0 X S ) and u ∈ C w ([−T 0 , T 0 ]; X S ) ∩ W 1,∞ (−T 0 , T 0 ; X * S ) such that u is a solution to (ACP). The solution satisfies also the pseudo-conservation laws:

TOSHIYUKI SUZUKI
Thus f (t) = h 0 (t) f 0 (t). Here (G4) and (G5) yield Re f 0 (t), i w(t) X * S ,X S = 0 a.a. t ∈ I. This implies the former half of (A7): Next assume further that w n (t) → w(t) (n → ∞) strongly in X a.a. t ∈ I. (G5) implies f 0 (t) = g 0 (w(t)) and hence f (t) = h 0 (t) g 0 (w(t)). Thus (A7) has been fully proved. Moreover, the vector-valued Sobolev space W 1,p (I; Y ) can be also defined: Here u denotes the weak derivative of u respect to time variable t ∈ I. It is wellknown that W 1,p (I; Y ) ⊂ C(Ī; Y ) (1 ≤ p ≤ ∞) is continuous (see [4,Corollary 1.4.36]). Now let A be a linear and maximal monotone operator in X, that is, R(1+A) = X and Re u, Au X ≥ 0. Then −A generates contraction C 0 -semigroups {e −tA |t ≥ 0} ⊂ B(X), the family of bounded linear operators on X. Now we solve (2.10) We use B M,X := {u ∈ X; u X ≤ M } for simple notation. Assume that g 0 satisfies (H1) Lipschitz continuity of g 0 in u: Next let S be a nonnegative and selfadjoint operator in the complex-valued Hilbert space X. Then ±iS are maximal monotone in X. Hence we can solve the following semilinear Schrödinger evolution equations: (H5) Gauge type condition: In a way similar to Cazenave [3, Theorem 3.3.1] for the self-excited case we can show the following: such that u is a global solution to (2.11). Moreover, the energy equalities are available.
Local existence (Proof of Theorem 2.1). First we consider the approximated problems of (ACP): Now we divide the proof of Theorem 2.1 into 5 steps. The story is a similar way to Okazawa-Suzuki-Yokota [16]. Here it is unknown that P a is m-accretive in L p (R N ) (p = 2 and a is near to a(N )). Thus the Cazenave approach [3, Theorem 3.3.5] cannot be applicable.
Step 1. Solve (ACP) ε globally in time t; Step 2. Evaluate S 1/2 u ε (t) X uniformly in t and ε; Step 3. Confirm the weak convergence of (ACP) ε to (ACP); Step 4. Derive the charge conservation and make a solution; Step 5. Derive the energy pseudo-conservation.
Proof of Theorem 2.1. We divide the proof into 5 steps as above.
Hence we have In the last equality we have used (2.13). Since (1 + ε(j)S) −1 u 0 → u 0 (j → ∞) strongly in X S and (2.27), we see that Therefore we conclude (2.5). This completes the proof of Theorem 2.1.

2.3.
Remarks on the global existence.
Theorem 2.7. Assume (A1)-(A7) and (A8) there exists ε > 0 such that Hence the Gronwall lemma implies that the uniform boundedness of S 1/2 u(t) 2 X : Extending the interval step by step, we get the global solution is unique, then the pseudo-conservation inequality (2.5) is exact equality. Moreover, we have Hence we can confirm that u belongs also to C([−T 0 , T 0 ]; X S ) ∩ C 1 ([−T 0 , T 0 ]; X * S ).

TOSHIYUKI SUZUKI
3. Scattering problems for (NLS) a with nonlocal term. Now we consider the scattering problems for the Hartree type equations where K is the integral operator whose kernel is k: If k(x, y) = W (x − y) (convolution type), then (HE) a is a Hartree equation. Especially, if W (x) = |x| −γ (k(x, y) = |x−y| −γ ), then (HE) a is a usual Hartree equation and analyzed by many people. In our setting, we can also consider general kernels, for example, k(x, y) = W (x + y) and k(x, y) = U (x)W (x − y)U (y). Suzuki [20] proved the global existence of solutions to (HE) a under the generalized kernels for a > a(N ). Thus we have the following results for global existence. Here we denote (K2) k can be divided into several symmetric kernels k j so that (K3) k − (x, y) := max{0, −k(x, y)} can be divided into several symmetric kernels k −j so that Then for every u 0 ∈ D there exists a unique global weak solution u to (HE) a . Moreover, u belongs to C(R; D) ∩ C 1 (R; D * ) and satisfies conservation laws where the "energy" E is defined as If assume further that u 0 ∈ Σ, then u belongs also to C(R; Σ).
We can also obtain the virial identity for (HE) a (see [21,24]). (K4) k(x, y) := 2k(x, y) + x · ∇ x k(x, y) + y · ∇ y k(x, y) can be divided into several symmetric kernels k j so that Then for every u 0 ∈ Σ the local weak solution u ∈ C([−T 1 , T 2 ]; Σ)∩C 1 ([−T 1 , T 2 ]; D * ) satisfies the virial identity Since the global well-posedness for (HE) a is well-studied, we consider the scattering problems for (HE) a . First we construct wave operators for (HE) a , that is, we solve Here applying the pseudo-conformal transform (1.5), (FV) a can be converted into the following Cauchy problems for nonautonomous semilinear Schrödinger equations.
where K 1/t is the integral operator whose kernel is t −2 k(t −1 x, t −1 y). Now we explain about the pseudo-conformal transform more precisely. Simple calculations yield that Thus we can rewrite the transformation (1.5) as Operating exp(−i(1 − t)P a ) we see Letting t → ∞ we conclude exp(−iP a )u + = i −N/2 M 1 exp(−iP a )v(0, x). Thus if we solve (NV) a , we can also solve (FV) a . Note that we need to set Σ not as D (usual energy space) but as D ∩ D(|x|) (weighted energy space) so that the transform C works well. In fact, we have following relations (K4a) k can be divided into several symmetric kernels so that
Next we consider the asymptotic free for (HE) a . Note that the existence the limit in L 2 (R N ) is proved by [23, Proposition 3.1] for a > a(N ) and by [24,Theorem 4.2] for a = a(N ) without the unsatisfactory restriction for γ ∈ (1, min{N, 4}). Here we see the discrepancy of the topology between the wave operators and the scattering states. This is resolved by applying the energy methods.
where the inclusion is continuous and dense. In particular, we have Another triplet is available. By virtue of the usual Hardy inequality (1.2), the energy class D((1 + P a ) 1/2 ) coincides with H 1 (R N ) for a > a(N ). But the critical case a = a(N ) implies that the energy class does not coincide with H 1 (R N ). So we define X 1 (R N ) := D((1 + P a(N ) ) 1/2 ). We can see the triplet as follows: Moreover, Suzuki [22, Theorem 3.2] ensures for 0 < s < 1 and N ≥ 3 that (Γ denotes the Gamma function). This implies that H 1 (R N ) ⊂ X 1 (R N ) ⊂ H s (R N ) (0 < s < 1). In particular, we have We see the Sobolev embedding inequalities in both cases: Note that the Rellich compactness lemma is available. We apply the so-called Strichartz estimates for the verifying the uniqueness of (HE) a . Here the estimates established Burq-Planchon-Stalker-Tahvildar-Zadeh [2, Theorem 3] for a > a(N ) and Suzuki [22,Proposition 4.8] for a = a(N ).
Then the mapping K is bounded from Hence we see that if k ∈ L β x (L α y ) is symmetric (k(y, x) = k(x, y)), On the one hand, we can prove the boundedness of integral operator whose kernel is the type of U (x)W (x ± y)U (y). It follows from the usual Hölder and Young inequalities.

3.2.
Wave operators for Hartree type equations with an inverse-square potential. Now we prove Theorem 3.3. Proof.
Next we show that f (t) = g(t, w(t)) by assuming further that w n (t) → w(t) (n → ∞) in L 2 (R N ) a.a. t ∈ I. Let M := sup n w n L ∞ (I;D) . It follows from (3.7) and (3.4) that Thus we see that g(t, w n (t)) → g(t, w(t)) (n → ∞) strongly in L ∞ (I; D * ) and (A7) is verified. Thus we can conclude the local existence of (HE) a by Theorem 2.1.
Remark 3.11. Assume the same as in Theorem 3.3. Then we can define the wave operator W + : u + → u 0 in Σ. We can also construct W − : u − → u 0 so that lim t→−∞ exp (−itP a )u(t) = u − strongly in Σ. (3.16) By virtue of u K(|u| 2 ) = u K(|u| 2 ), we see

3.3.
Asymptotic free for Hartree type equations with an inverse-square potential. Next we prove Theorem 3.4.
Here v satisfies the energy conservation: G t (s, v(s)) ds.
If U ∈ L p (R N ) and W ∈ L q (R N ), then k(x, y) := U (x)W (x ± y)U (y) belongs to L p x (L pq/(p+q) y ). Thus we can specialize the kernel k as the form U (x)W (x ± y)U (y).
(L1) U and W are real-valued and nonnegative. If we select W (x − y), we additionally impose that W is even: We can solve the scattering problems for (HE) a with a specified integral kernel. The proof is completed in ways similar to Theorems 3.3 and 3.4 with replacing (3.7) and (3.8) by (3.9) and (3.10), respectively.  (ii) Assume (L1)-(L4). Then for every global solution u ∈ C(R; Σ) ∩ C 1 (R; D * ) to (HE) a there exists u + ∈ Σ such that (3.3).

4.1.
Remarks on the energy methods: Generalization for systems. We can generalize the energy methods (Theorem 2.1) for the system. Let B : X * S → X * S be a bounded linear operator with the following conditions: • BSu = SBu for u ∈ X S ; • B is bounded and symmetric operator in X; • B is coercive in X: there exists ε > 0 such that Re Bu, u X ≥ ε u 2 X . By using B, (H5) is replaced with (H5a): Re g 0 (t, u), i Bu X = 0 ∀ t ∈ [−T, T ], ∀ u ∈ X.
Applying this lemma, we can generalize Theorem 2.1.
Then for every u 0 ∈ H 1 (R N ) there uniquely exists a global solution u ∈ C([0, T ]; H 1 (R N )) ∩ C 1 ([0, T ]; H −1 (R N )) to (4.1). Moreover, u satisfies for any s, t ∈ [0, T ] This follows from Theorems 2.1 and 2.7. Since (4.1) is a linear problem, we can see the uniqueness in a simple way. Here Corollary 4.2 is the energy-class version of Yajima [25]. Moreover, we can define the family of linear operators {U(t, s)|s, t ∈ [0, T ]} ⊂ B(H 1 (R N )) such that u(t) = U(t, s)u s satisfies i u t = −∆u + V (t)u in (0, T ), u(s) = u s ∈ H 1 (R N ).