# Journal of Nonlinear Mathematical Physics

Volume 25, Issue 4, July 2018, Pages 558 - 588

# Singular Hartree equation in fractional perturbed Sobolev spaces

Authors
Alessandro Michelangeli
SISSA – International School for Advanced Studies, Via Bonomea 265, 34136 Trieste (Italy),alemiche@sissa.it
Alessandro Olgiati
SISSA – International School for Advanced Studies, Via Bonomea 265, 34136 Trieste (Italy),aolgiati@sissa.it
Raffaele Scandone
SISSA – International School for Advanced Studies, Via Bonomea 265, 34136 Trieste (Italy),rscandone@sissa.it
Received 17 January 2018, Accepted 30 April 2018, Available Online 6 January 2021.
DOI
10.1080/14029251.2018.1503423How to use a DOI?
Keywords
Point interactions; Singular perturbations of the Laplacian; Regular and singular Hartree equation; Fractional singular Sobolev spaces; Strichartz estimates for point interaction Hamiltonians Fractional Leibniz rule; Kato-Ponce commutator estimates
Abstract

We establish the local and global theory for the Cauchy problem of the singular Hartree equation in three dimensions, that is, the modification of the non-linear Schrödinger equation with Hartree non-linearity, where the linear part is now given by the Hamiltonian of point interaction. The latter is a singular, self-adjoint perturbation of the free Laplacian, modelling a contact interaction at a fixed point. The resulting non-linear equation is the typical effective equation for the dynamics of condensed Bose gases with fixed point-like impurities. We control the local solution theory in the perturbed Sobolev spaces of fractional order between the mass space and the operator domain. We then control the global solution theory both in the mass and in the energy space.

Open Access

## 1. The singular Hartree equation. Main results

The Hartree equation in d dimension is the well-known semi-linear Schrödinger equation with cubic convolutive non-linearity of the form

itu=Δu+Vu+(w*|u|2)u (1.1)
in the complex-valued unknown uu(x, t), t ∈ ℝ, x ∈ ℝd, for given measurable functions V, w : ℝd → ℝ.

Among the several contexts of relevance of (1.1), one is surely the quantum dynamics of large Bose gases, where particles are subject to an external potential V and interact through a two-body potential w. In this case (1.1) emerges as the effective evolution equation, rigorously in the limit of infinitely many particles, of a many-body initial state that is scarcely correlated, say, Ψ(x1, …, xN) ~ u0(x1)…u0(xN), whose evolution can be proved to retain the approximate form Ψ(x1, …, xN; t) ~ u(x1, t)…u(xN, t), for some one-body orbital uL2 (ℝd) that solves the Hartree equation (1.1) with initial condition u(x, 0) = u0(x). The precise meaning of the control of the many-body wave function is in the sense of one-body reduced density matrices. The limit N → + ∞ is taken with a suitable rescaling prescription of the many-body Hamiltonian, so as to make the limit non-trivial. In the mean field scaling, that models particles paired by an interaction of long range and weak magnitude, the interaction term in the Hamiltonian has the form N–1j<k w(xjxk), and when applied to a wave function of the approximate form u(x1) · · · u(xN) it generates indeed the typical self-interaction term (w* |u|2)u of (1.1). This scenario is today controlled in a virtually complete class of cases, ranging from bounded to locally singular potentials w, and through a multitude of techniques to control the limit (see, e.g., [9, Chapter 2 ] and the references therein).

The Cauchy problem for (1.1) is extensively studied and understood too, including its local and global well-posedness and its scattering – for the vast literature on the subject, we refer to the monograph [12], as well as to the recent work [23]. Two natural conserved quantities for (1.1) are the mass and (as long as w(x) = w(−x)), the energy, namely,

(u)=d|u|2dx(u)=12d(|u|2+V|u|2)dx+14d×dw(xy)|u(x)|2|u(y)|2dxdy.

The natural energy space is therefore H1(ℝd), and the equation is energy sub-critical for wL1(ℝd) + L(ℝd) and mass sub-critical for wLq(ℝd) + L(ℝd), for qmax{1,d2} (q > 1 if d = 2).

In fact, irrespectively of the technique to derive the Hartree equation from the many-body linear Schrödinger equation (hierarchy of marginals, Fock space of fluctuations, counting of the condensate particles, and others), one fundamental requirement is that at least for the time interval in which the limit N → +∞ is monitored the Hartree equation itself is well-posed, which makes the understanding of the effective Cauchy problem an essential pre-requisite for the derivation from the many-body quantum dynamics.

In the quantum interpretation discussed above, the external potential V can be regarded as a confining potential or also as a local inhomogeneity of the spatial background where particles are localised in, depending on the model. In general, as long as V is locally sufficiently regular, this term is harmless both in the Cauchy problem associated to (1.1) and in its rigorous derivation from the many-body Schrödinger dynamics. This, in particular, allows one to model local inhomogeneities such as ‘bump’ -like impurities, but genuine ‘delta’-like impurities localised at some fixed points X1, … XM ∈ ℝ3 certainly escape this picture.

In this work we are indeed concerned with a so-called ‘delta-like singular’ version of the ordinary Hartree equation (1.1) where formally the local impurity V(x) = 𝒱 (xX) around the point X, for some locally regular potential 𝒱 is replaced by V(x) = δ(xX) and more concretely we study the Cauchy problem for an equation of the form

itu=Δu+δ(xX)u+(w*|u|2)u. (1.2)

There will be no substantial loss of generality, in all the following discussion, if we take one point centre X only, instead of X1, … , XM ∈ (ℝ3), and if we set X = 0 which we will do throughout.

The precise meaning in which the linear part in the r.h.s. of (1.2) has to be understood is the ‘singular Hamiltonian of point interaction’, that is, a singular perturbation of the negative Laplacian −Δ which, consistently with the interpretation of a local impurity that is so singular as to be supported only at one point, is a self-adjoint extension on L2(ℝd) of the symmetric operator Δ|C0(d) , and therefore acts precisely as −Δ on H2 -functions supported away from the origin. In fact, Δ|C0(d) is already essentially self-adjoint when d ⩾ 4, with operator closure given by the self-adjoint −Δ with domain H2(ℝd), therefore it only makes sense to consider the singular Hartree equation for d ∈ {1,2,3}, and the higher the co-dimension of the point where the singular interaction is supported, the more difficult the problem.

Our setting in this work will be with d = 3. We shall comment later on analogous results in the simpler case d = 1. In three dimensions one has the following standard construction, which we recall, for example, from [5, Chapter I. 1] and [24, Section 3].

The class of self-adjoint extensions in L2(ℝ3) of the positive and densely defined symmetric operator Δ|C0(3\{0}) is a one-parameter family of operators −Δα, α ∈ (−∞, +∞], defined by

𝒟(Δα)={ψL2(3)ψ=φλ+φλ(0)α+λ4πGλwithφλH2(3)}(Δα+λ)ψ=(Δ+λ)φλ, (1.3)
where λ > 0 is an arbitrarily fixed constant and
Gλ(x):=eλ|x|4π|x| (1.4)
is the Green function for the Laplacian, that is, the distributional solution to (−Δ + λ)Gλ = δ in 𝒟′(ℝ3).

The quadratic form of −Δα is given by

𝒟[Δα]=H1(3)+span{Gλ}(Δα)[φλ+κλGλ]=λφλ+κλGλ22+φλ22+λφλ22+(α+λ4π)|κλ|2. (1.5)

The above decompositions of a generic ψ𝒟(− Δα) or ψ𝒟[− Δα] are unique and are valid for every chosen λ. The extension −Δα = is the Friedrichs extension and is precisely the self-adjoint −Δ on L2(ℝ3) with domain H2(ℝ3).

The operator −Δα is reduced with respect to the canonical decomposition

L2(3)L=02(3)=1L2(3)
in terms of subspaces L2(3) of definite angular symmetry, and it is a non-trivial modification of the negative Laplacian in the spherically symmetric sector only, i.e.,
(Δα)|𝒟(Δα)L2(3)=(Δ)|H2(3),0 (1.6)

Each ψ𝒟(− Δα) satisfies the short range asymptotics

ψ(x)=cψ(1|x|1a)+o(1)asx0,a:=(4πα)1, (1.7)
or also, in momentum space,
p3|p|<Rψ^(p)dp=dψ(R+2π2α)+o(1)asR+, (1.8)
for some cψ, dψ ∈ ℂ. Equations (1.7) and (1.8) are referred to as, respectively, the Bethe-Peierls contact condition [11] and the Ter-Marryrosyan-skornyakov condition [28], and express a boundary condition for the wave function in the vicinity of the origin, which is indeed the characteristic behaviour of the low-energy bound state for a Schrödinger operator −Δ + V where V has almost zero support and s-wave scattering length a = −(4πα)−1. Thus, −Δα is recognised to be the Hamiltonian of point interaction in the s-wave channel, localised at x = 0, and with inverse scattering length α in suitable units.

The spectrum of −Δα is given by

σess(Δα)=σac(Δα)=[0,+),σsc(Δα)=,σp(Δα)={ifα[0,+]{(4πα)2}ifα(,0). (1.9)

The negative eigenvalue −(4πα)2, when it exists, is simple and the corresponding eigenfunction is |x|−1e−4π|α||x|. Thus, α ⩾ 0 corresponds to a non-confining, ‘repulsive’ contact interaction.

We can now make (1.2) unambiguous and therefore consider the singular Hartree equation

itu=Δαu+(w*|u|2)u. (1.10)

In order to avoid non-essential additional discussions, we restrict ourselves once and for all to positive α’s. In fact, −Δα is semi-bounded from below for every α ∈ (−∞, +∞], as seen in (1.9) above, thus shifting it up by a suitable constant one ends up with studying a modification of (1.10) with a trivial linear term that does not affect the solution theory of the equation.

Owing to the self-adjointness of −Δα and to its positivity for α ⩾ 0, the ‘singular (or perturbed) Schrödinger propagator’ teitΔα leaves the domain of each power of −Δα invariant. In complete analogy to the non-perturbed case, where the free Schrödinger propagator teitΔ leaves the Sobolev space Hs(ℝ3) = 𝒟(−Δ)s/2 invariant, and the solution theory for the ordinary Hartree equation is made in Hs(ℝ3), including the energy space H1(ℝ3), now the meaningful spaces of solutions where to settle the Cauchy problem for (1.10) are of the type H˜αs(3) , the ‘singular Sobolev space’ of order s, namely the Hilbert space

H˜αs(3):=𝒟((Δα)s/2) (1.11)
equipped with the ‘fractional singular Sobolev norm’
ψH˜αs:=(1Δα)s/2ψ2 (1.12)

It is worth remarking that whereas the kernel of the propagator teitΔα is known since long [4,27], the characterisation of the singular fractional Sobolev space H˜αs(3) is only a recent achievement [17], and we shall review it in Section 2.

In view of the preceding discussion, we consider the Cauchy problem

{iiu=Δau+(w*|u|2)uu(0)=fH˜αs(3). (1.13)

We are going to discuss its local solution theory both in a regime of low (i.e., s[0,12) ), intermediate (i.e., s(12,32) ), and high (i.e., s(32,2] ) regularity. Then, exploiting the conservation of the mass and the energy, we are going to obtain a global theory in the mass space (s = 0) and the energy space (s = 1).

We deal with strong H˜αs -solutions of the problem (1.13), meaning, functions u ∈ ( 𝒞(I,H˜αs(𝕉3)) for some interval I ⊆ ℝ with I ∋ 0, which are fixed points for the solution map

Φ(u)(t):=eitΔafi0tei(tτ)Δa(w*|u(τ)|2)u(τ)dτ (1.14)

Let us recall the notion of local and global well-posedness (see [12, Section 3.1]).

### Definition 1.1.

We say that the Cauchy problem (1.13) is locally well-posed in H˜αs (ℝ3) if the following properties hold:

1. (i)

For every fH˜αs(3) , there exists a unique strong H˜αs -solution u to the equation

u(t)=eitΔαfi0tei(tτ)Δa(w*|u(τ)|2)u(τ)dτ (1.15)
defined on the maximal interval (−T, T), where T, T ∈ (0, + ∞) depend on f only.

2. (ii)

There is the blow-up alternative: if T* < +∞ (resp., if T* < +∞), then limtT*u(t)H˜αs=+ (resp., limtT*u(t)H˜αs=+ ).

3. (iii)

There is continuous dependence on the initial data: if fnn+f in H˜αs(3) , and if I ⊂ (−T*, T*) is a closed interval, then the maximal solution un to (1.13) with initial datum fn is defined on I for n large enough, and satisfies unn+u in 𝒞(I,H˜αs(3)) .

If T = T = +∞ we say that the solution is global. If (1.13) is locally well-posed and for every fH˜αs(3) the solution is global, we say that (1.13) is globally well-posed in H˜αs(3) .

Let us emphasize the following feature of solutions to (1.15): if both f and w are spherically symmetric, so too is u. This follows at once from the symmetry of the non-linear term of (1.15) together with the previously mentioned fundamental property that the subspaces of L2(ℝ3) of definite rotational symmetry are invariant under the propagator eitΔα. This makes the above definitions of strong solutions and well-posedness meaningful also with respect to the spaces

equipped with the H˜αs -norm. Part of the solution theory we found is set in such spaces.

We can finally formulate our main results. Let us start with the local theory.

### Theorem 1.1

(L2 - theory - local well-posedness). Let α ⩾ 0 Let wL3γ,(3) for γ[0,32) . Then the Cauchy problem (1.13) is locally well-posed in L2(ℝ3).

### Theorem 1.2

(Low regularity - local well-posedness). Let α ⩾ 0 and s(0,12) . Let wL3γ,(3) for γ ∈ [0, 2s]. Then the Cauchy problem (1.13) is locally well-posed in H˜αs(3) , which in this regime coincides with Hs(ℝ3).

### Theorem 1.3

(Intermediate regularity - local well-posedness). Let α ⩾ 0 and s(12,32). . Let wWs,p (ℝ3) for p ∈ (2, + ∞). Then the Cauchy problem (1.13) is locally well-posed in H˜αs(3) .

### Theorem 1.4

(High regularity - local well-posedness). Let α ⩾ 0 and s(32,2]. . Let wWs,p(ℝ3) for p ∈ (2, + ∞) and spherically symmetric. Then the Cauchy problem (1.13) is locally well-posed in H˜α,rads(3) .

The transition cases s=12 and s=32 are not covered explicitly for the mere reason that the structure of the perturbed Sobolev spaces H˜α1/2(3) and H˜α3/2(3) is not as clean as that of H˜αs(3) when s{12,32} – see Theorem 2.1 below for the general case and Remark 2.2 for the peculiarities of the transition cases.

Let us remark that for s > 0 we have an actual ‘continuity’ in s of the assumption on w in the three Theorems 1.2, 1.3, and 1.4 above – in the low regularity case our proof does not require any control on derivatives of w and therefore we find it more informative to formulate the assumption in terms of the Lorentz space corresponding to Ws,p(ℝ3).

Such a ‘continuity’ is due to the fact that under the hypotheses of Theorems 1.2, 1.3, and 1.4 we can work in a locally-Lipschitz regime of the non-linearity. When instead s = 0 we have a ‘jump’ in the form of an extra range of admissible potentials w, which is due to the fact that for the L2 -theory we are able to make use of the Strichartz estimates for the singular Laplacian.

Next, we investigate the global theory in the mass and in the energy spaces.

### Theorem 1.5

(Global solution theory in the mass space). Let α ⩾ 0, and let wL (ℝ3) ∩ w1,3 (ℝ3), or wL3γ,(3) for γ(0,32). . Then the Cauchy problem (1.13) is globally well-posed in L2(ℝ3).

### Theorem 1.6

(Global solution theory in the energy space). Let α ⩾ 0, wWrad1,p(3) for p ∈ (2, +∞), and fH˜α,rad1(3) .

1. (i)

There exists a constant Cw > 0, depending only on ‖wW1,p, such that ‖fL2Cw, then the unique strong solution in H˜α,rad1(3) to (1.13) with initial data f is global.

2. (ii)

If w ⩾ 0, then the Cauchy problem (1.13) is globally well-posed in H˜α,rad1(3) .

As stated in the Theorems above, part of the local and of the global solution theory is set for spherically symmetric potentials w and solutions u. In a sense, this is the natural solution theory for the singular Hartree equation, for sufficiently high regularity. In particular, the spherical symmetry needed for the high regularity theory is induced naturally by the special structure of the space H˜αs(3) (as opposite to Hs(ℝ3), or also to H˜αs(3) for small s), where a boundary (‘contact’) condition holds between regular and singular component of H˜αs -functions. In the concluding Section 7 we comment on this phenomenon, with the proofs of our main Theorems in retrospective.

Before concluding this general introduction, it is worth mentioning that the one-dimensional version of the non-linear Schrödinger equation with point-like pseudo-potentials is much more deeply investigated and better understood, as compared to the so far virtually unexplored scenario in three dimensions.

On L2(ℝ) the Hamiltonian of point interaction “ d2dx2+δ(x) ,” is constructed in complete analogy to − Δα, namely as a self-adjoint extension of (d2dx2)|C0(\{0}) , which results in a larger variety (a two-parameter family) of realisations, each of which is qualified by an analogous boundary condition at x = 0 [5, Chapters I .3 and I .4]. In fact, such an analogy comes with a profound difference, for the one-dimensional Hamiltonians of point interaction are form-bounded perturbations of the Laplacian, unlike − Δα with respect to − Δ on L2(ℝ3), and hence much less singular and with a more easily controllable domain. For instance, among the other realisations, one can non-ambiguously think of d2dx2+δ(x) as a form sum, δ(x) now denoting there the Dirac distribution.

In the last dozen years a systematic analysis was carried out of the non-linear Schrödinger equation in one dimension, mainly with local non-linearity, of the form

itu=-(d2dx2+δ(x))u+α|u|γ-1u,
or the analogous equation with δ′-interaction instead of δ-interaction, initially motivated by phenomenological models of short-range obstacles in non-linear transport [29]. This includes local and global well-posedness in operator domain and energy space and blow-up phenomena [1–3], weak Lp -solutions [6], scattering [8], solitons [19,21], as well as more recent modifications of the nonlinearity [7]. None of such works has a three-dimensional counterpart.

## 2. Preparatory materials

In this Section we collect an amount of materials available in the literature, which will be crucial for the following discussion.

We start with the following characterisation, proved by two of us in a recent collaboration with V. Georgiev, of the fractional Sobolev spaces and norms naturally induced by − Δα, that is, the spaces H˜αs(3) introduced in (1.11)(1.12)

### Theorem 2.1

(Perturbed Sobolev spaces and norms, [17]). Let α ⩾ 0, λ > 0, and s ∈ [0,2]. The following holds.

1. (i)

If s[0,12) , then

H˜αs(3)=Hs(3) (2.1)
and
ψH˜αsψHs (2.2)
in the sense of equivalence of norms. The constant in (2.2) is bounded, and bounded away from zero, uniformly in α

2. (ii)

If s(12,32) , then

H˜αs(3)=Hs(3)+˙span{Gλ} (2.3)
where Gλ is the function (1.4), and for arbitrary ψ=φλ+κλGλH˜αs(3)
φ+κλGλH˜asφHs+(1+α)|κλ| (2.4)

3. (iii)

If s(32,2] , then

H˜αs(3)={ψL2(3)ψ=φλ+φλ(0)α+λ4πGλwithφλHs(3)} (2.5)
and for arbitrary ψ=ϕλ+ϕ2(0)α+λ4πGλH˜αs(3)
φλ+φλ(0)α+λ4πGλH˜asφλHs. (2.6)

The constant in (2.6) is bounded, and bounded away from zero, uniformly in α.

### Remark 2.1.

The case s = 0 is trivial, the case s = 1 reproduces the form domain of − Δα given in (1.3) above, the case s = 2 reproduces the operator domain (1.5).

### Remark 2.2.

Separating the three regimes above, two different transitions occur (see [17, Section 8]). When s decreases from larger values, the first transition arises at s=32 , namely the level of Hs- regularity at which continuity is lost. Correspondingly, the elements in H˜α3/2(3) still decompose into a regular H32 -part plus a multiple of Gλ (singular part), and the decomposition is still of the form ϕλ + κλGλ, except that now ϕλ cannot be arbitrary in H32(3) : indeed, ϕλ has additional properties, among which the fact that its Fourier transform is integrable (a fact that is false for generic H32 -functions), and for such ϕλ ’s the constant κλ has a form that is completely analogous to the constant in (2.5), that is

κλ=1α+λ4π1(2π)323dpφλ^(p)
(see [17, Prop. 8.2]). Then, for s>32 , the link between the two components disappears completely. Decreasing s further, the next transition occurs at s=12 , namely the level of Hs -regularity below which the Green’s function itself belongs to Hs(ℝ3) and it does not necessarily carry the leading singularity any longer. At the transition s=12 , the elements in H˜α1/2(3) still exhibit a decomposition into a regular H12 -part plus a more singular H12 -part, except that H12 -singularity is not explicitly expressed in terms of the Green’s function Gλ (see [17, Prop. 8.1]). Then, for s<12 , only Hs - functions form the fractional domain. Remarkably, yet in the same spirit, such transition thresholds s=12 and s=32 (and their analogues) emerge in several other contexts, such as the regularity of solutions to the cubic non-linear Schrodinger equation on the half-line with Bourgain’s restricted norm methods [15] or the regime of rank-one singular perturbations of the fractional Laplacian [25,26].

### Remark 2.3.

In the limit α → ∞ (recall that Δα = ∞ is the self-adjoint Laplacian on L2(ℝ3)) the equivalence of norms (2.4) tends to be lost, consistently with the fact that the function Gλ does not belong to Hs(ℝ3). Instead, the norm equivalences (2.2) and (2.6) remain valid in the limit α → ∞, which is also consistent with the structure of the space H˜αs(3) in those two cases.

The second class of results we want to make use of were recently proved by two of us in collaboration with Dell’Antonio, Iandoli, and Yajima, and concern the dispersive properties of the propagator teitΔα associated with − Δα, quantified both by (dispersive) pointwise-in-time estimates and by (Strichartz-like) space-time estimates.

To this aim, let us define a pair of exponents (q, r) admissible for − Δα if

r[2,3)and02q=3(121r)<12, (2.7)
that is, q=4r3(r2)(4,+] .

### Theorem 2.2

([14,20]

1. (i)

There is a constant C > 0 such that, for each r ∈ [2, 3),

eiΔtauLr(R3)C|t|3(121r)uL(3),t0 (2.8)

2. (ii)

Let (q, r) and (s, p) be two admissible pairs for − Δα. and rʹ, sʹ are dual exponents of r, s. Then, for a constant C > 0,

eitΔαfLq(t,Lr(x3))CFL2(3) (2.9)
and
0tei(tτ)ΔaF(τ)dτLq(t,Lr(x3))CFLs(t,Lp(x3)). (2.10)

### Remark 2.4.

The dispersive estimate (2.8) has a precursor in [13] in the form of the weighted Lt1Lx -estimate

w1eitΔauL(3)C|t|32wuL1(3),t0 (2.11)
where w(x) ≔ 1 + |x|−1. The weight w is needed to compensate the |x|−1 singularity naturally emerging in eiα u for any t ≠ 0, as typical for a generic element of the energy space H˜αs(3) – see (2.3) above. By interpolation between (2.11) and eiΔtau2=u2 one can then obtain a weighted version of (2.8) in the whole regime r ∈ [2, +∞] (see [20, Prop. 4] or [14, Eq. (1.19)]. It was a merit of [20] to have observed that as long as r ∈ [2, 3), the dispersive estimate (2.8) is valid also without weights. In parallel, in [14] the same estimate (2.8) was obtained as a corollary of the much stronger result of the Lr -boundedness, r ∈ (1, 3), of the wave operators
Wα±=stronglimt+eitΔαeitΔ
associated to the pair (− Δα, − Δ) and of the intertwining properties of W±, which allow one to deduce (2.8) in the regime r ∈ [2, 3), from the analogous and well-known dispersive estimate for the free propagator teitΔ.

The third class of properties that we need to recall concern fundamental tools of fractional calculus. One is the following fractional Leibniz rule by Kato and Ponce, also in the generalised version by Gulisashvili and Kon.

### Theorem 2.3

(Generalised fractional Leibniz rule, [18,22]).Suppose that r ∈ (1, +∞] and p1, p2, q1, q2 ∈ (1, + ∞] with 1pj+1qj=1r , j ∈ {1,2}, and suppose that s, μ, v ∈ [0, +∞). Let d ∈ ℕ, then

𝒟s(fg)Lr(d)𝒟s+μfLp1(d)𝒟μgLq1(d)+𝒟vfLp2(d)𝒟s+vgLq2(d), (2.12)
where 𝒟s=(Δ)s2 , the Riesz potential. The same result holds when 𝒟s is the Bessel potential (𝕝Δ)s2 .

### Remark 2.5.

As a direct consequence of Mihlin multiplier theorem [10, Section 6.1], the estimate (2.12) holds as well for 𝒟s=(Δ+λ𝕝)s2 for any λ ⩾ 0.

We also need a more versatile re-distribution of the derivatives among the two factors f and g in (2.12): the following recent result by Fujiwara, Georgiev, and Ozawa provides a very useful refinement of the fractional Leibniz rule and is based on a careful treatment of the correction term

[f,g]s:=f𝒟sg+g𝒟sf (2.13)

### Theorem 2.4

(Higher order fractional Leibniz rule, [16]). Suppose that p, q, r ∈ (1, +∞) with 1p+1q=1r and let d ∈ ℕ

1. (i)

Let s1, s2 ∈ [0, 1] and set ss1 + s2. Then

𝒟s(fg)[f,g]sLr(d)𝒟s1fLp(d)𝒟s2gLq(d). (2.14)

2. (ii)

Let s1 ∈ [0, 2], ∈ s2 ∈ [0, 1] be such that ss1 + s2 ⩾ 1. Then

𝒟s(fg)[f,g]s+s𝒟s2(fg)+s𝒟s2(gΔf)sg𝒟s2ΔfLr(d)𝒟s1fLp(d)𝒟s2gLq(d). (2.15)

Moreover, since

𝒟s2(fg)+𝒟s2(gΔf)g𝒟s2Δf=𝒟s2(gf)+g𝒟sf

we can rewrite (2.15) in the more compact form

𝒟s(fg)f𝒟sg+(s1)g𝒟sf+s𝒟s2(gf)Lr(d)𝒟s1fLp(d)𝒟s2gLq(d). (2.16)

For the fractional derivative of |x|−1e−λ|x| we need, additionally, a point-wise estimate.

### Lemma 2.1.

Let λ > 0, s ∈ (0, 2]. We have the estimate

|𝒟seλ|x||x||eλ|x||x|+eλ|x||x|1+s,x0. (2.17)

Proof. Obvious when s = 2, and a straightforward consequence of the identity

𝒟sGλ=1(|p|sG^λ(p))=1(1(2π)32λp2+λ)+1(1(2π)32|p|s+λp2+λ)
when s ∈ (0, 2).

In the second part of this Section, based on the preceding properties, we derive two useful estimates that we are going to apply systematically in our discussion when s>12 . Let us recall that in this case

g1g2Ws,qg1Ws,qg2Ws,q(s>12,q(6,+)) (2.18)
as follows, for example, from the fractional Leibniz rule (2.12) and Sobolev’s embedding Ws,qL(ℝ3).

We start with the estimate for the regime s(12,32) , for which we recall Sobolev’s embedding

Ws,q(3)𝒞0,ϑ(s,q)(3)ϑ(s,q):=min{s3q,1}(s(12,32),q(6,+). (2.19)
.

### Proposition 2.1.

Let s(12,32) and hWs,p (ℝ3) for some q ∈ (6, +∞). Then

𝒟s((hh(0))Gλ)L2hWs,q(3), (2.20)
where Gλ is the function (1.4) for some λ > 0.

Proof. It is not restrictive to fix λ= 1. For short, we set G(x) ≔ |x|−1e−|x| and G˜(x):=|x|1e12|x|. .

By means of the commutator bound (2.14) we find

𝒟S((hh(0))G)L2=𝒟S(e12|x|(hh(0))G˜)L2𝒟S(e12|x|(hh(0))G˜)[e12|x|(hh(0)),G˜]SL2+e12|x|(hh(0))𝒟sG˜L2+G˜𝒟S(e12|x|(hh(0)))L2𝒟S1(e12|x|(hh(0)))Lq1𝒟s2G˜Lq2+e12|x|(hh(0))𝒟SG˜L2+G˜𝒟S(e12|x|(hh(0)))L2=𝒭1+𝒭2+𝒭3 (i)
for every s1, s2 ∈ [0, 1] with s1 + s2 = s and every q1, q2 ∈ [2, +∞] such that q11+q21=21 .

Let us estimate the term 3. Since, by (2.18) and Sobolev’s embedding,

𝒟s(e12||x|(hh(0)))Lq(𝕝Δ)s2(e12||x|(hh(0)))Lqe12|x|hWs,q+|h(0)|e12||x|Ws,qhWs,q+hLhWs,q, (ii)
and since G˜2q42<+ because 2qq2<3 for q > 6, then Holder’s inequality yields
𝒭3𝒟s(e12||x|(hh(0)))LqG˜L2qq2hWs,q. (iii)

Next, let us estimate 1. When s(12,1] , we choose s1 = s, s2 = 0, q1 = q, and q2=2qq2 and we proceed exactly as for 3. When instead s(1,32) , we choose s1 = 1, s2=s1(0,12) , q1=(1qs13)1 , and q2=(121q1)1 . Then Sobolev’s embedding ws,q(ℝ3) ↪ W1,q1 (ℝ3) and estimate (ii) above imply

𝒟s1(e12|x|(hh(0)))Lq1(𝕝Δ)s2(e12|x|(hh(0)))LqhWs,q,
whereas estimate (2.17) and the fact that q2 < sq2 < 3 for s(1,32) imply
𝒟s2G˜Lq2e12|x|(1|x|+1|x|s)Lq2<+.

Thus, in either case s(12,1) and s(1,32) ,

𝒭1hWs,q. (iv)

Last, let us estimate 2. Because of the embedding (2.19),

e12|x|(hh(0))|x|ϑ(s,q)LhWs,q

Moreover, since ϑ(s,q)>s12 and hence 2(1 − ϑ(s,q)) < 2(1 + sϑ(s,q)) < 3 for every s(12,32) , estimate (2.17) implies

|x|ϑ(s,q)𝒟sG˜L2e12|x|(|x|(1ϑ(s,q))+|x|(1+sϑ(s,q)))L2<+.

Thus,

𝒭2e12|x|(hh(0))|x|ϑ(s,q)L|x|ϑ(s,q)𝒟sG˜L2hWs,q. (v)

Plugging (iii), (iv), and (v) into (i) the thesis follows.

We establish now an analogous estimate for the regime s(32,2] , for which we recall Sobolev’s embedding

Ws,q(3)𝒞1,ϑ(s,q)(3)ϑ(s,q):=s13q(s(32,2],q(6,+)). (2.21)

### Proposition 2.2.

Let s(32,2] and hWs,q(ℝ3) for some q ∈ (6, + ∞). Assume further that h is spherically symmetric and that (∇h) (0) = 0. Then

𝒟s((hh(0))Gλ)L2hWs.q(3), (2.22)
where Gλ is the function (1.4) for some λ > 0.

Prior to proving Proposition 2.2 let us highlight the following property.

### Lemma 2.2.

Under the assumptions of Proposition 2.2,

|h(x)h(0)|hWs,q|x|1+ϑ(s,q), (2.23)
where ϑ(s,q)=s13q , as fixed in (2.21).

Proof. By assumption, h(x)=h˜(|x|) for some even function h˜: . Owing to the embedding (2.21), h˜𝒞1,ϑ(s,q)() , whence h˜𝒞0,ϑ(s,q)(). . Moreover, h˜(0)=0 , because (∇h)(0) = 0.

Therefore,

|h˜(ρ)|=|h˜(ρ)h˜(0)|hWs,|ρ|ϑ(s,q).

As a consequence,

|h(x)h(0)|0|x||h˜(ρ)|dρhWs,q0|x|ρϑ(s,q)dϑhWs,q|x|1+ϑ(s,q),
which completes the proof.

Proof. [Proof of Proposition 2.2] It is not restrictive to fix λ = 1. For short, we set G(x) ≔ |x|−1e−|x| and

G˜(x):=|x|1e12|x|
.

Let us split

𝒟s((hh(0))G)L2=𝒟s(e12|x|(hh(0))G˜)L2||𝒟s(e12|x|(hh(0))G˜)e12|x|(hh(0))𝒟sG˜+(s1)G˜𝒟s(e12|x|(hh(0)))+s𝒟s2(G˜(e12|x|(hh(0)))||L2+e12|x|(hh(0))𝒟sG˜L2+(s1)G˜𝒟s(e12|x|(hh(0)))L2+s𝒟s2(G˜(e12|x|(hh(0)))L2 (i)

We estimate the term 1 by means of the commutator bound (2.16) with s1 = s and s2 = 0 and of (2.18), namely

𝒭1𝒟s(e12|x|(hh(0)))LqG˜L2qq2hWs,q (ii)
(since 2qq2[2,3) ) and G˜2qq2<+ ).

For the estimate of 2, we observe that sϑ(s,q)<32 and hence (2.17) implies

|x|1+ϑ(s,q)𝒟sG˜L2e12|x|(|x|ϑ(s,q)+|x|(sϑ(s,q)))L2<+;
this and the bound (2.23) yield
𝒭2hh(0)|x|1+ϑ(s,q)L|x|1+ϑ(s,q)𝒟sG˜L2hWs,q. (iii)

For 3, Hölder’s inequality, the property (2.18), and Sobolev’s embedding yield

𝒭3𝒟s(e12|x|(hh(0)))LqG˜L2qq2e12|x|(hh(0))Ws,qhWs,q+hLhWs,q. (iv)

For 4, one has

𝒭4𝒟s1(G˜(e12|x|(hh(0)))L2(e12|x|(hh(0))Ws1,qhWs,q, (v)
where we used the estimate (2.20) in the second inequality (indeed, s1(12,1) ), and the property (2.18) and Sobolev’s embedding in the last inequality.

Plugging the bounds (ii)-(v) into (i) completes the proof.

## 3. L2 -theory and low regularity theory

In this Section we prove Theorems 1.1 and 1.2. Let us start with Theorem 1.2 and then discuss the adaptation for s = 0. The proof for s(0,12) is based on a fixed point argument in the complete metric space 𝒳T,M,d defined by

𝒳T,M:={uL([T,T],H˜αs(3))uL(IT,T],H^(s(3)M}d(u,v):=uvL([T,T],L2(3)) (3.1)
for given T, M > 0. This is going to be the same space for the contraction argument in the intermediate regularity regime s(12,32) (Section 4), whereas for the high regularity regime s(32,2) (Section 5) we are going to only use the spherically symmetric sector of the space (3.1).

Proof. [Proof of Theorem 1.2]

Since by assumption s(0,12) , the spaces Hs(ℝ3) and H˜αs(3) coincide and their norms are equivalent ( Theorem 2.1), so we can interchange them in the computations that follow.

From the expression (1.14) for the solution map Φ(u) one finds

Φ(u)LH˜αsfH˜αs+T(w*|u|2)uLH˜αs
and applying the fractional Leibniz rule (2.12) ( Theorem 2.3), Hölder’s inequality, and Young’s inequality one also finds
(w*|u|2)uLH˜αs=(w*|u|2)uLHsw*|u|2LL𝒟suLL2+𝒟s(w*|u|2)LL6γ,uLL63γwL3γ,(3)u2LL63γ2𝒟suLL2.

Sobolev’s embedding H˜α(3)=Hs(3)Hγ2(3)L63γ(3) then yields

Φ(u)LH˜αs(3)fH˜αs+C1TwL3γ,uLH˜as3. (i)
for some constant C1 > 0.

On the other hand, again by Hölder’s and Young’s inequality,

Φ(u)Φ(v)LL2T(w*|u|2)u(w*|v|2)vLL2T((w*|u|2)(uv)LL2+w*(|u|2|v|2)vLL2)T(wL3γ,uLL63γ2uvLL2+wL3γ,,uvLL2|u|+|v|LL63γvLL63γ),
whence, by the same embedding H˜α(3)L63γ(3) as before,
d(Φ(u),Φ(v))C2wL3γ,(uLH˜α22+vLH˜α22)Td(u,v) (ii)
for some constant C2 > 0.

Thus, choosing T and M such that

M=2fH˜αs,T=14(max{C1,C2}M2wWs,p)-1,
estimate (i) reads Φ(u)LH˜asM and shows that Φ maps the space 𝒳T,M defined in (3.1) into itself, whereas estimate (ii) reads d(Φ(u),Φ(v))12d(u,v) and shows that Φ is a contraction on 𝒳T,M. By Banach’s fixed point theorem, there exists a unique fixed point u𝒳T,M of Φ and hence a unique solution u𝒳T,M to (1.15), which is therefore also continuous in time.

Furthermore, by a customary continuation argument we can extend such a solution over a maximal interval for which the blow-up alternative holds true. Also the continuous dependence on the initial data is a direct consequence of the fixed point argument. We omit the standard details, they are part of the well-established theory of semi-linear Schrödinger equations.

We move now to the proof of Theorem 1.1. Crucial for this case are the Strichartz estimates of Theorem 2.2. To this aim, we modify the contraction space (3.1) to the complete metric space (𝒴T,M,d) defined by

𝒴T,M:={uL([T,T],L2(3))Lq(γ)([T,T],Lr(γ)(3))s.t.uL([T,T],L2(3))+uLq(γ)([T,T],Lr(γ)(3))M}d(u,v):=uvL([T,T],L2(3))+uvLq(γ)([T,T],Lr(γ)(3)) (3.2)
for given T, M > 0, where
q(γ):=6γ,r(γ):=1892γ (3.3)
are defined so as to form an admissible pair (q(γ), r(γ)) for − Δα, in the sense of (2.7). For the rest of the proof let us drop the explicit dependence on γ in (q, r).

Proof. [Proof of Theorem 1.1] Clearly, when γ = 0 the very same argument used in the proof of Theorem 1.2 applies.

When γ(0,32) we exploit instead a contraction argument in the modified space (3.2).

One has

Φ(u)LL2+Φ(u)LqLr0tei(tτ)Δα((w*|u|2)u)(τ)dτLL2+0tei(tτ)Δα((w*|u|2)u)(τ)dτLqLr+fL2+eitΔαfLqLr,
from which, by means of the Strichartz estimates (2.9)(2.10), one deduces
Φ(u)LL2+Φ(u)LqLrC(fL2+(w*|u|2)uLqLr)
for some constant C > 0.

By Hölder’s and Young’s inequalities,

(w*|u|2)uLqLrw*|u|2LqL9γuLL2wL3γ,u2L2qLruLL2
and
uL2qLr2(2T)1γ2uLqLr2,
whence
Φ(u)LL2+Φ(u)LqLrC1(fL2+T1γ2wL3γ,uLqLr2uLL2) (i)
for some constant C1 > 0.

Following the very same scheme, one finds

Φ(u)Φ(v)LL2+Φ(u)Φ(v)LqLr(w*|u|2)u(w*|v|2)vLqLr(w*|u|2)(uv)LqLr+w*(|u|2|v|2)vLqLr,
and moreover
(w*|u|2)(uv)LqLrT1γ2wL3γ,uLqLr2uvLL2
and
w*(|u|2|v|2)vLqLrT1γ2wL3γ,uvLqLr(uLqLr+vLqLr)vLL2.

Thus,

dΦ(u),Φ(v)C2wL3γ,(uLL22+uLqLr2+vLL22+vLqLr2)×T1γ2d(u,v) (ii)

Therefore, choosing T and M such that

M=2C1fL2,T=(8max{C1,C2}M2wL3γ,)1+γ2,
estimate (i) reads ‖Φ(u)‖LL2 + ‖Φ(u)‖LqLr and shows that Φ maps 𝒴T,M into itself, whereas estimate (ii) reads d(Φ(u),Φ(v))12d(u,v) and shows that Φ is a contraction on 𝒴T,M.

The thesis then follows by Banach’s fixed point theorem through the same arguments outlined in the end of the proof of Theorem 1.2.

For later purposes, let us conclude this Section with the following stability result.

### Proposition 3.1.

Let α ⩾ 0. For given wL3γ,(3),γ[0,32) , and f ∈ L2(ℝ3) let u be the unique strong L2 -solution to the Cauchy problem (1.13) in the maximal interval (−T*,T*). Consider moreover the sequences (wn)n and (fn)n of potentials and initial data such that wnn+w in L3γ,(3) and fnn+f in L2 (ℝ3). Then there exists a time T:=T(wL3γ,,fnL2)>0 , with [−T,T] ⊂ (−T*,T*), such that, for sufficiently large n, the Cauchy problem (1.13) with potential wn and initial data fn admits a unique strong L2 -solution un in the interval [−T, T]. Moreover,

unn+uin𝒞([T,T],L2(3)). (3.4)

Proof. As a consequence of Theorem 1.1, there exist an interval [−Tn, Tn] for some Tn:=Tn(wnL3γ,,fnL2)>0 and a unique un𝒞([−Tn, Tn], L2(ℝ2)) such that

un(t)=eitΔafni0tei(tτ)Δa(wn*|un(τ)|2)un(τ)dτ. (i)

Since wnL3γ, and ‖fnL2 are asymptotically close, respectively, to wL3γ, and ‖fnL2, then there exists T:=T(wL3γ,,fL2 such that TTn eventually in n, which means that un is defined on [−T, T]. Let us set ϕnuun

By assumption u solves (1.15), thus subtracting (i) from (1.15) yields

φn=eitΔα(ffn)i0tei(tτ)Δα((w*|u|2)u(wn*|un|2)un)(τ)dτ=eitΔα(ffn)i0tei(tτ)Δα{((wwn)*|u|2)u+(wn*|u|2)φn+(wn*(u¯φn+φn¯un))un}(τ)dτ. (ii)

Let us first discuss the case γ = 0. From (ii) above, using Hölder’s and Young’s inequality in weak spaces, one has

φnLL2ffnL2+TwwnLuLL23+TwnL(uLL22+unLL22)φnLL2.

Since ‖wnL and ‖unLL2 are bounded uniformly in n, then the above inequality implies, decreasing further T if needed,

φnLL2ffnL2+wwnL3γ,n+0,
which proves the proposition in the case γ = 0.

Let now γ(0,32) . In this case, owing to Theorem 1.1, u, unLq([−T, T], Lr(ℝ3)), where (q,r)=(6γ,1892γ) is the admissible pair defined in the proof therein. We can then argue as in the proof of Theorem 1.1. Applying the Strichartz estimates (2.9)(2.10) to the identity (ii) above, one gets

φnLL2+φnLqLrffnL2+((wwn)*|u|2)uLqLr+(wn*|u|2)φnLqLr+(|wn|*(|un|+|u|)|φn|)unLqLr. (iii)

By means of Hölder’s and Young’s inequality in weak spaces one finds

((wwn)*|u|2)uLqrT1γ2wwnL3γ,unLqLr2uLL2(wn*|u|2)φnLqrT1γ2wnL3γ,uLqLr2φnLL2(|wn|*(|un|+|u|)|φn|)unLqrT1γ2wnL3γ,(unLqLr+uLqLr)××φnLqLruLL2. (iv)

Since

wnL3γ,
and ‖unLqLr‖ are bounded uniformly in n, then inequalities (iii) and (iv) imply, decreasing further T if needed,
φnLL2+φnLqLrffnL2+wwnL3γ,n+0,
which completes the proof.

## 4. Intermediate regularity theory

In this Section we prove Theorem 1.3. The proof is based again on a contraction argument in the complete metric space 𝒳T,M, for suitable T, M > 0, defined in (3.1), now with s(12,32) .

As a consequence, in the energy space (s = 1) we shall deduce that the solution to the integral problem (1.15) is also a solution to the differential problem (1.13).

We conclude the Section with a stability result of the solution with respect to the initial datum f and the potential w.

### Lemma 4.1.

Let α⩾ and s(12,32) . Let wWs,p (ℝ3) for p ∈ (2, + ∞). Then

w*(ψ1ψ2)L(3)w*(ψ1ψ2)ws,3p(3)wWs,p(3)ψ1H˜αs(3)ψ2H˜αs(3) (4.1)
for any H˜αs -functions ψ1, ψ2, and ψ3,

Proof. The first inequality in (4.1) is due to Sobolev’s embedding

Ws,3p(3)L(3).

For the second inequality, let us observe preliminarily that

H˜αs(3)L6p3p2(3).

Indeed, decomposing by means of (2.3) a generic ψH˜αs(3) as ψ = ϕλ + kλGλ for some ϕλHs(ℝ3) and some κλ ∈ ℂ, one has

ψL6p3p2φλL6p3p2+|κλ|GλL6p3p2φλHs+|κλ|ψH˜αs,
the second step following from Sobolev’s embedding Hs(3)L6p3p2(3) and from GλL6pp2(3) , because 6pp2[2,3) for p ∈ (2, + ∞) the last step being the norm equivalence (2.4) Therefore Young’s inequality yields
w*(ψ1ψ2)Ws,3p(𝕝Δ)s2(w*(ψ1ψ2))L3p=((𝕝Δ)s2w)*(ψ1ψ2)L3p(𝕝Δ)s2wLpψ1L6p3p2ψ2L6p3p2wWs,p(3)ψ1H˜αs(3)ψ2H˜αs(3),
thus proving (4.1).

### Lemma 4.2.

Let α ⩾ 0 and s(12,32) . Let hWs,q(ℝ3) for q ∈ (6, + ∞). Then hψH˜αs(3) for each ψH˜αs(3) and

hψH^αs(3)hWs,q(3)ψH˜αr(3). (4.2)

Proof. Let us decompose ψH˜αs(3) as ψ = ϕλ + ∈ κλGλ for some ϕλHs(ℝ3) and κλ ∈ ℂ, according to (2.3). On the other hand, by the embedding (2.19) the function h is continuous and |h(0)|hL(3)hWs,q(3) . Thus,

hg=hφλ+κλ(hh(0))Gλ+κλh(0)Gλ. (i)

Applying the fractional Leibniz rule (2.12) and using Sobolev’s embedding,

hφλ Hs (𝕝-Δ)s2(hφλ) L2 (𝕝-Δ)s2h Lq φλ L2qq-2+ h L (𝕝-Δ)s2φλ L2 h Ws,q φλ Hs. (ii)

Moreover, since GλL2(ℝ3),

(hh(0))GλL2hh(0)LhWs,q;
this, together with the estimate (2.20), gives
κλ(hh(0))GλHJ|κλ|hWs,q. (iii)

The bounds (ii) and (iii) imply that is the sum of the function λ + κλ (hh(0))GλHs(ℝ3) and of the multiple κλh(0) Gλ of Gλ : as such, owing to (2.3), belongs to H˜αs(3) and its H˜αs - norm is estimated, according to the norm equivalence (2.4), by hψH˜αs(3)hφλ+κλ(hh(0))GλH˜αs+|κλ||h(0)|hWs,q(φλHs+|κλ|)+|κλ|hWs,qhWs,qψH˜αs, which completes the proof.

Combining Lemmas 4.1 and 4.2 one therefore has the trilinear estimate

(w*(u1u2))u3H˜αs(R3)wWs,p(3)j=13ujH˜αs(3) (4.3)

Let us now prove Theorem 1.3.

Proof. [Proof of Theorem 1.3 ]

From the expression (1.14) for the solution map Φ(u) and from the bound (4.3) one finds

Φ(u)LH˜αsfH˜αs+T(w*|u|2)uLH˜αsfH˜αs+C1TwWs,puLH˜αs3 (i)
for some constant C1 > 0.

Moreover,

Φ(u)Φ(v)LL2T(w*|u|2)u(w*|v|2)vLL2T((w*|u|2)(uv)LL2+w*(|u|2|v|2)vLL2. (ii)

For the first summand in the r.h.s. above estimate (4.1) and Hölder’s inequality yield

(w*|u|2)(uv)LL2w*|u|2LLuvLL2wWs,puLH˜as2uvLL2. (iii)

For the second summand, let us observe preliminarily that

H˜αs(3)L3,(3). (iv)

Indeed, decomposing by means of (2.3) a generic ψH˜αs(3) as ψ = ϕλ + κλGλ for some ϕλHs(ℝ3) and some κ2 ∈ ℂ, one has

ψL3,φλL3,+|κλ|GλL3,φλHs+|κλ|ψH˜αs,
the second step following from the Sobolev’s embedding Hs(ℝ3) → L3(ℝ3), the last step being the norm equivalence (2.4). Then (iv) above, Sobolev’s embedding Ws,p(ℝ3) → L3(ℝ3), and an application of Holder’s and Young’s inequality in Lorentz spaces, yield
(w*(|u|2|v|2))vLL2w*(|u|2|v|2)LL6,2vLL3,wL3u+vLL3,uvLL2vLL3,wws,pu+vLH˜αsuvLL2vLH˜αs.. (v)

Thus, (ii), (iii), and (v) together give

d(Φ(u),Φ(v))C2TwWs,p(uLH˜n22+vLH˜αs2)d(u,v) (vi)
for some constant C2 > 0.

Now, setting C ≔max{C 1, C 2) and choosing T and M such that.

M=2fH˜αs,T=14(CM2wWs,p)1,
estimate (i) reads Φ(u)LH˜αsM and shows that Φ maps the space 𝒳T,M defined in (3.1) into itself, whereas estimate (vi) reads d(Φ(u),Φ(v))12d(u,v) and shows that Φ is a contraction on 𝒳T,M. By Banach’s fixed point theorem, there exists a unique fixed point u𝒳T,M of Φ and hence a unique solution u𝒳T,M to (1.15), which is therefore also continuous in time.

Furthermore, by a standard continuation argument we can extend such a solution over a maximal interval for which the blow-up alternative holds true. Also the continuous dependence on the initial data is a direct consequence of the fixed point argument.

A straightforward consequence of Theorem 1.3 when s =1 concerns the differential meaning of the local strong solution determined so far.

### Corollary 4.1

(Integral and differential formulation). Let α ⩾ 0. For given wW1,p(ℝ3), p ∈ (2, +∞), and fH˜α1(3) , let u be the unique solution in the class 𝒞([T,T]H˜α1(3)) to the integral equation (1.15) in the interval [−T, T] for some T > 0, as given by Theorem 1.3. Then u(0) = f and u satisfies the differential equation (1.10) as an identity between H˜α1 -functions, H˜α1(3) being the topological dual of H˜α1(3) .

Proof. The bound (4.3) shows that the non-linearity defines a map u ↦ (w*|u|2)u that is continuous from H˜α1(3) into itself, and hence in particular it is continuous from H˜α1(3) to H˜α1(3) . Then the thesis follows by standard facts of the theory of linear semi-groups (see [12, Section 1.6]).

For later purposes, let us conclude this Section with the following stability result.

### Proposition 4.1.

Let α ⩾ 0 and s(12,32) . For given wWs,p(ℝ3), p ∈ (2, + ∞) and fH˜αs(3) , let u be the unique strong H˜αs -solution to the Cauchy problem (1.13) in the maximal interval (−T*,T*). Consider moreover the sequences (wn)n and (fn)n of potentials and initial data such that wnn+w in Ws,p(ℝ3) and fnn+f in H˜αs(3). . Then there exists a time T:=T(wWs,p,fH˜αs)>0 , with [−T, T] ⊂(T*, T *), such that, for sufficiently large n, the Cauchy problem (1.13)with potential wn and initial data (fn) admits a unique strong H˜αs -solution un in the interval [−T, T]. Moreover,

unn+uin𝒞([T,T],H˜αs(3)) (4.4)

Proof. As a consequence of Theorem 1.3, there exist an interval [−Tn, Tn] for some Tn:=Tn(wnWssp,fnH˜αs)>0 and a unique un𝒞([Tn,Tn],H˜α1(3)) such that

un(t)=eitΔαfni0tei(tτ)Δa(wn*|un(τ)|2)un(τ)dτ. (*)

Since ‖wnWs,p‖ and fnH˜αs are asymptotically close, respectively, to ‖wnWs,p‖ and fH˜αs , then there exists T:=T(wWs,p,fH˜αs) such that TTn eventually in n, which means that un is defined on [−T, T]. Let us set ϕnuun. By assumption u solves (1.15), thus subtracting (*) from (1.15) yields

φn=eitΔα(ffn)i0tei(tτ)Δα((w*|u|2)u(wn*|un|2)un)(τ)dτ=eitΔα(ffn)i0tei(tτ)Δα{((wwn)*|u|2)u+(wn*|u|2)φn+(wn*(u¯φn+φn¯un))un}(τ)dτ.

From the above identity, taking the H˜αs -norm of ϕn boils down to repeatedly applying the estimate (4.3) to the summands in the integral on the r.h.s., thus yielding

φnLH˜αsffnH˜αs+TwwnWs,puLH˜αs3+TwnWs,p(uLH˜αs2+unLH˜αs2)φnLH˜αs.

Since by assumption ‖wnWs,p and unLH˜αs2 are bounded uniformly in n, then the above inequality implies, decreasing further T if needed,

φnLH˜αsffnH˜αs+wwnWs,pn+0,
which completes the proof.

## 5. High regularity Theory

In this Section we prove Theorem 1.4 for the regime s(32,2] . The approach is again a contraction argument, that we now set in the spherically symmetric sector of the space 𝒳T,M introduced in (3.1), namely in the complete metric space (𝒳T,M(0),d) with

for suitable T, M > 0.

A very much useful by-product of such a contraction argument will be the proof that when s = 2 the solution to the integral problem (1.15) is also a solution to the differential problem (1.13), as we shall show in a moment.

### Lemma 5.1.

Let α ⩾ 0 and s(32,2]. . Let wWs,p (ℝ3) for p ∈ (2, + ∞) and assume that w is spherically symmetric. Then

1. (i)

one has the estimate

w*(ψ1ψ2)L(3)w*(ψ1ψ2)Ws,3p(3)wWs,p(3)ψ1H˜αs(3)ψ2H˜αs(3) (5.2)

for any H˜αs -functions ψ1, ψ2 and ψ3;

2. (ii)

if in addition ψ1, ψ2 are spherically symmetric, so too is w*(ψ1 ψ2) and

((w*(ψ1ψ2)))(0)=0.

Proof. The first inequality in (5.2) is due to Sobolev’s embedding

Ws,3p(3)FFFDL(3).

For the second inequality, let us observe preliminarily that

H˜αs(3)L6p3p2(3).

Indeed, decomposing by means of (2.5) a generic ψH˜αs(3) as ψ=φλ+φλ(0)α+λ4πGλ for some ϕλHs(ℝ3) , one has

ψL6p3p2φλL6p3p2+|φλ(0)|GλL6p3p2φλHsψH˜αs,
the second step following from Sobolev’s embedding Hs(3)L6p3p2(3)L(3) and from GλL6pp2(3) , because 6p3p2[2,3) for p ∈ (2, + ∞) the last step being the norm equivalence (2.6). Therefore Young’s inequality yields
w*(ψ1ψ2)Ws,3p(𝕝Δ)s2(w*(ψ1ψ2))L3p=((𝕝Δ)s2w)*(ψ1ψ2)L3p(𝕝Δ)s2wLpψ1L6p3p2ψ2L6p3p2wWs,p(3)ψ1H˜αs(3)ψ2H˜αs(3),
thus proving (5.2).
1. (ii)

The spherical symmetry of w*(ψ1 ψ2) in this second case is obvious. From Sobolev’s embedding Ws;3p(ℝ3) ↪ 𝒞1(ℝ3) we deduce that ∇ (w*(ψ1 ψ2))(x) is well defined for every x ∈ ℝ3; moreover,

(w*(ψ1ψ2))(0)=((w)*(ψ1ψ2))(0)=3(w)(y)ψ1(y)ψ2(y)dy=0,
the above integral vanishing because the integrand is of the form R(y)y|y| for some spherically symmetric function R.

### Lemma 5.2.

Let α ⩾ 0 and s(32,2] . Let hWrads,q(3) for some q ∈ (6, + ∞) and assume that (∇h)(0) = 0. Then hψH˜αs(3) for each ψH˜αs(3) and

hψH^αs(3)hW3,q(3)ψH˜αs(3). (5.3)

Proof. Let us decompose ψH˜αs(3) as ψ=φλ+φλ(0)α+λ4πGλ for some ϕλHs(ℝ3), according to (2.5). On the other hand, by the embedding (2.21) the function h is continuous and |h(0)| ⩽ ‖hL(ℝ3) ≲ ‖hWs,q(ℝ3). Thus,

hψ=hφλ+φλ(0)α+λ4π(hh(0))Gλ+φλ(0)α+λ4πh(0)Gλ. (i)

Applying the fractional Leibniz rule (2.12) and using Sobolev’s embedding,

hφλHs(𝕝Δ)s2(hφλ)L2(𝕝Δ)s2hLqφλL2qq2+hL(𝕝Δ)s2φλL2hWs,qφλHs. (ii)

Moreover, since GλL2(ℝ3),

φλ(0)α+λ4π(hh(0))GλL2hh(0)LφλLhWs,qφλHs;
this, together with the estimate (2.22) (which requires indeed spherical symmetry), gives
φλ(0)α+λ4π(hh(0))GλHshWs,qφλHs. (iii)

The bounds (ii) and (iii) above imply that Fλ:=hφλ+φλ(0)α+λ4π(hh(0))Gλ belongs to Hs(ℝ3) with

FHshWs,qφλHs. (iv)

In particular, Fλ is continuous. One has

Fλ(0)=h(0)φλ(0)+φλ(0)α+λ4πlim|x|0h(x)h(0)|x|=h(0)φλ(0)
because by assumption (∇h)(0) = 0. In turn, (i) now reads hψ=Fλ+Fλ(0)α+λ4πGλ , which means, in view of the domain decomposition (2.5), that belongs to H˜αs(3) . Owing to (iv) above and to the norm equivalence (2.6), we conclude
hψH˜αsFλHshWs,qφλHs,
which completes the proof.

Combining Lemmas 5.1 and 5.2 one therefore has the trilinear estimate

Let us now prove Theorem 1.4.

Proof. [Proof of Theorem 1.4]

From the expression (1.14) for the solution map Φ(u) and from the bound (5.4) one finds

for some constant C1 > 0.

Moreover,

Φ(u)Φ(v)LL2T(w*|u|2)u(w*|v|2)vLL2T((w*|u|2)(uv)LL2+w*(|u|2|v|2v)LL2. (ii)

For the first summand in the r.h.s. above the bound (5.2) and Hölder’s inequality yield

(w*|u|2)(uv)LL2w*|u|2LLuvLL2wWspuLH˜a22uvLL22. (iii)
For the second summand, let us observe preliminarily that the embedding
H˜αs(3)L3,(3) (iv)
valid for s(12,32) and established in the proof of Theorem 1.3 holds true even more when s(32,2] . Then (iv) above, Sobolev’s embedding Ws,p(ℝ3) ↪ L3(ℝ3) and an application of Holder’s and Young’s inequality in Lorentz spaces, yield
(w*(|u|2|v|2))vLL2w*(|u|2|v|2)LL6,2vLL3,wL3u+vLL3,uvLL2vLL3,wWs,pu+vLH˜αsuvLL2vLH˜αs. (v)

Combining (ii), (iii), and (v) we get

d(Φ(u),Φ(v))C2TwWs,p(uLH˜αs2+vLH˜αs2)d(u,v) (vi)
for some constant C2 > 0.

Thus, choosing T and M such that

d(Φ(u),Φ(v))C2TwWs,p(uLH˜αs2+vLH˜αs2)d(u,v)
estimate (i) reads Φ(u)LH˜α,radsM and shows that Φ maps the space 𝒳T,M(0) into itself, whereas estimate (vi) reads d(Φ(u),Φ(v))12d(u,v) and shows that Φ is a contraction on 𝒳TM(0) . By Banach’s fixed point theorem, there exists a unique fixed point u𝒳T,M(0) of Φ and hence a unique solution u𝒳T,M(0) to (1.15), which is therefore also continuous in time.

Furthermore, by a standard continuation argument we can extend such a solution over a maximal interval for which the blow-up alternative holds true. Also the continuous dependence on the initial data is a direct consequence of the fixed point argument.

A straightforward, yet crucial for us, consequence of Theorem 1.4 when s = 2 concerns the differential meaning of the local strong solution determined so far.

### Corollary 5.1

(Integral and differential formulation). Let α ⩾ 0 and wW2,p(ℝ3), p ∈ (2, + ∞) a spherically symmetric potential. Assume moreover fH˜α,rad2(3) . Let u𝒞([T,T],H˜α,rad2(3)) the unique local to the Cauchy problem (1.13) in the interval [−T, T], for some T > 0, i.e. u satisfies the Duhamel formula (1.15). Then u(0, .) = f and u satisfies the equation itu = −Δαu + (w*|u|2)u as an identity in between L2 (ℝ3) - functions.

Proof. The bound (5.4) shows that the non-linearity defines a map u ↦ (w*|u|2) that is continuous from H˜α,rad2(3) into itself, and hence in particular it is continuous from H˜α,rad2(3) to L2(ℝ3). Then the thesis follows by standard fact on the theory of linear semi-groups (see [12, Section 1.6 ]).

## 6. Global solutions in the mass and in the energy space

In order to study the global solution theory of the Cauchy problem (1.13) when s =0 (the mass space L2(ℝ3)) and s =1 (the energy space H˜α1(3) ), we introduce the following two quantities, that are formally conserved in time along the solutions.

### Definition 6.1

1. (i)

Let uL2(ℝ3). We define the mass of u as

(u):=uL22.

2. (ii)

Let λ > 0 and let u=φλ+κλGλH˜α1(3) , according to (2.3) We define the energy of u as

(u):=12(Δα)[u]+143(w*|u|2)|u|2dx=12(λφλL22+φλL22+(α+λ4π)|κλ|2λuL22)+143(w*|u|2)|u|2dx.

### Remark 6.1.

For given u, the value of (−Δα) [u] (the quadratic form of (−Δα)) is independent of λ, and so too is the energy (u).

We shall establish suitable conservation laws in order to prolong the local solution globally in time. The mass is conserved in L2(ℝ3) in the following sense.

### Proposition 6.1

(Mass conservation in L2(ℝ3)). Let α ⩾ 0 and let w belong either to the class L(ℝ3) ∩ W1,3(ℝ3) or to the class wL3γ,(3) , for γ(0,32) . For a given fL2(ℝ3), let u be the unique local solution in 𝒞 ((−T*,T*), L2(ℝ3)) to the Cauchy problem (1.15) in the maximal interval (−T*, T*) as given by Theorem 1.3. Then (u(t)) is constant for t ∈ (−T*, T*).

Proof. Let us discuss first the case wL(ℝ3) ∩ W1,3(ℝ3). Consider preliminarily an initial data fH˜α1(3) . Owing to Corollary 4.1, for each t ∈ (−T*, T*)u satisfies itu = −Δαu + (w * |u|2)u as an identity between H˜α1 -functions, whence

itu+Δαu(w*|u|2)u,uH˜α1,H˜α1=0.

The imaginary part of the above identity gives

ddtu(t)L22=0,
which implies that (u(t)) is constant on (−T*,T*). For arbitrary fL2(ℝ3) we use a density argument. Let (fn)n be a sequence in H˜α1(3) such that fnnf in L2(ℝ3), and denote by un the solution to the Cauchy problem (1.13) with initial datum fn. Because of the continuous dependence on the initial data, we have that unu in 𝒞(I,L2(ℝ3)), for every closed interval I ⊂(−T*,T*). Since (un(t)) = (un(0)) = (fn) for every n, we deduce that (ut) = (f) for tI.

Owing to the continuity of the t(u(t)), , we conclude that (u(t)) = (f) for t ∈ (−T*, T*).

Let us discuss now the case wL3γ,(3),γ(0,32) . Consider preliminarily an initial data fH˜α1(3) and a Schwartz potential w. Owing to Corollary 4.1 , for each t ∈ (− T*,T*)u satisfies iδtu = −Δαu+(w*|u|2)u as an identity between H˜α1(3) -functions, and reasoning as above we deduce that ℳ(u(t)) is constant on (−T*, T*). For arbitrary fL2(ℝ3) and wL3γ,(3),γ(0,32) , we use a density argument. Let (fn)n be a sequence in H˜α1(𝕉3) such that fnnf in L2(ℝ3), (wn)n be a sequence of Schwart potentials such that wnnw in L3γ,(3) , and denote by un the L2 strong solution to the Cauchy problem (1.13) with initial datum fn and potential wn. The stability result given by Proposition 3.1 guarantees that unn+u in 𝒞([−T, T], L2(ℝ2)) for some T > 0, whence (un(t))n+(u(t)) for t ∈ [− T, T]. Using the mass conservation for un we deduce that ℳ(u(t)) = ℳ(f) for t ∈ [−T, T]. Repeating the above argument with f replaced by u(t0) for some t0 ∈ (−T*,T*) yields the property that t(u(u)) is constant in a suitable interval around (−T*, T*) t0, and hence, by the arbitrariness of t0, it is locally constant on the whole (−T*,T*). But (−T*;T*) ∋ t(u(t)) is also continuous, whence the conclusion.

We therefore conclude the following.

Proof. [Proof of Theorem 1.5] An immediate consequence of the conservation of the mass, i.e., conservation of the L2 -norm, and of the blow up alternative in L2.

Let us move now to the conservation of mass and energy in the energy space. We observe the following.

### Lemma 6.1.

Let α ⩾ 0 and let wW1,p (ℝ3) for some p > 2. If vnn+v in H˜α1(3) , then (vn)n+(v). As a consequence, if u𝒞([T,T],H˜α1(3)) for some T > 0, then t(u(t)) is continuous on [−T, T].

Proof. The limit (vn)n+(v) follows from the inequality

|(v)(vn)||(Δα)[v](Δα)[vn]|+(w*|v|2)|v|2(w*|vn|2)|vn|2L1
combined with the estimates
|(Δα)[v](Δα)[vn]|vvnH˜a1(vH˜a1+vnH˜α1)
and (w*|v|2)|v|2(w*|vn|2)|vn|2L1(w*|v|2)(|v|2|vn|2)L1+w*(|v|2|vn|2)|vn|2L1w*|v|2Lvvn2(v2+vn2)+|w|*(|vvn|(|v|+|vn|))LvnL22wW1,pvvnH˜α1(vH˜α12+vnH˜α12), the last two steps above following from Hölder’s and Young’s inequality, and from the inequality (4.1).

We then see that mass and energy are conserved in the spherically symmetric component of the energy space.

### Proposition 6.2

(Mass and energy conservation in H˜αrad1(3) ). Let α ⩾ 0. For a given wWrad1,p(3) , p ∈ (2, + ∞) and a given fH˜αrad1(3) , let u be the unique local solution in 𝒞 ((−T*;T*), L2(ℝ3)) 𝒞((T*,T*),H˜α1(3)) to the Cauchy problem (1.15) in the maximal interval(−T*,T*) as given by Theorem 1.3. Then ℳ(u(t)) and ℰ (u(t)) are constant for t ∈ (−T*,T*).

Proof. We start proving the statement for the mass. Owing to Corollary 4.1, for each t ∈ (−T*, T*) u satisfies itu = −Δαu + (w*|u|2)u as an identity in H˜α1(3) , whence

itu+Δαu(w*|u|2)u,uH˜α1,H˜α1=0.

The imaginary part of the above identity gives

ddtu(t)L22=0,
which implies that (u(t)) is constant on (−T*, T*).

Let us prove now that the energy is conserved, first in the special case fH˜α,rad2(3) and wWrad2,p(3) , for p ∈ (2, + ∞). Owing to Corollary 5.1, u satisfies itu = −Δαu+ (w*|u+2)u as an identity in L2(ℝ3), whence

itu+Δαu(w*|u|2)u,tuL2=0.

The real part in the above identity gives

ddt(12Δαu,uL2143(w*|u|2)|u|2dx)=0,
which implies that (u(t)) is constant on (−T*, T*).

For arbitrary fH˜α,rad1(3) and wWrad1,p(3) we use the stability result of Proposition 4.1. Let (fn)n be a sequence in H˜αrad2(3) and (wn)n be a sequence in Wrad2,p(3) such that fnn+f in H˜α1(3) and wnn+w in W1,p(ℝ3), and denote by un the solution to the Cauchy problem (1.13) with initial datum fn and potential wn. Then Proposition 4.1 guarantees that unn+u in 𝒞([−T,T]), 𝒞([T,T],H˜α1(3)) for some T > 0, and Lemma 6.1 implies that (un(t))n+(u(t)) for t ∈ [−T, T]. Using the energy conservation for un we deduce that (u(t)) = (f) for t ∈ [−T, T]. Repeating the above argument with f replaced by u(t0) for some t0 ∈ [−T*, T*] yields the property that t(u(t)) is constant in a suitable interval around t0 and hence, by the arbitrariness of t0, it is locally constant on the whole (−T*, T*). But u𝒞((T*,T*),H˜α,rad1(3)) is also continuous, whence the conclusion.

We are now ready to prove our result on the solution theory for the Cauchy problem (1.13).

Proof. [Proof of Theorem 1.6]

Let u𝒞((−T*, T*), wWrad1,p(3) be the unique local strong solution to (1.13), on the maximal time interval (−T*, T*), with given initial datum f=φλ+cGλH˜α,rad1(3) , for some λ > 0, and given potential wWrad1,p(3) , for some p ∈ (2, + ∞) as provided by Theorem 1.3. Then (−T*,T*) ∋ t(u(t))+(u(t)) is the constant map, as follows from Propositions 6.2. Decomposing u(t) = ϕλ(t) + κλ(t)G λ for each t ∈ [−T*, T*] and using (2.4) we find

u(t)H˜α12φλ(t)H12+|κλ(t)|2(λ+1)u(t)L2+(λφλ(t)L22λu(t)L22+φλ(t)L22+(α+λ4π)|κλ(t)|2)(u(t))+12(Δα)[u(t)]. (*)

For part (i) of the statement, we observe that

supt(T*,T*)u(t)H˜α12supt(T*,T*)((u(t))+12(Δα)[u(t)])supt(T*,T*)((u(t))+(u(t))+(w*|u(t)|2)|u(t)|2Lx1)1+supt(T*,T*)w*|u|2Lxu)Lx221+supt(T*,T*)wWs,puH˜α12fL22,
having used (*) the estimate (4.1), and the mass and energy conservation. Therefore, if ‖fL2 is sufficiently small (depending only on ‖wWs,p), then
supt(T*,T*)u(t)H˜a121,
and we conclude that solution is global, owing to the blow up alternative.

For part (ii) of the statement, the additional assumption w ⩾ 0 implies

12(Δα)[u(t)]12(Δα)[u(t)]+143(w*|u(t)|2)|u(t)|2dx=(u(t)),
which, combined with (*) and the mass and energy conservation yields
supt(T*,T*)u(t)H˜α12supt(T*,T*)((u(t))+(u(t)))1.

Therefore, the solution is global, by the blow up alternative. Since this is true for every initial datum fH˜α,rad1(3) , we deduce global well-posedness for (1.13).

## 7. Comments on the spherically symmetric solution theory

As initially mentioned in the Introduction and then shown in the preceding discussion, part of the solution theory was established for spherically symmetric potentials and solutions (Theorems 1.4 and 1.6) and in this Section we collect our remarks on the emergence of such a feature.

This is indeed a natural phenomenon both for the local high regularity theory and for the global theory in the energy space, as we are now going to explain. Of course, the spherically symmetric solution theory is the most relevant in the study of the singular Hartree equation, since the linear part, namely the operator −Δα, differs from the ordinary −Δ precisely in the L2 -sector of rotationally symmetric functions.

For the local theory, one ineludible ingredient of the fixed point argument is the treatment of the non-linear part of the solution map (1.14) with a H˜αs -estimate that we close by means of the trilinear estimate (4.3)/(5.4).

This estimate is designed for functions of the form hu, where h = w*|u|2, and it is crucially sensitive to the specific structure of the space H˜αs(3) for s>12 (Theorem 2.1 (ii)-(iii)). In particular, in order to recognise that the regular part hu is indeed a Hs -function, one must show that (h–h(0))GλHs(ℝ3). Technically this is dealt with by means of the fractional Leibniz rule, suitably generalised so as to avoid the direct Lp -estimate of s derivatives of each factor hh(0) and Gλ; already heuristically it is clear that this only works with a sufficient vanishing rate of hh(0) as |x| →0 in order to compensate the local singularity of Gλ.

For intermediate regularity ( Proposition 2.1 and Lemmas 4.14.2) the vanishing rate h(x) − h(0) ~|x|θ that can be deduced from the embedding w*|u2| ∈ 𝒞0, θ(ℝ3) is enough to close the argument and no spherical symmetry is required. For high regularity ( Proposition 2.2 - Lemma 2.2, and Lemmas 5.15.1) the embedding w*|u2| ∈ 𝒞1, θ(ℝ3) would only guarantee an insufficient vanishing rate h(x) −h(0) ~ |x|; since one needs h(x) − h(0) ~|x|1+θ, this requires the additional condition (∇h)(0) = 0. For the latter condition to hold for h = w*|u|2, as shown in the proof of Lemma 5.1 (ii), the spherical symmetry of both w and u appears as the most natural and explicitly treatable assumption.

In fact, the condition (∇h)(0) = 0 is even more crucial and apparently unavoidable in one further point of the argument huH˜αs(3) , because unlike the intermediate regularity case, where it suffices to prove that the regular component of hu is a Hs-function, in the high regularity case one must also prove that such regular component satisfies the correct boundary condition in connection with the singular component. As shown in the proof of Lemma 5.2, the correct boundary condition is equivalent to |x|−1(h(x) −h(0)) →0 as |x| →0, for which ∇h(0) = 0 is again necessary.

Concerning the global theory in the energy space, the emergence of a solution theory for spherically symmetric functions is due to one further mechanism. As usual, globalisation is based upon the mass and energy conservation. In the theory of semi-linear Schrödinger equations it is typical that the conservation laws are deduced from a suitably regularised problem (see, e.g., the proof of [12, Theorem 3.3.5]. In the present context ( Proposition 6.2) we follow this scheme showing first the conservation laws at the level of H˜α2 - regularity, and then controlling the stability of a density argument which is set for H˜α1 -regularity. Clearly the first step appeals to the local H˜α2 -theory, which is derived only for the spherically symmetric case, thus the stability argument can only work in the spherically symmetric sector of the energy space.

## Acknowledgements

We are deeply indebted to G. Dell’Antonio for many enlightening discussions on the subject and to V. Georgiev for his precious advices on his work [16] and on Theorem 2.4.

## Bibliography

[5]S Albeverio, F Gesztesy, R Høegh-Krohn, and H Holden, Solvable Models in Quantum Mechanics, Springer-Verlag, New York, 1988. Texts and Monographs in Physics
[6]J Angulo Pava and LCF Ferreira, On the Schrödinger equation with singular potentials, Differential Integral Equations, Vol. 27, No. 7–8, 2014, pp. 767-800.
[9]N Benedikter, M Porta, and B Schlein, Effective evolution equations from quantum dynamics, Springer, Cham, 2016. Springer Briefs in Mathematical Physics, Vol. 7
[10]J Bergh and J Löfström, Interpolation spaces. An introduction, Springer-Verlag, Berlin-New York, 1976. Grundlehren der Mathematischen Wissenschaften, No. 223.
[12]T Cazenave, Semilinear Schrödinger equations, Courant Lecture Notes in Mathematics, Vol. 10, New York University Courant Institute of Mathematical Sciences, New York, 2003.
[16]K Fujiwara, V Georgiev, and T Ozawa, Higher Order Fractional Leibniz Rule, Journal of Fourier Analysis and Applications, Apr 2017.
[17]V Georgiev, A Michelangeli, and R Scandone, On fractional powers of singular perturbations of the Laplacian, Journal of Functional Analysis, 2018. https://doi.org/10.1016/j.jfa.2018.03.007
[20]F Iandoli and R Scandone, Dispersive estimates for Schrödinger operators with point interactions in 𝕉3, A Michelangeli and G Dell’Antonio (editors), Advances in Quantum Mechanics: Contemporary Trends and Open Problems, Springer International Publishing, pp. 187-199. Springer INdAM Series, vol. 18
[23]C Miao, G Xu, and L Zhao, The Cauchy problem of the Hartree equation, J. Partial Differential Equations, Vol. 21, No. 1, 2008, pp. 22-44.
[25]A Michelangeli, A Ottolini, and R Scandone, Fractional powers and singular perturbations of differential operators, 2018. arXiv:2018.01
[26]A Michelangeli and R Scandone, Point-like perturbed fractional Laplacians through shrinking potentials of finite range, Vol. 10191, 2018. arXiv:1803.
[28]GV Skornyakov and KA Ter-Martirosyan, Three Body Problem for Short Range Forces. I. Scattering of Low Energy Neutrons by Deuterons, Sov. Phys. JETP, Vol. 4, 1956, pp. 648-661.
Journal
Journal of Nonlinear Mathematical Physics
Volume-Issue
25 - 4
Pages
558 - 588
Publication Date
2021/01/06
ISSN (Online)
1776-0852
ISSN (Print)
1402-9251
DOI
10.1080/14029251.2018.1503423How to use a DOI?
Open Access

TY  - JOUR
AU  - Alessandro Michelangeli
AU  - Alessandro Olgiati
AU  - Raffaele Scandone
PY  - 2021
DA  - 2021/01/06
TI  - Singular Hartree equation in fractional perturbed Sobolev spaces
JO  - Journal of Nonlinear Mathematical Physics
SP  - 558
EP  - 588
VL  - 25
IS  - 4
SN  - 1776-0852
UR  - https://doi.org/10.1080/14029251.2018.1503423
DO  - 10.1080/14029251.2018.1503423
ID  - Michelangeli2021
ER  -