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

in the complex-valued unknown u ≡ u(x, t), t ∈ ℝ, x ∈ ℝ^d, for given measurable functions V, w : ℝ^d → 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, …, x_N) ~ u₀(x₁)…u₀(x_N), whose evolution can be proved to retain the approximate form Ψ(x₁, …, x_N; t) ~ u(x₁, t)…u(x_N, t), for some one-body orbital u ∈ L² (ℝ^d) 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^–1∑_j<k w(x_j – x_k), and when applied to a wave function of the approximate form u(x₁) · · · u(x_N) 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,

The natural energy space is therefore H¹(ℝ^d), and the equation is energy sub-critical for w ∈ L¹(ℝ^d) + L^∞(ℝ^d) and mass sub-critical for w ∈ L^q(ℝ^d) + L^∞(ℝ^d), for q≥max{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, … X_M ∈ 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 (x−X) around the point X, for some locally regular potential 𝒱 is replaced by V(x) = δ(x−X) and more concretely we study the Cauchy problem for an equation of the form

There will be no substantial loss of generality, in all the following discussion, if we take one point centre X only, instead of X₁, … , X_M ∈ (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(ℝ^d) of the symmetric operator −Δ|C0∞(ℝd) , and therefore acts precisely as −Δ on H² -functions supported away from the origin. In fact, −Δ|C0∞(ℝd) is already essentially self-adjoint when d ⩾ 4, with operator closure given by the self-adjoint −Δ with domain H²(ℝ^d), 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∞(ℝ3\{0}) is a one-parameter family of operators −Δ_α, α ∈ (−∞, +∞], defined by

where λ > 0 is an arbitrarily fixed constant and is the Green function for the Laplacian, that is, the distributional solution to (−Δ + λ)G_λ = δ in 𝒟′(ℝ³).

The quadratic form of −Δ_α is given by

The above decompositions of a generic ψ ∈ 𝒟(− Δ_α) or ψ ∈ 𝒟[− Δ_α] are unique and are valid for every chosen λ. The extension −Δ_{α = ∞} is the Friedrichs extension and is precisely the self-adjoint −Δ on L²(ℝ³) with domain H²(ℝ³).

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

in terms of subspaces Lℓ2(ℝ3) of definite angular symmetry, and it is a non-trivial modification of the negative Laplacian in the spherically symmetric sector only, i.e.,

Each ψ ∈ 𝒟(− Δ_α) satisfies the short range asymptotics

or also, in momentum space, for some c_ψ, d_ψ ∈ 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, −Δ_α 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 −Δ_α is given by

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

In order to avoid non-essential additional discussions, we restrict ourselves once and for all to positive α’s. In fact, −Δ_α 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 −Δ_α and to its positivity for α ⩾ 0, the ‘singular (or perturbed) Schrödinger propagator’ t ↦ e^itΔ_α leaves the domain of each power of −Δ_α invariant. In complete analogy to the non-perturbed case, where the free Schrödinger propagator t ↦ e^itΔ leaves the Sobolev space H^s(ℝ³) = 𝒟(−Δ)^s/2 invariant, and the solution theory for the ordinary Hartree equation is made in H^s(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(ℝ3) , the ‘singular Sobolev space’ of order s, namely the Hilbert space

equipped with the ‘fractional singular Sobolev norm’

It is worth remarking that whereas the kernel of the propagator t ↦ e^itΔ_α is known since long [4,27], the characterisation of the singular fractional Sobolev space H˜αs(ℝ3) 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

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

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

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 e^itΔ_α. 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.

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(ℝ3) and H˜α3/2(ℝ3) is not as clean as that of H˜αs(ℝ3) 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 W^s,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.

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(ℝ3) (as opposite to H^s(ℝ³), or also to H˜αs(ℝ3) 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 − Δ_α, namely as a self-adjoint extension of (−d2dx2)|C0∞(ℝ\{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 − Δ_α 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

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 L^p -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 − Δ_α, that is, the spaces H˜αs(ℝ3) introduced in (1.11)–(1.12)

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 t ↦ e^itΔ_α associated with − Δ_α, 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 − Δ_α if

that is, q=4r3(r−2)∈(4,+∞] .

Let us estimate the term ℛ₃. Since, by (2.18) and Sobolev’s embedding,

and since ‖G˜‖2q4−2<+∞ because 2qq−2<3 for q > 6, then Holder’s inequality yields

Next, let us estimate ℛ₁. When s∈(12,1] , we choose s₁ = s, s₂ = 0, q₁ = q, and q2=2qq−2 and we proceed exactly as for ℛ₃. When instead s∈(1,32) , we choose s₁ = 1, s2=s−1∈(0,12) , q1=(1q−s−13)−1 , and q2=(12−1q1)−1 . Then Sobolev’s embedding w^s,q(ℝ³) ↪ W^1,q₁ (ℝ³) and estimate (ii) above imply

whereas estimate (2.17) and the fact that q₂ < sq₂ < 3 for s∈(1,32) imply

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

Last, let us estimate ℛ₂. Because of the embedding (2.19),

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

Thus,

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

Therefore,

As a consequence,

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

.

Let us split

We estimate the term ℛ₁ by means of the commutator bound (2.16) with s₁ = s and s₂ = 0 and of (2.18), namely

(since 2qq−2∈[2,3) ) and ‖G˜‖2qq−2<+∞ ).

For the estimate of ℛ₂, we observe that s−ϑ(s,q)<32 and hence (2.17) implies

this and the bound (2.23) yield

For ℛ₃, Hölder’s inequality, the property (2.18), and Sobolev’s embedding yield

For ℛ₄, one has

where we used the estimate (2.20) in the second inequality (indeed, s−1∈(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.