Abstract
We study the Hamiltonian for a system of three identical bosons in dimension three interacting via zero-range forces. In order to avoid the fall to the center phenomenon emerging in the standard Ter-Martirosyan–Skornyakov (TMS) Hamiltonian, known as Thomas effect, we develop in detail a suggestion given in a seminal paper of Minlos and Faddeev in 1962 and we construct a regularized version of the TMS Hamiltonian which is self-adjoint and bounded from below. The regularization is given by an effective three-body force, acting only at short distance, that reduces to zero the strength of the interactions when the positions of the three particles coincide. The analysis is based on the construction of a suitable quadratic form which is shown to be closed and bounded from below. Then, domain and action of the corresponding Hamiltonian are completely characterized and a regularity result for the elements of the domain is given. Furthermore, we show that the Hamiltonian is the norm resolvent limit of Hamiltonians with rescaled non-local interactions, also called separable potentials, with a suitably renormalized coupling constant.
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1 Introduction
In a system of nonrelativistic quantum particles at low temperature, the thermal wavelength is typically much larger than the range of the two-body interactions and therefore the details of the interactions are irrelevant. In these conditions, the effective behavior of the system is well described by a Hamiltonian with zero-range forces, where the only physical parameter characterizing the interaction is the scattering length.
The mathematical construction of such Hamiltonians as self-adjoint and, possibly, lower-bounded operators is straightforward in dimension one since standard perturbation theory of quadratic forms can be used. Moreover, the Hamiltonian can be obtained as the resolvent limit of approximating Hamiltonians with rescaled two-body smooth potentials (see [4] for the case of three particles and [18] for n bosons). On the contrary, in dimensions two and three the interaction is too singular and more refined techniques are required for the construction. The two-dimensional case is well understood ([11, 12], see also [19] for applications to the Fermi polaron model), and it has been recently shown [17] that the Hamiltonian is the norm resolvent limit of Hamiltonians with rescaled smooth potentials and with a suitably renormalized coupling constant.
In dimension three, the problem is more subtle due to the fact that a natural construction in the case of \(n \geqslant 3\) particles, obtained following the analogy with the one particle case, leads to the so-called TMS Hamiltonian [35] which is symmetric but not self-adjoint. Furthermore, all its self-adjoint extensions are unbounded from below. Such instability property, known as Thomas effect, can be seen as a fall to the center phenomenon, and it is due to the fact that the interaction becomes too strong and attractive when (at least) three particles are very close to each other. This phenomenon does not occur in dimension two because the singularity of the wave function at the coincidence hyperplane is a mild logarithmic one.
The Thomas effect was first noted by Danilov [10] and then rigorously analyzed by Minlos and Faddeev [28, 29], and it makes the Hamiltonian unsatisfactory from the physical point of view (for some recent mathematical contributions see, e.g., [5, 6, 14, 22] with references therein). For other approaches to the construction of many-body contact interactions in \(\mathbb {R}^3\), we refer to [2, 31, 36].
We note that a different situation occurs in the case (which is not considered here) of a system made of two species of fermions interacting via zero-range forces, where it happens that for certain regime of the mass ratio the TMS Hamiltonian is in fact self-adjoint and bounded from below (for mathematical results in this direction see, e.g., [7,8,9, 15, 24,25,26,27, 30, 32]). For a general mathematical approach to the construction of singularly perturbed self-adjoint operators in a Hilbert space, we refer to [33, 34].
Inspired by a suggestion contained in [28], in this paper we propose a regularized version of the TMS Hamiltonian for a system of three bosons and we prove that it is self-adjoint and bounded from below. Furthermore, we show that the Hamiltonian is the norm resolvent limit of approximating Hamiltonians with rescaled non-local interactions, also called separable potentials, and with a suitably renormalized coupling constant.
We stress that a more interesting problem from the physical point of view would be the approximation in norm resolvent sense by a sequence of Hamiltonians with (local) rescaled potentials as in [17] for the two-dimensional case and [4, 18] for the one-dimensional case. Such a result is more difficult to prove, and we plan to approach it in a forthcoming work.
We also believe that our approach and results can be generalized to the case of three different particles. For the case of a system made of N bosons in interaction with another particle, see [13].
In the rest of this section, we introduce our Hamiltonian at heuristic level and discuss some of its properties. Let us consider a system of three identical bosons with masses 1/2 in the center of mass reference frame and let \(\mathbf{x}_1, \mathbf{x}_2\) and \( \mathbf{x}_3 = -\mathbf{x}_1 - \mathbf{x}_2\) be the Cartesian coordinates of the particles. Let us introduce the Jacobi coordinates \( \mathbf{r}_{23} \equiv \mathbf{x}, \; \mathbf{r}_1 \equiv \mathbf{y}\), namely
The other two pairs of Jacobi coordinates in position space are \(\, \mathbf{r}_{31}= -\frac{1}{2}\mathbf{x}+\mathbf{y}\,, \;\mathbf{r}_2 = - \frac{3}{4} \mathbf{x}-\frac{1}{2} \mathbf{y}\,\) and \(\, \mathbf{r}_{12}= - \frac{1}{2} \mathbf{x}-\mathbf{y}\,, \; \mathbf{r}_3 = \frac{3}{4} \mathbf{x}- \frac{1}{2} \mathbf{y}\). Due to the symmetry constraint, the Hilbert space of states is
Notice that the symmetry conditions in (1.1) correspond to the exchange of particles 2, 3 and 3, 1 and they also imply the symmetry under the exchange of particles 1, 2, i.e., \( \psi (\mathbf{x},\mathbf{y})= \psi \big ( \frac{1}{2} \mathbf{x}- \mathbf{y}, - \frac{3}{4} \mathbf{x}-\frac{1}{2} \mathbf{y}\big )\).
The formal Hamiltonian describing the three boson system in the Jacobi coordinates reads
where \( \mu \in \mathbb {R}\) is a coupling constant and \(H_0\) is the free Hamiltonian
Our aim is to construct a rigorous version of (1.2) as a self-adjoint, and possibly bounded from below, operator in \(L^2_{\text {sym}}(\mathbb {R}^6)\). In other words, we want to define a self-adjoint perturbation of the free Hamiltonian (1.3) supported by the coincidence hyperplanes
Following the analogy with the one particle case [1], a natural attempt is to define the TMS operator acting as the free Hamiltonian outside the hyperplanes and characterized by a (singular) boundary condition on each hyperplane. Specifically, on \(\pi _{23}\) one imposes
where \(x:= |\mathbf{x}|\), \(\xi \) is a function depending on \(\psi \) and
has the physical meaning of two-body scattering length (and it can be related to \(\mu \) via a renormalization procedure). Notice that, due to the symmetry constraint, (1.4) implies the analogous boundary conditions on \(\pi _{31}\) and \(\pi _{12}\).
As already recalled, the TMS operator defined in this way is symmetric but not self-adjoint and its self-adjoint extensions are all unbounded from below. Therefore, the natural problem arises of figuring out if and how one can modify the boundary condition (1.4) to obtain a bounded from below Hamiltonian. In a comment on this point, at the end of the paper [28] the authors claim that it is possible to find another physically reasonable realization of \(\tilde{H}\) as self-adjoint and bounded from below operator. They also affirm that the recipe consists in the replacement
in the boundary condition (1.4), where K is a convolution operator in the Fourier space with a kernel \(K(\mathbf{p}-\mathbf{p}')\) satisfying
with \(p=|\mathbf{p}|\) and the positive constant \(\gamma \) sufficiently large. The authors do not explain the reason of their assertion neither they clarify the physical meaning of the boundary condition (1.6). They only conclude: “A detailed development of this point of view is not presented here because of lack of space” and, strangely enough, their idea has never been developed in the literature.
Almost 20 years later, Albeverio, Høegh-Krohn and Wu [3] have proposed an apparently different recipe to obtain a bounded from below Hamiltonian, i.e., the replacement
with \(y=|\mathbf{y}|\), in the boundary condition (1.4), where again the positive constant \(\gamma \) is chosen sufficiently large. Also, the proof of this statement has been postponed to a forthcoming paper which has never been published. Even if it has not been explicitly noted by the authors of [3], it is immediate to realize that the two proposals contained in [28] and [3] essentially coincide in the sense that in [28] the term added in the boundary condition is the Fourier transform of the term added in [3]. It is also important to stress that, according to the claim in [28], only the asymptotic behavior of \(K(\mathbf{p})\) for \(|\mathbf{p}| \rightarrow \infty \) (see (1.7)) is relevant to obtain a lower-bounded Hamiltonian. Correspondingly, it must be sufficient to require only the asymptotic behavior \(\gamma y^{-1} + O(1)\) for \(y \rightarrow 0\) for the boundary condition in position space in [3].
The above considerations suggest to define our formal regularized TMS Hamiltonian \(\tilde{H}_{\text {reg}}\) as an operator in \(L^2_{\text {sym}}(\mathbb {R}^6)\) acting as the free Hamiltonian outside the hyperplanes and characterized by the following boundary condition on \(\pi _{23}\)
where \(\Gamma _{\text {reg}}\) is defined by
and \(\theta \) is a real cutoff function. Due to the symmetry constraint, (1.8) implies the boundary condition on \(\pi _{31}\) and \(\pi _{12}\)
We assume different hypothesis on \(\theta \) depending on the situation. The first possible hypothesis is
The simplest choice satisfying (H1) is the characteristic function
The second possible hypothesis requires some minimal smoothness
Examples satisfying (H2) are \(\theta (r)= e^{-r/b}\) or \(\theta \in C_0^\infty (\mathbb {R}^+)\) such that \(\theta (r) = 1\) for \(r \leqslant b\), \(b>0\).
We stress that (H2) implies (H1). Hypothesis (H2) will be used only in Sect. 6, where we study the approximation with separable potentials, and in Appendix A.2.
Note that the crucial point is the behavior of \(\theta \) at the origin, which represents the minimal requirement for the regularization of the dynamics at short distances. The support of the function \(\theta \) is not relevant, in particular a simple choice would be \(\theta (r)=1\).
Needless to say, the operator \(\tilde{H}_{\text {reg}}\) is only formally defined since its domain and action are not clearly specified. Our aim is to construct an operator which represents the rigorous counterpart of \(\tilde{H}_{\text {reg}}\) using a quadratic form method. The main idea of the construction has been announced and outlined in [14], where a more detailed historical account of the problem is given. We also mention the recent paper [22], where the construction is approached using the theory of self-adjoint extensions.
Let us make some comments on the formal operator \(\tilde{H}_{\text {reg}}\).
As we already remarked, the singular behavior of \((\Gamma _{\text {reg}}\xi )(\mathbf{y})\) for \(y \rightarrow 0\) corresponds in the Fourier space to a convolution operator whose kernel has the asymptotic behavior (1.7). It will be clear in the course of the proofs in Sect. 3 that such a behavior is chosen in order to compensate the singular behavior of the off-diagonal term appearing in the quadratic form. In this sense, one can say that the singularity of \((\Gamma _{\text {reg}}\xi )(\mathbf{y})\) for \(y \rightarrow 0\) is the minimal one required to obtain a self-adjoint and bounded from below Hamiltonian.
Concerning the physical meaning of our regularization, we recall that we have replaced the parameter \(\beta \) in (1.4) with \(\Gamma _{\text {reg}}\) in (1.8). By analogy with the definition (1.5), we can introduce an effective, position-dependent scattering length
which can be interpreted as follows. For simplicity, let us fix \(\beta >0\) and choose the cutoff (1.10). Consider the zero-range interaction between the particles 2, 3 which takes place when \(\mathbf{x}_2=\mathbf{x}_3\), i.e., for \(\mathbf{x}=0\). In these conditions, the coordinate y is the distance between the third particle 1 and the common position of particles 2, 3. Then one has
i.e., the effective scattering length associated with the interaction of particles 2, 3 is equal to \(\mathfrak {a}\) if the third particle 1 is at a distance larger than b while for distance smaller than b the scattering length depends on the position of the particle 1 and it decreases to zero, i.e., the interaction vanishes, when the distance goes to zero. In other words, we introduce a three-body interaction which is a common procedure in certain low-energy approximations in nuclear physics. Such three-body interaction reduces to zero the two-body interaction when the third particle approaches the common position of the first two. This is precisely the mechanism that prevents in our model the fall to the center phenomenon, i.e., the Thomas effect.
The paper is organized as follows.
In Sect. 2, starting from the formal Hamiltonian \(\tilde{H}_{\text {reg}}\), we construct a quadratic form which is the initial point of our analysis and we formulate our main results.
In Sect. 3, we prove that the quadratic form is closed and bounded from below for any \(\gamma \) larger than a threshold explicitly given.
In Sect. 4, we characterize the self-adjoint and bounded from below Hamiltonian H uniquely associated with the quadratic form which is the rigorous counterpart of \(\tilde{H}_{\text {reg}}\).
In Sect. 5, we introduce a sequence of approximating Hamiltonians \(H_{\varepsilon }\) with rescaled separable potentials and a renormalized coupling constant and we prove a uniform lower bound on the spectrum.
In Sect. 6, we show that the Hamiltonian H is the norm resolvent limit of the sequence of approximating Hamiltonians \(H_{\varepsilon }\).
In the Appendix, we prove a technical regularity result for the elements of the domain of H.
In conclusion, we collect here some of the notation frequently used throughout the paper.
- \(\mathbf{x}\) is a vector in \(\mathbb {R}^3\) and \(x=|\mathbf{x}|\).
- \(\hat{f} \equiv {\mathcal {F}} f\) is the Fourier transform of f.
- For a linear operator A acting in position space, we denote by \(\hat{A} = {\mathcal {F}} A {\mathcal {F}}^{-1}\) the corresponding operator in the Fourier space.
- \(H^s(\mathbb {R}^n)\) denotes the standard Sobolev space of order \(s>0\) in \(\mathbb {R}^n\).
- \(\Vert \cdot \Vert \) and \((\cdot , \cdot )\) are the norm and the scalar product in \(L^2(\mathbb {R}^n)\), \(\Vert \cdot \Vert _{L^p}\) is the norm in \(L^p(\mathbb {R}^n)\), with \(p\ne 2\), and \(\Vert \cdot \Vert _{H^s}\) is the norm in \(H^s(\mathbb {R}^n)\). It will be clear from the context if \(n=3\) or \(n=6\).
- \(f \big |_{\pi _{ij}} \in H^{s} (\mathbb {R}^3)\) is the trace of \(f \in H^{3/2 + s}(\mathbb {R}^6)\), for any \(s>0\).
- \(\mathcal {B}({\mathcal {K}}, {\mathcal {H}})\) is the Banach space of the linear bounded operators from \({\mathcal {K}}\) to \({\mathcal {H}}\), where \({\mathcal {K}}\), \({\mathcal {H}}\) are Hilbert spaces and \(\mathcal {B}( {\mathcal {H}})=\mathcal {B}({\mathcal {H}}, {\mathcal {H}})\).
- c will denote numerical constant whose value may change from line to line.
2 Construction of the Quadratic Form and Main Results
Here, we describe a heuristic procedure to construct the quadratic form \( (\psi , \tilde{H}_{\text {reg}} \psi )\) associated with the formal Hamiltonian \( \tilde{H}_{\text {reg}}\) defined in the introduction.
Since we mainly work in the Fourier space, we introduce the coordinates \( \mathbf{k}_{23} \equiv \mathbf{k},\;\) \( \mathbf{k}_1 \equiv \mathbf{p}\), conjugate variables of \(\mathbf{x},\mathbf{y}\)
where \(\mathbf{p}_1, \mathbf{p}_2\) and \(\mathbf{p}_3= -\mathbf{p}_1 - \mathbf{p}_2\) are the momenta of the particles. The other two pairs of Jacobi coordinates in momentum space are \(\, \mathbf{k}_{31}= - \frac{1}{2} \mathbf{k}+ \frac{3}{4} \mathbf{p}\), \(\,\mathbf{k}_2= -\mathbf{k}-\frac{1}{2} \mathbf{p}\;\) and \(\;\mathbf{k}_{12} = - \frac{1}{2} \mathbf{k}- \frac{3}{4} \mathbf{p}\), \(\,\mathbf{k}_3 = \mathbf{k}- \frac{1}{2} \mathbf{p}\). In the Fourier space, the Hilbert space of states is equivalently written as
The symmetry conditions in (2.1) correspond to the exchange of particles 2, 3 and 3, 1 and they also imply the symmetry under the exchange of particles 1, 2, i.e., \(\psi (\mathbf{k},\mathbf{p})= \hat{\psi }\big (\frac{1}{2}\mathbf{k}- \frac{3}{4} \mathbf{p}, -\mathbf{k}-\frac{1}{2} \mathbf{p}\big )\). Moreover, the free Hamiltonian is
We also introduce the “potential” produced by the “charge density” \(\xi \) distributed on the hyperplane \(\pi _{23}\) by
and one can verify that the function \( \mathcal G_{23}^{\lambda } \xi \) satisfies the equation
in distributional sense. Analogously, we have
and the potential produced by the three charge densities is
Note that the function \( \widehat{{\mathcal {G}}^{\lambda }} \hat{\xi }\) is symmetric under the exchange of particles; hence, it belongs to \(L^2_{\text {sym}}(\mathbb {R}^6)\), see Eq. (2.1).
These potentials exhibit the same singular behavior required in the boundary conditions (1.8). Indeed, we have for \( x \rightarrow 0, \; \mathbf{y}\ne 0\)
Taking into account of the contribution of the other two charge densities, we have for \(x \rightarrow 0, \; \mathbf{y}\ne 0\)
The asymptotic behavior (2.8) suggests to represent an element \(\psi \) satisfying (1.8) as
where \(w^{\lambda }\) is a smooth function. Using (2.8), the boundary condition (1.8) is rewritten as
where \(w^{\lambda }(0,\cdot ) \equiv w^{\lambda } \big |_{\pi _{23}}\) denotes the trace of \(w^{\lambda }\) on the hyperplane \(\pi _{23}\).
We are now ready to construct the energy form \( (\psi , \tilde{H}_{\text {reg}} \psi )\) associated with the formal Hamiltonian \(\tilde{H}_{\text {reg}}\) defined in the Introduction. For \(\varepsilon >0\), let us consider the domain \(D_{\varepsilon }= \{ (\mathbf{x},\mathbf{y}) \in \mathbb {R}^6 \;\; s.t.\;\; |\mathbf{x}_i - \mathbf{x}_j|> \varepsilon \; \; \forall \; i\ne j\}\). Taking into account that \(\tilde{H}_{\text {reg}}\) acts as the free Hamiltonian in \(D_{\varepsilon }\), the decomposition (2.9) and the fact that \((H_0 + \lambda ) {\mathcal {G}}^{\lambda } \xi =0\) in \(D_{\varepsilon }\), we have
Using (2.3), (2.4), (2.5) and the symmetry properties of \(w^{\lambda }\), the last term in (2.11) reduces to
By (2.11), (2.12) and the boundary condition (2.10), we finally arrive at the definition of the following quadratic form.
Definition 2.1
where \((\hat{\Gamma }_{reg } \hat{\xi })(\mathbf{p})\) is the Fourier transform of the function defined in Eq. (1.9). We define the quadratic form F on the domain
Remark 2.2
From the explicit expression of the potential (2.6), one immediately sees that for any \(\xi \in H^{1/2}(\mathbb {R}^3)\)
Therefore, we have \({\mathcal {D}}(F) \supset H^1(\mathbb {R}^6)\) and, for fixed \(\lambda \), the decomposition \(\psi =w^{\lambda }+\mathcal {G}^{\lambda } \xi \) is unique.
In the rest of the paper, we assume Definition 2.1 as the starting point of our rigorous analysis. Let us conclude this section collecting the main results we prove in the paper. First, we show that for any \(\gamma >\gamma _c,\) where
the quadratic form F on the domain \({\mathcal {D}}(F)\) is closed and bounded from below. This is the content of the next theorem whose proof is presented in Sect. 3.
Theorem 2.3
Assume (H1) and let \(\gamma >\gamma _c\), then:
-
(i)
there exists \(\lambda _0>0\) such that for all \(\lambda >\lambda _0\) the quadratic form \(\Phi ^\lambda \) in \(L^2(\mathbb {R}^3)\) defined in (2.15), is coercive and closed on the domain \({\mathcal {D}} (\Phi ^\lambda ) = H^{1/2}(\mathbb {R}^3)\);
-
(ii)
the quadratic form \(F, {\mathcal {D}}(F)\) on \(L_{\text {sym}}^2(\mathbb {R}^6)\) introduced in Definition 2.1 is bounded from below and closed.
Theorem 2.3 implies that \(F,{\mathcal {D}}(F)\) defines a self-adjoint and bounded from below Hamiltonian \(H, {\mathcal {D}}(H)\) in \(L^2_\mathrm {sym}(\mathbb {R}^6)\). If we denote by \(\Gamma ^\lambda ,{\mathcal {D}}(\Gamma ^\lambda )\) the positive, self-adjoint operator associated with the quadratic form \(\Phi ^\lambda \), \({\mathcal {D}} (\Phi ^\lambda ) = H^{1/2}(\mathbb {R}^3)\), then domain and action of the Hamiltonian are characterized in the following proposition.
Theorem 2.4
Under the same assumptions as in Theorem 2.3, we have
The proof of Theorem 2.4 is deferred to Sect. 4.
The next question we address is the approximation through a regularized Hamiltonian \(H_\varepsilon ,{\mathcal {D}}(H_\varepsilon )\) with non-local interactions, also known as separable potentials. In order to define the approximating model, we need to first introduce some notation.
Let \(\chi \in L^2(\mathbb {R}^3, (1+x) \text {d}\mathbf{x}) \cap L^1(\mathbb {R}^3, (1+x) \text {d}\mathbf{x})\), spherically symmetric, real-valued, nonnegative and such that \(\int \text {d}\mathbf{x}\, \chi (x) = 1\). Moreover, set
For all \(\varepsilon >0\), we define the scaled function \(\chi _\varepsilon \) as
and the operator \(g_\varepsilon \) on \(L^2(\mathbb {R}^3)\)
with \(\Gamma _{{\text {reg}}}\) given in (1.9). Then, the approximating Hamiltonian \(H_\varepsilon ,{\mathcal {D}}(H_\varepsilon )\) on \(L^2_{{\text {sym}}}(\mathbb {R}^6)\) is defined as
where S is the permutation operator exchanging the triple of labels (1, 2, 3) in the triple (2, 3, 1). So that,
Taking into account that \(\chi _{\varepsilon } \rightarrow \delta \,\) for \(\varepsilon \rightarrow 0\) in distributional sense, one sees that the three interaction terms in (2.23) for \(\varepsilon \rightarrow 0\) formally converge to zero-range interactions supported on the hyperplanes \(\pi _{23}\), \(\pi _{31}\), \(\pi _{12}\).
Moreover, the operator \(g_{\varepsilon }\) plays the role of renormalized coupling constant. We also note that in position space \(g_{\varepsilon }\) reduces to the multiplication operator by \(g_{\varepsilon }(y)\) which for \(\varepsilon \) small behaves as
In particular, if we assume for simplicity that \(\theta \) is the characteristic function (1.10) then we find that for \(y>b\) we have the standard behavior required to approximate a point interaction in dimension three with scattering length \( - \beta ^{-1}\) (see [1], chapter II.1.1, pages 111–112), while for \(y\leqslant b\) we have introduced a dependence on the position y such that the modified scattering length \(- \big (\beta +\frac{\gamma }{y}\big )^{-1}\) goes to zero as \(y \rightarrow 0\).
In Sect. 5 (see Theorem 5.7), we prove a uniform lower bound for the spectrum of \(H_{\varepsilon }\), i.e., we show that there exists \(\lambda _1 >0\), independent of \(\varepsilon \), such that \(\inf \sigma (H_\varepsilon )>-\lambda _1\).
Finally, we prove the norm resolvent convergence of \(H_\varepsilon ,{\mathcal {D}}(H_\varepsilon )\) to \(H, {\mathcal {D}}(H)\) as \(\varepsilon \rightarrow 0\). More precisely, the following result holds true and it is proved in Sect. 6.
Theorem 2.5
Assume (H2) and \(\gamma >\max \{\gamma _0,2\}\). Moreover, let us define \({\lambda _{{\text {max}}}}:= \max \{\lambda _1, -\inf \sigma (H)\}\). Then, for all \(z\in \mathbb {C}\backslash [-{\lambda _{{\text {max}}}},\infty )\) there holds true
3 Analysis of the Quadratic Form
In this section, we prove closure and boundedness of the quadratic form F defined by (2.13)–(2.16) for \(\gamma >\gamma _c\) with \(\gamma _c\) defined in (2.17). To this end we first study the quadratic form \(\Phi ^\lambda \) in \(L^2(\mathbb {R}^3)\) given by (2.15) and acting on the domain \({\mathcal {D}}(\Phi ^\lambda )=H^{1/2}(\mathbb {R}^3)\). Recalling the definition of \(\Gamma _\mathrm {reg}\) given in (1.9) and using the fact that
we write
where
with
Note that \(a \in L^{\infty } (\mathbb {R}^3)\) if we assume (H1) and \(a, \nabla a \in L^{\infty }(\mathbb {R}^3)\) if we choose (H2).
We will show that \(\Phi ^{\lambda }\) is equivalent to the \(H^{1/2}\)-norm. First, we prove that \(\Phi ^\lambda (\xi )\) can be bounded from above by \(\Vert \xi \Vert _{H^{1/2}}^2\). This is the content of the next proposition which ensures that \(\Phi ^\lambda \) is well defined on \({\mathcal {D}}(\Phi ^\lambda )=H^{1/2}(\mathbb {R}^3)\).
Proposition 3.1
Assume (H1), \(\lambda >0\) and \(\gamma >0\). Then, there exists \(c>0\) such that
Proof
Using the bound (see [20, Remark 5.12] or [21, 37])
we immediately get
Moreover (see [15, Lemma 2.1]),
Since \(\Phi ^\lambda _\mathrm {diag}\) is clearly bounded by \(\Vert \xi \Vert _{H^{1/2}}^2\), the thesis immediately follows from (3.1). \(\square \)
The next step is to bound from below \(\Phi ^\lambda \), which is our main technical result for the construction of the Hamiltonian. Our main tool is the decomposition of the function \({\hat{\xi }}\) into partial waves (for an alternative approach see, e.g., [30]). Then, we write
where \(Y^\ell _m\) denotes the spherical harmonics of order \(\ell ,m\) and \(\mathbf{p}=(p,\theta _\mathbf{p},\varphi _\mathbf{p})\) in spherical coordinates. Accordingly, we find the following decomposition of the quadratic form \(\Phi ^\lambda \)
where \(\phi ^\lambda _\ell \) is the quadratic form whose action on \(g\in L^2((0,+\infty ),p^2\sqrt{p^2+1} \text {d}p)\) is given by (see, e.g., [8, Lemma 3.1])
where
with \(P_\ell (y)=\frac{1}{2^\ell \ell !}\frac{d^\ell }{\text {d}y^\ell }(y^2-1)^\ell \) the Legendre polynomial of degree \(\ell \).
In the next lemma, we investigate the sign of \(\phi ^\lambda _{\mathrm {off},\ell .}\)
Lemma 3.2
Let \(g\in L^2(\mathbb {R}^+,p^2\sqrt{p^2+1}\,\text {d}p)\) and \(\lambda >0\). Then,
Proof
The proof follows [8]. For the sake of completeness, we give the details below. First, we rewrite
where in the last line we integrated by parts \(\ell \) times. Next, we note that
hence,
with
It is easy to see that \(B_{\ell j}=0\) if \(\ell \) and j do not have the same parity. Moreover, \(B_{\ell j}\leqslant 0\) if \(\ell ,j\) are even and \(B_{\ell j}\geqslant 0\) if \(\ell ,j\) are odd. Then, (3.8) yields (3.7). \(\square \)
Thanks to Lemma 3.2, in order to obtain a lower bound we can neglect \(\phi ^\lambda _{\mathrm {off},\ell }\) with \(\ell \) odd and focus on \(\phi ^0_{\mathrm {off},\ell }\) which control \(\phi ^\lambda _{\mathrm {off},\ell }\) with \(\ell \) even. In the next lemma we show that \(\phi ^0_{\mathrm {off},\ell }\) and \(\phi _{\mathrm {reg},\ell }\) can be diagonalized, note that we also include the analysis of \(\phi ^0_{\mathrm {off},\ell }\) with \(\ell \) odd for later convenience.
Lemma 3.3
Let g be a real analytic function whose Taylor expansion near the origin
has a radius of convergence bigger or equal than one. Define
with \(P_\ell \) being the Legendre polynomials. Then for any \(\ell \in {\mathbb {N}}\), we have
Proof
Using \(P_\ell (y)=\frac{1}{2^\ell \ell !}\frac{d^\ell }{\text {d}y^\ell }(y^2-1)^\ell \) and integrating by parts, one easily prove the first claim. Integrating by parts again, one finds
and the monotonicity follows from the first claim. \(\square \)
Lemma 3.4
Let \(g\in L^2(\mathbb {R}^+,p^2\sqrt{p^2+1}\text {d}p)\). Then,
where
and
Moreover,
and
Proof
The proof is similar to [8, Lemma 3.3, 3.5]. For reader’s convenience we give the details below. With the change of variables \(p_1=e^{x_1}\) and \(p_2=e^{x_2}\), we rewrite
Taking the Fourier transform, we get
with (see, e.g., [16, p. 511])
Proceeding analogously for \(\phi _{\mathrm {reg},\ell }(g)\), we find
where
Finally, noting that \(\sinh (k\pi )=2\sinh (k\frac{\pi }{2})\cosh (k\frac{\pi }{2} )\) and
and recalling that \(P_\ell \) has the same parity of \(\ell \) we get (3.9) and (3.10).
In order to prove the monotonicity properties (3.11), it is sufficient to notice that the Taylor expansions of
have positive coefficients and invoke Lemma 3.3. A dilation of a factor 2 preserve the positivity of the coefficients and then also (3.12) follows from Lemma 3.3. \(\square \)
Notice that
Then, we rewrite the quadratic form as
where
Notice also, see (3.15), that if \({\hat{\xi }} (\mathbf{p})= g(p) \, Y^{\ell }_m (\theta _\mathbf{p}, \varphi _\mathbf{p})\) then \(\Vert g^\sharp \Vert _{L^2(\mathbb {R})}= \Vert \xi \Vert _{\dot{H}^{1/2}(\mathbb {R}^3)}\). Equations (3.16) and (3.17) suggest that we need to ensure that \(S_\ell \) is positive for \(\gamma >\gamma _c\).
From Lemma 3.4 and [6, Lemma 3.2], it follows that for any \(\ell \geqslant 2\) even
where
This is enough to control \(\phi ^0_{\mathrm {off},\ell }\) for \(\ell \geqslant 2\) even with \(\phi ^\lambda _\mathrm {diag}\) as we will show in Proposition 3.6. It remains therefore to study what happens in \(s-\)wave. To this end, we introduce the auxiliary quadratic form \(\Theta ^\lambda _s(g)\), \(s\in (0,1)\) acting on \(L^2(\mathbb {R}^+,p^2\sqrt{p^2+1}\text {d}p)\) and defined by
A key ingredient of the proof of closure and boundedness from below of \(\Phi ^\lambda \) is the following lemma.
Lemma 3.5
Let \(g\in L^2(\mathbb {R}^+,p^2\sqrt{p^2+1}\text {d}p)\) and \(\gamma >\gamma _c\). Then, there exists \(s^* \in (0,1)\) such that \(\Theta ^{\lambda }_{s^*}(g) \geqslant 0\) for any \(\lambda >0\).
Proof
Since \(\phi ^\lambda _{\mathrm {diag}}(g) \geqslant \phi ^0_{\mathrm {diag}}(g)\), by Lemma 3.4 we have
The explicit computation of the two integrals in (3.19) yields
where
It remains to show \(f(k)\geqslant 0\) to conclude. We note that f(k) is even and thus it is enough to consider \(k\geqslant 0.\) We have \(f(0)>0\), \(f_2(k) \geqslant 0\) and
For \(\gamma >\gamma _c\), let us choose \(s^*\) such that \(\max \{0,1-\frac{\pi }{\sqrt{3}}(\gamma - \gamma _c) \}< s^* <1\). Hence, we also have \(f_1(k) \geqslant 0\) and the proof is complete. \(\square \)
We are ready to prove the lower bound for \(\Phi ^\lambda (\xi )\). This is the content of the next proposition which together with Proposition 3.1 shows that \(\Phi ^\lambda \) defines a norm equivalent to \(\Vert \cdot \Vert _{H^{1/2}}\).
Proposition 3.6
Assume (H1) and \(\gamma >\gamma _c\). Then, there exist \(\lambda _0>0\) and \(c_0>0\) such that
for any \(\lambda > \lambda _0\).
Proof
By (3.5), (3.6) and Lemma 3.2, we get
Then, using Lemmata 3.4, 3.5, we obtain
Now, we note that Lemma 3.4 and the estimate (3.18) yield
where \(\big (1-\frac{2}{\sqrt{3}}B\big )>0.\) Hence, setting \(\Lambda =\min \{1-s^*,1-\frac{2}{\sqrt{3}}B\}>0\), we get
To conclude, we note
From (3.20) and (3.21) choosing \(\lambda \) large enough we get the thesis since clearly \(\Phi ^\lambda _\mathrm {diag}({\hat{\xi }})\geqslant c\Vert \xi \Vert _{H^{1/2}}^2\). \(\square \)
We are now in position to prove Theorem 2.3 formulated in Sect. 2.
Proof of Theorem 2.3
Point (i) is a consequence of Propositions 3.1 and 3.6. For the proof of point (ii), we follow a standard strategy (see, e.g., [8]). For the convenience of the reader, we give the details below. By Proposition 3.6, we have
for any \(\lambda >\lambda _0\) and then the form is bounded from below. Moreover, let us fix \(\lambda >\lambda _0\) and define
on the domain \({\mathcal {D}}(F)\). Let us consider a sequence \(\{\psi _n=w^\lambda _n+{\mathcal {G}}^\lambda \xi _n\}_{n\geqslant 0} \subset {\mathcal {D}}(F)\), and \(\psi \in L^2_{\text {sym}}(\mathbb {R}^6)\) such that \(\lim _n \Vert \psi _n - \psi \Vert _{L^2}=0\) and \(\lim _{n,m} F^{\lambda }(\psi _n - \psi _m)=0\). By (3.22) we have \(\lim _{n,m} {\mathcal {F}}^{\lambda } (w^{\lambda }_n - w^{\lambda }_m)=0\) and \(\lim _{n,m} \Phi ^{\lambda }(\xi _n -\xi _m)=0\) or, equivalently, \(\{w^{\lambda }_n\}_{n\geqslant 0}\) is a Cauchy sequence in \(H_{\text {sym}}^1(\mathbb {R}^6)\) and, by Proposition 3.6, \(\{\xi _n\}_{n\geqslant 0}\) is a Cauchy sequence in \(H^{1/2}(\mathbb {R}^3)\). Then, there exist \(w^{\lambda } \in H_{\text {sym}}^1(\mathbb {R}^6)\) and \(\xi \in H^{1/2}(\mathbb {R}^3)\) such that
Moreover, we also have
Formulas (3.23) and (3.24) imply that \(\psi _n = w^{\lambda }_n + {\mathcal {G}}^{\lambda } \xi _n\) converges in \(L^2(\mathbb {R}^6)\) to \(w^{\lambda } + {\mathcal {G}}^{\lambda } \xi \). By uniqueness of the limit, we have that \(\psi = w^{\lambda } + {\mathcal {G}}^{\lambda } \xi \) and then \(\psi \in D(F)\). Furthermore, by (3.23) we have
Thus, we have shown that \(F^{\lambda }\) and, a fortiori, F are closed quadratic forms and this concludes the proof. \(\square \)
Remark 3.7
By (3.21) one has
We underline that \(\lambda _0\) depends on \(\gamma \) both via \(\Vert a\Vert _{L^\infty } \) and via \(\Lambda \). In particular, as \(\gamma \rightarrow \gamma _c\) we have \(s^*\rightarrow 1\), so that \(\Lambda \rightarrow 0\) and \(\lambda _0\rightarrow \infty \). In the concrete case (1.10), we can take
From the proof of Theorem 2.3, it is clear that \(-\lambda _0\) is a lower bound for the infimum of the spectrum of H.
Remark 3.8
We expect \(\gamma _c\) to be optimal that is, if \(\gamma < \gamma _c\) one could argue as in [15] and prove that F is unbounded from below.
4 Hamiltonian
In this section, we explicitly construct the Hamiltonian of our three bosons system. Let us first consider the quadratic form \(\Phi ^\lambda \), \({\mathcal {D}} (\Phi ^\lambda ) = H^{1/2}(\mathbb {R}^3)\) in \(L^2(\mathbb {R}^3)\) (see (2.15)). As a straightforward consequence of Point (i) of Theorem 2.3, such a quadratic form is closed and positive, and therefore it uniquely defines a positive, self-adjoint operator \(\Gamma ^\lambda \) in \(L^2(\mathbb {R}^3)\) for \(\lambda >\lambda _0\) characterized as follows
In the appendix , we prove that \({\mathcal {D}}(\Gamma ^{\lambda }) = H^1(\mathbb {R}^3)\) for \(\gamma > \gamma _c^*\) (see Proposition A.2) and
Let us now consider the quadratic form F, \({\mathcal {D}}(F)\) in \(L^2_{\text {sym}}(\mathbb {R}^6)\). By Theorem 2.3, such quadratic form uniquely defines a self-adjoint and bounded from below Hamiltonian H, \({\mathcal {D}}(H)\) in \(L^2_{\text {sym}}(\mathbb {R}^6)\), next we prove Theorem 2.4 which characterizes its domain and action.
Proof of Theorem 2.4
Let us assume that \(\psi = w^{\lambda } + {\mathcal {G}}^{\lambda } \xi \in {\mathcal {D}}(H)\). Then, there exists \(f \in L^2_{\text {sym}}(\mathbb {R}^6)\) such that \(F(v,\psi ) = (v,f)\) for any \(v = w^{\lambda }_v + {\mathcal {G}}^{\lambda } \xi _v \in {\mathcal {D}}(F)\) and \(f=H\psi \). Let us consider \(v\in H^1(\mathbb {R}^6)\), so that \(\xi _v =0\) and
Hence, \(w^{\lambda } \in H^2(\mathbb {R}^6)\) and \((H_0 + \lambda ) w^{\lambda } = f + \lambda \psi = (H + \lambda ) \psi \) which is equivalent to (2.19).
Let us consider \(v \in {\mathcal {D}}(F)\) with \(\xi _v \ne 0\). Then,
Taking into account that
equation (4.4) is rewritten as
It remains to compute the right hand side of (4.5). Using (2.6) and the symmetry properties of \(w^{\lambda }\) (see (2.1)), we have
and by (4.5) we find the equation
for any \(\xi _v \in H^{1/2}(\mathbb {R}^3)\). By definition of the operator \(\Gamma ^{\lambda }\) (see (4.1), (4.2)), we conclude that \(\xi \in {\mathcal {D}}(\Gamma ^{\lambda })\) and \(\Gamma ^{\lambda } \xi = w^{\lambda }\big |_{\pi _{23}}\).
Let us now assume that \(\psi \in {\mathcal {D}}(F)\) with \( w^{\lambda } \in H^2(\mathbb {R}^6)\), \(\xi \in {\mathcal {D}}(\Gamma ^{\lambda })\) and \( \Gamma ^{\lambda } \hat{\xi } = w^{\lambda }\big |_{\pi _{23}} \). For any \(v = w^{\lambda }_v + {\mathcal {G}}^{\lambda } \xi _v \in {\mathcal {D}}(F)\) we have
It is now sufficient to define \(f = H_0 w^{\lambda } - \lambda \mathcal G^{\lambda } \xi \) to obtain that \(\psi \in {\mathcal {D}}(H)\) and \(f=H\psi \) and thus to conclude the proof. \(\square \)
Remark 4.1
We emphasize that the Hamiltonian \(H, {\mathcal {D}}(H)\) is the rigorous counterpart of the formal regularized TMS Hamiltonian introduced in Sect. 1. Indeed, for any \(\psi \in L^2_{sym }(\mathbb {R}^6) \cap C_0^{\infty }( \mathbb {R}^6 {\setminus } \cup _{i<j} \pi _{ij})\) we have \(\psi \in {\mathcal {D}}(H)\) and \(H\psi =H_0\psi \), i.e., the Hamiltonian acts as the free Hamiltonian outside the hyperplanes. Moreover, we show that the boundary condition (1.8) is also satisfied. Let us consider \(\psi \in {\mathcal {D}}(H)\) and let us recall that the corresponding charge \(\xi \) belongs to \(H^1(\mathbb {R}^3)\). For \(\mathbf{x}\ne 0\) we write
and we compute the limit of the above expression for \(x\rightarrow 0\) in the \(L^2\)-sense. Taking into account of (2.7), we have
which, by dominated convergence theorem, converges to zero for \(x\rightarrow 0\). Moreover, for any \(\eta \in C_0^{\infty }(\mathbb {R}^3)\), with \(\Vert \eta \Vert _{L^2}=1\), we estimate the difference \( (\mathcal G^{\lambda }_{31} \xi )(\mathbf{x},\mathbf{y})-({\mathcal {G}}^{\lambda }_{31} \xi )(0,\mathbf{y})\) proceeding as in (3.4)
Notice, see Remark A.3 that
then we conclude that \( ({\mathcal {G}}^{\lambda }_{31} \xi )(\mathbf{x},\mathbf{y})-({\mathcal {G}}^{\lambda }_{31} \xi )(0,\mathbf{y}) \rightarrow 0\) for \(x \rightarrow 0\) in \(L^2(\mathbb {R}^3)\) by dominated convergence theorem and the same is true for \( ({\mathcal {G}}^{\lambda }_{12} \xi )(\mathbf{x},\mathbf{y})-({\mathcal {G}}^{\lambda }_{12} \xi )(0,\mathbf{y})\). Note that \(({\mathcal {G}}^{\lambda }_{31} \xi )(0,\mathbf{y}) + ({\mathcal {G}}^{\lambda }_{12} \xi )(0,\mathbf{y}) = -( \Gamma ^{\lambda }_{off } \xi )(\mathbf{y})\). For the last term in (4.6) we have
and then \(w^{\lambda }(\mathbf{x},\mathbf{y}) - w^{\lambda }(0,\mathbf{y}) \rightarrow 0\) for \(x \rightarrow 0\) in \(L^2(\mathbb {R}^3)\). Taking into account the above estimates, the condition \(\Gamma ^{\lambda } \xi = w^{\lambda }\big |_{\pi _{23}}\) and the decomposition \(\Gamma ^{\lambda }= \Gamma ^{\lambda }_{diag } + \Gamma ^{\lambda }_{off } + \Gamma _{reg } \) we conclude
which is precisely the boundary condition (1.8) satisfied in the \(L^2\)-sense.
Let us characterize the resolvent of our Hamiltonian. We first introduce the shorthand notation for the operator \(\mathcal G^{\lambda }_{23} = G^{\lambda }\) (see (2.2)), i.e.,
its adjoint is
Next, we prove the following preliminary result.
Proposition 4.2
For any \(\lambda >0 \), there holds \(G^\lambda \in \mathcal {B}(H^{s-\frac{1}{2}}(\mathbb {R}^3),H^s(\mathbb {R}^6))\) for all \(s<1/2\), hence, \(G^{\lambda *} \in \mathcal {B}(H^{-s}(\mathbb {R}^6), H^{-s+\frac{1}{2}}(\mathbb {R}^3))\) for all \(s<1/2\). In particular, \(G^\lambda \in \mathcal {B}(L^2(\mathbb {R}^3),L^2(\mathbb {R}^6))\) and \(G^{\lambda *} \in \mathcal {B}(L^2(\mathbb {R}^6),L^2(\mathbb {R}^3))\).
Proof
We note that
Hence,
So that
\(\square \)
Let us recall the definition of the operator S given in (2.24), additionally we notice that
If \({\hat{\psi }} \in L^2_{\text {sym}}(\mathbb {R}^6)\), it holds true
We note that the second equality is a consequence of the first one. Furthermore, we have \(S^* = S^2\). Taking into account of (2.6), (4.7), (2.24), (4.8), we can write
We claim that the resolvent \((H+\lambda )^{-1}\) of \(H, {\mathcal {D}}(H)\) computed in \(z = -\lambda < -\lambda _0 \) is given by
where \(R_0^{\lambda } = (H_0 + \lambda )^{-1}\) and \((\Gamma ^\lambda )^{-1}\) is a well-defined and bounded operator in \(L^2(\mathbb {R}^3)\) since \(\Phi ^\lambda \) is coercive. Indeed, let us consider \(\;R^{\lambda } f\) for \(f \in L^2_{\text {sym}}(\mathbb {R}^6)\). We have \(R^{\lambda } f = w^{\lambda } + {\mathcal {G}}^{\lambda } \xi \), where \(\;w^{\lambda } \equiv R_0^{\lambda } f \in H^2(\mathbb {R}^6)\), \(\;\xi \equiv (4 \pi )^{-1} (\Gamma ^\lambda )^{-1} \, G^{\lambda *}f \in {\mathcal {D}}( \Gamma ^{\lambda })\) and \(\Gamma ^{\lambda } \xi = (4 \pi )^{-1} G^{\lambda *} f = R_0^{\lambda } f \big |_{23}\). Hence, \(R^{\lambda }f \in {\mathcal {D}}(H) \) and \((H+\lambda ) R^{\lambda }f= (H_0 + \lambda ) R_0^{\lambda } f =f\). Therefore, we conclude that \(R^{\lambda }= (H+\lambda )^{-1}\).
5 Approximating Hamiltonian
In this section, we prove a uniform bound on the infimum of the spectrum of \(H_\varepsilon \) introduced in Sect. 2 and obtain the Konno–Kuroda formula for its resolvent (Theorem 5.7).
Remark 5.1
Let us recall the scaled function \(\chi _\varepsilon \) defined in (2.21), and the definitions of the constants \(\ell \) and \(\ell '\) in (2.20). The assumptions on \(\chi \) imply that \({\hat{\chi }}\) is real-valued, Lipschitz, that \(\ell ,\ell '<\infty \), and that
Moreover, we recall the definition of the infinitesimal, position-dependent coupling constant in (2.22). In the position space \(g_\varepsilon \) is just the multiplication operator for the function (which we denote by the same symbol)
where a(y) was introduced in (3.2). From now on, we always assume that \(\varepsilon < \ell /(2\Vert a\Vert _{L^\infty })\) so that \( 1 + \frac{\varepsilon }{\ell } a(y) + \frac{\varepsilon }{\ell } \frac{\gamma }{y}>1/2\) and \(g_\varepsilon \), as a function, is bounded, in particular \(\Vert g_\varepsilon \Vert _{L^\infty }\leqslant 8\pi \varepsilon /\ell \).
Let us consider the Hamiltonian \(H_\varepsilon \) defined in (2.23). We remark that the term \(\sum _{j=0}^2\,S^j \big (|\chi _\varepsilon \rangle \langle \chi _\varepsilon | \otimes g_\varepsilon \big ) {S^j}^*\) is bounded (although not uniformly in \(\varepsilon \)) in \(L^2(\mathbb {R}^6)\), with norm bounded by \(3 \Vert \chi _\varepsilon \Vert ^2 \Vert g_\varepsilon \Vert _{L^{\infty }}\leqslant (24\pi /\ell ) \varepsilon ^{-2} \Vert \chi \Vert ^2\), and therefore \(H_{\varepsilon }\), \({\mathcal {D}}(H_{\varepsilon })\) is self-adjoint and bounded from below for any \(\varepsilon >0\).
As a first step, we introduce the following operators which will play a crucial role in writing the Konno–Kuroda formula for the resolvent of \(H_\varepsilon \) (see Theorem 5.7).
Definition 5.2
For any \(\lambda >0\), let us define
where \(\Gamma ^\lambda _{{\text {diag}},\varepsilon } \) and \(\Gamma ^\lambda _{{\text {off}},\varepsilon }\) are the bounded operators (see the remark below)
with
and
We also define the quadratic form associated with \(\Gamma ^\lambda _{\varepsilon }\)
We will show in the proof of Lemma 6.2 that the operators \( \Gamma ^\lambda _{{\text {diag}},\varepsilon } \) and \( \Gamma ^\lambda _{{\text {off}},\varepsilon }\) converge, as \(\varepsilon \rightarrow 0\), to the corresponding limiting operators \( \Gamma ^\lambda _{{\text {diag}}} \) and \( \Gamma ^\lambda _{{\text {off}}}\), defined in Eq. (4.3).
Remark 5.3
We observe that \( \Vert \hat{\chi }\Vert _{L^\infty } \leqslant (2\pi )^{-3/2} \Vert \chi \Vert _{L^1} = (2\pi )^{-3/2} \), so that \(r(s)\leqslant 1\) and also \(r(0)=1\). Additionally, we note the trivial bound \(s r(s) \leqslant \ell \). The latter implies \(\Vert \Gamma _{{\text {diag}},\varepsilon }\Vert _{\mathcal {B}(L^2)}\leqslant \ell /\varepsilon \).
We also note that sr(s) as a function of \(s\in [0,+\infty )\) is strictly increasing, this is an immediate consequence of the identity
Since \({\hat{\chi }}\) is a Lipschitz function, interpolating with the \(L^\infty \) estimate, we immediately obtain
Moreover, by the Cauchy–Schwarz inequality, we find
Hence, \(\Vert \Gamma _{{\text {off}},\varepsilon }\Vert _{\mathcal {B}(L^2)} \leqslant C \lambda ^{-1/4} \varepsilon ^{-3/2} \Vert \chi \Vert \) for some numerical constant C.
We want to obtain a lower bound for \(\Phi _{\varepsilon }^\lambda (\xi )\). In the next lemma, we first analyze \(( \xi , \Gamma ^\lambda _{{\text {off}},\varepsilon } \xi )\).
Lemma 5.4
Let \(\xi \in {\mathcal {D}}(\Phi _{\varepsilon }^\lambda )\), \(\lambda >0\) and \(\gamma _0\) as in (2.20). Then,
Proof
By (5.1), the change of variable \(\mathbf{k}=-\mathbf{q}- \frac{1}{2} \mathbf{p}\) and the action of the operator S (see (2.24)), we find
It is convenient to write the r.h.s. of (5.6) in the position space. To this aim, we denote by \(R_0^\lambda (\mathbf{x},\mathbf{y};\mathbf{x}',\mathbf{y}')\) the integral kernel of the operator \(R_0^\lambda \). Its explicit expression is given by the formula
where \(\mathscr {K}_2\) is the modified Bessel function of the third kind and it is a nonnegative function. By the definition of S, see (2.24), we obtain the formula
To proceed, we add and subtract to \(\xi (\mathbf{y}')\) the function \(\xi (\mathbf{y})\) and obtain
where
and
We claim that
To prove inequality (5.9), we reason as follows. The integral kernel of \(R_0^\lambda \) is (pointwise) positive and \(\chi \) is a nonnegative function; hence, \({\widetilde{J}}_\varepsilon > 0\). Moreover, the expression
is invariant if one changes \((\mathbf{x},\mathbf{y};\mathbf{x}', \mathbf{y}') \rightarrow (-\mathbf{x}',\mathbf{y}';-\mathbf{x},\mathbf{y})\) (as one can check with a straightforward calculation), and \(R_0^\lambda \Big (- \frac{1}{2} \mathbf{x}+\mathbf{y}, -\frac{3}{4}\mathbf{x}- \frac{1}{2}\mathbf{y};\mathbf{x}',\mathbf{y}'\Big )\) shares the same property. Taking also into account the fact that \(\chi \) is spherically symmetric, it follows that \({\widetilde{J}}_\varepsilon (\mathbf{y},\mathbf{y}') = {\widetilde{J}}_\varepsilon (\mathbf{y}',\mathbf{y})\). The symmetry of \({\widetilde{J}}_\varepsilon \) implies
from which the inequality (5.9) immediately follows. By (5.8) and (5.9), we find
To conclude the proof, we are left to show that (5.10) implies (5.5). The following identity can be obtained by integration starting from identity (5.7)
Hence,
To proceed, we use the identity
It is easy to check that for \(k>0\) the function \(\int _0^k ds \frac{\sin s}{s}\) has maxima in \(k= n\pi \) for \(n\in {\mathbb {N}}\) odd, minima in \(k=n\pi \) for \(n\in {\mathbb {N}}\) even, it is positive and has an absolute maximum in \(k = \pi \). Hence, \(|\int _0^k ds \frac{\sin s}{s}| \leqslant \int _0^\pi ds \frac{\sin s}{s} \leqslant \pi \). By the latter considerations, we infer
Using the latter bound in (5.11) gives
with the explicit expression for \(\gamma _0\). By (5.10) and (5.12), we conclude the proof of the lemma. \(\square \)
Using the previous result, we can now establish a uniform lower bound for \(\Phi _\varepsilon ^\lambda ( \xi )\).
Lemma 5.5
Assume (H1), \(\xi \in {\mathcal {D}}(\Phi _{\varepsilon }^\lambda )\) and \(\gamma > \gamma _0\) (see (2.20)). Then, there exist \(\varepsilon _0,\lambda _1,c>0\) such that
for all \(\lambda >\lambda _1\) and \(0<\varepsilon <\varepsilon _0\).
Proof
We recall that in the position space the operator \(\Gamma _{{\text {reg}}}\) (see (1.9) and (3.2)) is just the multiplication by the function (denoted by the same symbol)
So that
By the above inequality and Lemma 5.4, we infer
Since \(s \rightarrow s \,r(s)\) is strictly increasing (see Remark 5.3) one has that for all \(\lambda \geqslant \lambda _1\)
Now, fix \(\lambda _1\) so that \(\sqrt{\lambda _1} > \Vert a\Vert _{L^\infty }\) and \(\varepsilon _0\) so that for all \(0<\varepsilon <\varepsilon _0\) there holds \(\sqrt{\lambda _1} \,r \big (\varepsilon \sqrt{\lambda _1}\big ) > \Vert a\Vert _{L^\infty }\) (which is possible since \(\lim _{s\rightarrow 0} r(s) = 1\)). Then,
for some positive constant \({\tilde{c}}\). Hence, from inequality (5.14), taking \(\gamma >\gamma _0\), we infer the lower bound (5.13) with \(c = \min \{{\tilde{c}}, \gamma -\gamma _0\}\). \(\square \)
To proceed, we need some further notation. We denote by \(\frac{\gamma }{|\,\cdot \,|}\) the multiplication operator for \(\frac{\gamma }{y}\) and define the operator
Similarly to \(g_\varepsilon \), in the position space the operator \(\nu _\varepsilon \) acts as the multiplication by the function (denoted by the same symbol)
Obviously, we have
Moreover, there holds the identity
and the bounds \(\Vert \nu _\varepsilon \Vert _{\mathcal {B}(L^2)} = \Vert \nu _\varepsilon ^*\Vert _{\mathcal {B}(L^2)} \leqslant \sqrt{2} \) for all \(\varepsilon \leqslant \ell /(2\Vert a\Vert _{L^\infty })\).
In the next lemma we study the invertibility of the operator \(\nu _\varepsilon ^* \Gamma _\varepsilon ^\lambda \nu _\varepsilon \).
Lemma 5.6
For any \(\lambda >0 \), there holds true the identity
where \(B_\varepsilon ^\lambda \) is the bounded (although not uniformly in \(\varepsilon \)) operator
Moreover, assume (H1) and \(\gamma > \gamma _0\). Then, \(\nu _\varepsilon ^* \Gamma _\varepsilon ^\lambda \nu _\varepsilon \) is invertible in \(L^2(\mathbb {R}^3)\) for all \(\lambda >\lambda _1\) and \(\varepsilon \) small enough, with inverse uniformly bounded in \(\varepsilon \) and \(\lambda \).
Proof
We start by pointing out the identity (note that, obviously, \(\nu _\varepsilon \), \(\nu _\varepsilon ^*\), and \(\Gamma _{{\text {reg}}}\) commute)
hence, \(\nu _\varepsilon ^* \Gamma _{{\text {reg}}}\nu _\varepsilon \) is well defined in \(L^2(\mathbb {R}^3)\) with norm bounded by \(3\ell /\varepsilon \). Let us consider the quadratic form associated with \(\frac{1}{4\pi }\frac{\ell }{\varepsilon }- B_{\varepsilon }^\lambda \),
We set
so that
and
We first study the term \( \frac{1}{4\pi }\frac{\ell }{\varepsilon } \Vert \xi \Vert ^2 - (\xi , B_{{\text {diag}},\varepsilon }^\lambda \xi )\) making use of two identities. The first one is
and gives
The second one is \( \nu _\varepsilon ^* \, \nu _\varepsilon = \mathbb {I}- \frac{\varepsilon }{\ell }\nu _\varepsilon ^* \, \Gamma _{{\text {reg}}} \, \nu _\varepsilon . \) Now, we can compute
Next, we study the term \((\xi , B_{{\text {off}},\varepsilon }^\lambda \xi )\). We have that
which concludes the proof of identity (5.15).
We are left to prove the second part of the lemma. We use Lemma 5.5 by noticing the identity \({\widetilde{\Phi }}_\varepsilon ^\lambda ( \xi ) = \frac{1}{4\pi } \Phi _\varepsilon ^\lambda ( \nu _\varepsilon \xi )\) with \(\Phi _\varepsilon ^\lambda (\xi )\) defined in (5.2)–(5.3). We stress that the identity makes sense for all \(\xi \in L^2(\mathbb {R}^3)\) since \(\nu _\varepsilon \xi \in D(\Phi _\varepsilon ^\lambda )\) (recall the remark after (5.16)).
Then, from Lemma 5.5 we obtain
where we used the inequality
To see that the latter inequality holds true, recall that we are assuming \(\varepsilon < \ell /(2\Vert a\Vert _{L^\infty })\) so that \(0<1 + \frac{\varepsilon }{\ell } \beta + \frac{\varepsilon }{\ell } \frac{\gamma }{y} \theta (y) = 1 + \frac{\varepsilon }{\ell } a(y) + \frac{\varepsilon }{\ell } \frac{\gamma }{y} \leqslant \frac{3}{2}+ \frac{\varepsilon }{\ell } \frac{\gamma }{y} \), and
Since \({\widetilde{\Phi }}_\varepsilon ^\lambda \) is the quadratic form associated with \(\frac{1}{4\pi }\frac{\ell }{\varepsilon }- B_{\varepsilon }^\lambda \), the bound (5.19) implies that \(\frac{1}{4\pi }\frac{\ell }{\varepsilon }- B_{\varepsilon }^\lambda \) is invertible, with an inverse bounded by 1/c, and this concludes the proof. \(\square \)
The last preparatory step is the definition of the bounded operator
In Fourier transform,
Hence (recall that \(|{\hat{\chi }}(k)| \leqslant (2\pi )^{-3/2}\) so that \( 4\pi |{\hat{\chi }}(k)| \leqslant \sqrt{2/\pi }\))
\(\Vert A_\varepsilon ^\lambda \xi \Vert \leqslant \Vert G^\lambda \Vert _{\mathcal {B}(L^2(\mathbb {R}^3),L^2(\mathbb {R}^6))}\Vert \nu _\varepsilon \xi \Vert \leqslant \sqrt{2} \Vert G^\lambda \Vert _{\mathcal {B}(L^2(\mathbb {R}^3),L^2(\mathbb {R}^6))}\Vert \xi \Vert \), and \(A_\varepsilon ^\lambda \) is uniformly bounded in \(\varepsilon \).
In what follows, we will use the identity
and write
We are now ready to formulate and prove the main result of this section.
Theorem 5.7
Assume (H1) and \(\gamma _0\), \(\lambda _1\), and \(\varepsilon _0\) as in Lemma 5.6. Then, for all \(\gamma >\gamma _0\), \(0<\varepsilon <\varepsilon _0\), and \(\lambda >\lambda _1\), the operator \(H_\varepsilon +\lambda \) has a bounded inverse in \(L^2_{sym }(\mathbb {R}^6)\). Moreover, denoting its inverse by \(R_\varepsilon ^\lambda \), one has the Konno–Kuroda formula
Remark 5.8
As a matter of fact, see, e.g., [18, Theorem B.1], the Konno–Kuroda formula holds true for all the complex \(\lambda \) such that \(-\lambda \in \rho (H_0) \cap \rho (H_\varepsilon )\). The relevant information of Th. 5.7 is that there exists a real \(\lambda _1\), independent from \(\varepsilon \), such that \(R_\varepsilon ^\lambda \) is a well-defined bounded operator for all \(\lambda >\lambda _1\). This is equivalent to the lower bound \(\inf \sigma (H_\varepsilon ) \geqslant - \lambda _1\).
Proof of Theorem 5.7
The action of \(H_\varepsilon \) on a (symmetric) wave function in its domain is given by
We describe how to obtain formula (5.21). For a given function \(\phi \in L^2_{{\text {sym}}}(\mathbb {R}^6)\), and \(\lambda \) large enough, assume that \( \psi _\varepsilon \in {\mathcal {D}}(H_\varepsilon )\subset L^2_{{\text {sym}}}(\mathbb {R}^6)\) is a solution in of the equation
The latter gives
and, recalling that \(R_0^\lambda =(H_0+\lambda )^{-1}\) is a well defined bounded operator,
where we used the fact that \(R_0^\lambda \) and S commute. Hence,
Set
and rewrite (5.22) as
We want to obtain a formula for \(h_\varepsilon \). To this aim, apply the operator \(\big ( \langle \chi _\varepsilon | \otimes \nu _\varepsilon ^* \big )\) to (the left of) identity (5.22). By simple algebraic manipulations, it follows that
By Lemma 5.6, the operator at the l.h.s. is invertible and
where we used the identity
Using the identity (5.24) in (5.23) we obtain the formula
Now, as suggested from the formula above, one can define \(R_\varepsilon ^\lambda \) as in (5.21) and show by a straightforward calculation that \((H_\varepsilon +\lambda )R_\varepsilon ^\lambda =\mathbb {I}\) on \(L^2(\mathbb {R}^6)\) and \( R_\varepsilon ^\lambda (H_\varepsilon +\lambda ) = \mathbb {I}\) on \({\mathcal {D}}(H_\varepsilon )\), from which it follows that \(R_\varepsilon ^\lambda = (H_\varepsilon +\lambda )^{-1}\). \(\square \)
6 Norm Resolvent Convergence
In this section, we prove that \(H_\varepsilon \) converges to H in the norm resolvent sense and we give an estimate of the rate of convergence.
Proof of Theorem 2.5
It is enough to prove the statement for some fixed \(z = -\lambda < -{\lambda _{{\text {max}}}}\), then it holds true for a generic \(z\in \mathbb {C}\backslash [-{\lambda _{{\text {max}}}},\infty )\) by analytic continuation.
Since
we need to show that for some \(\lambda \) large enough there holds
Without loss of generality, here as well as in the proof of the Lemmata 6.1 and 6.2, we can assume \(\lambda >1\). All the generic constants denoted by c are independent from \(\lambda \) for \(\lambda >1\). We start with the trivial identity
By Lemma 6.1, and since \(( \nu _\varepsilon ^* \Gamma _\varepsilon ^\lambda \nu _\varepsilon )^{-1}\) and \(A_\varepsilon ^\lambda \) are uniformly bounded in \(\varepsilon \) (see Lemma 5.6 and the remark after (5.20)), we infer
and
We are left to prove that
We note the identity
So that, taking into account the fact that \(G^\lambda \) is bounded and \(( \nu _\varepsilon ^* \Gamma _\varepsilon ^\lambda \nu _\varepsilon )^{-1}\) is uniformly bounded in \(\varepsilon \), we infer that (6.1) is a consequence of
which holds true by Lemma 6.2. \(\square \)
Lemma 6.1
There exists \(c>0\) such that
for all \(0<\delta < 1/2\) and for all \(\lambda >1\).
Proof
We use the identity
and write
with
For all \(\xi \in L^2(\mathbb {R}^3)\), one has
Concerning \(A_{2,\varepsilon }^\lambda \), we note that one has
We have
Hence, taking into account Remark 5.1 and the inequality \(\big |1- (1+s)^\frac{1}{2}\big | \leqslant |s| \) for all \(s\geqslant -1 \), one has
Setting \(O:= |\cdot |^{-1/2} (-\Delta +1)^{-\frac{1}{4}} \), inequality (3.3) reads \(\Vert Of \Vert ^2 \leqslant \frac{\pi }{2} \Vert f\Vert ^2. \) which implies also
and we arrive at
Moreover,
so that
Hence,
which concludes the proof of the lemma. \(\square \)
Clearly we have also
for all \(0<\delta < 1/2\) and for all \(\lambda >1\).
Lemma 6.2
Assume (H2) and \(\lambda >1\). For any \(\phi \in L^2_{{\text {sym}}}(\mathbb {R})\), there holds true
Proof
In this proof we set
and we know by Proposition A.4 that
We use again the identity (6.2) and obtain
We are going to prove that for \(0<\delta <1/2\), we have:
Let us prove (6.4). We have that
Concerning the first couple of operators, using (5.4), we have for \(0<\delta <1/2\)
On the other hand, using (5.4) and the boundedness of \({\hat{\chi }}\), we have
Due to (6.3), it is sufficient to prove that
is the integral kernel of an \(L^2\)-bounded operator for \( 0<\delta < 1/2\). If we put \(f(\mathbf{p})= (p^2+1)^{-3/4}\), it is straightforward to prove that for \(0<\delta <1/2\) we have
then the claim follows from Schur’s test.
Let us prove (6.5). We claim that for all \(s\in [0,1/2)\) there exists \(C>0\) such that
Due to Remarks A.3 and A.5, it is sufficient to control each term in \( \Gamma _\varepsilon ^\lambda \xi \) with corresponding limit uniformly in \(\varepsilon \). It holds true for \({ \Gamma }^{ \lambda }_{\text {diag},\varepsilon }\xi \), since \(0\leqslant r(s) \leqslant 1\); see also (6.3). It holds true for \({\Gamma }^{ \lambda }_{\text {off},\varepsilon }\xi \) since
Therefore, estimate (6.8) holds true. We have
with
and, due to Remark 5.1, we have the estimates
Moreover, with straightforward calculations, one has for \(p>6\)
By Sobolev embedding and (6.8), we have\( \Vert \Gamma _\varepsilon ^\lambda \xi \Vert _{L^q}\leqslant C \Vert \phi \Vert \) for \(\forall q \in [2,3)\). Using Hölder inequality, (6.9) and (6.10), we have
and (6.5) is proved.
Let us prove (6.6). Due to estimate (6.3), it is sufficient estimate \(\Vert (\nu _\varepsilon -1)\xi \Vert _{H^1}\). We start from
The second term in (6.11) can be estimated as (6.5); also the first term, even if it is more regular, can be estimated in the same way. We discuss the third term. First notice that
with
Taking into account that
is bounded, we have \(\Vert f_3\Vert _{L^\infty } \leqslant c \, \varepsilon \). Moreover, we also have for \(2<p <6 \)
By Sobolev embedding \(\xi \in L^q\) for \(2\leqslant q <\infty \), then using Hölder inequality, we have
where \(p^{-1}+q^{-1}= 1/2\), and (6.6) is proved.
Let us prove (6.7). This estimate immediately reduces to (6.6) since \(\Vert \nu _\varepsilon ^*-1\Vert _{L^\infty } \leqslant c \) uniformly in \(\varepsilon \). \(\square \)
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Appendix A. Regularity of the Charge
Appendix A. Regularity of the Charge
In this appendix, we characterize \({\mathcal {D}}(\Gamma ^\lambda )\) and we prove a regularity result for the charge associated with \(\psi \in {\mathcal {D}}(H)\).
1.1 A.1 Domain of \(\Gamma ^\lambda \)
Here, we prove that \({\mathcal {D}}(\Gamma ^{\lambda }) = H^1(\mathbb {R}^3)\).
Remark A.1
By Proposition 3.6, the spectrum of \(\Gamma ^{\lambda }\) is contained in \([c_0, \infty )\), \(c_0 >0\). Therefore, \((\Gamma ^{\lambda } )^{-1}\) exists and it is a bounded operator in \(L^2(\mathbb {R}^3)\) with norm less than \(c_0^{-1}\). That is, for any \(f\in L^2 (\mathbb {R}^3)\) there exists \( \xi \in L^2(\mathbb {R}^3)\) solution of the equation \(\Gamma ^\lambda \xi = f\) and \(\Vert \xi \Vert \leqslant c_0^{-1} \Vert f\Vert \).
We want to prove that \(\xi \in H^1(\mathbb {R}^3)\), that is \((\Gamma ^{\lambda } )^{-1}\in {\mathcal {B}}(L^2(\mathbb {R}^3),H^1(\mathbb {R}^3)).\)
For the sake of notation, we introduce the operator T defined as:
Let
Proposition A.2
Assume (H1) and \(\gamma >\gamma _c^*\), let \(f\in L^2(\mathbb {R}^3)\) and let \(\xi \) be the solution of
then \(\xi \in H^1\) and \( \Vert \xi \Vert _{H^1}\leqslant c \Vert f\Vert _{ L^2(\mathbb {R}^3)}\).
Proof
Set \(f^\lambda := f- (\Gamma ^\lambda - T) \xi \), and recast (A.2) as
By (4.3) and (A.1), there follows
We start by noticing that \(f^\lambda \in L^2(\mathbb {R}^3)\). To see that this is indeed the case recall that by our previous remark \(\xi \in L^2(\mathbb {R}^3)\) and notice that all the terms at the r.h.s. of identity (A.4) are in \(L^2\). To convince oneself that this is true also for the integral term (for all the others it is obvious), it is sufficient to notice that the integral kernel is an Hilbert–Schmidt operator. To this aim, one can use the inequality
and introducing polar coordinates in \(\mathbb {R}^6\) verify that the latter integral is finite.
To conclude the proof of the proposition we need to show that
Decomposing \({\hat{\xi }}\) and \({\hat{f}}^\lambda \) on the basis of spherical harmonics, and setting \(p=e^x\), \(\zeta _{\ell m}(x) = e^{5/2 \,x} {\hat{\xi }}_{\ell m}(e^x)\), and \(h_{\ell m}(x)= e^{3/2 \,x} {\hat{f}}_{\ell m}(e^x)\) we obtain:
and
here, clearly, \( \hat{f}_{\ell m}^\lambda (p)\in L^2(\mathbb {R}^+, p^2\,\text {d}p)\) for \(\ell \in {\mathbb {N}}\) and, \(m=-\ell , \ldots , \ell \,\). We look for an inequality between the \(L^2\)-norms of the functions \(\zeta _{\ell m}\) and \(h_{\ell m}\) of the form \(\Vert \zeta _{\ell m} \Vert \leqslant c\Vert h_{\ell m}\Vert \) with c independent on \(\ell \) and m. To proceed, we decompose (A.3) on the basis of Spherical Harmonics and obtain:
Then, we multiply the latter equation by \(p^{3/2}\) and change variables as above, with \(p=e^x\) and \(q=e^y\), to obtain:
The latter equation can be seen as a convolution equation on \(L^2 (\mathbb {R})\) and discussed by Fourier transform, to this aim we note the identities:
and
(which hold true because \(S_{\mathrm {off},\ell }(k) \) and \(S_{\mathrm {reg},\ell }(k)\), defined in (3.13) and (3.14), admit an holomorphic extension to the strip \(\{ |{\text {Im}}k| <1\}\)); therefore (see (3.17))
Then, (A.6) is equivalent to
To conclude the proof of the proposition, it is sufficient to prove that \(\left| S_\ell \left( k+ \frac{i}{2} \right) \right| \geqslant c>0\). We shall focus on the real part of \(S_\ell \) and prove that
Starting from (3.9) and (3.10), with some straightforward calculations one arrives at
and
We analyze separately the cases \(\ell \geqslant 1\) and \(\ell =0\). We start with \(\ell \geqslant 1\). Notice that \(\frac{\cosh \big (k\arccos (-y)\big )}{\sqrt{1-y}}\) has a series expansion with positive coefficients. To see that this is indeed the case recall that: (see [16, (1.112.4)])
and \(\arccos (-y)= \pi /2 +\arcsin (y)\) as well as \(\cosh (x)\) have positive coefficients series. Then,
and
by Lemma 3.3.
Now, we prove some suitable lower bounds for \({\text {Re}}S_{\mathrm {off},\ell }\). By (A.11), we infer
so that we cannot apply directly Lemma 3.3 to \({\text {Re}}S_{\mathrm {off},\ell }\). However, due to the parity properties of the Legendre polynomials, we can apply the lemma for \(\ell \) even and \(\ell \) odd separately. We start with the analysis of the case \(\ell \) odd. We have that
where the latter inequality is a consequence of Lemma 3.3.
On the other hand, for \(\ell \) even we have
Hence, using again Lemma 3.3, we infer
To get the lower bound in the second to last line, we restricted the integral where the second Legendre polynomial is positive, so that we can infer the bound (A.14) by using the monotonicity properties of the integrand. The remaining steps are elementary inequalities.
Therefore, by (A.7), together with the lower bounds (A.12), (A.13), and (A.14) we have
These give the lower bound (A.8) for \(\ell \geqslant 1\). We remark that the lower bound for \(\ell \geqslant 1\) holds true whenever \(\gamma \geqslant 0\), hence, for these values of \(\ell \) the regularizing three-body interaction does not play any role.
To obtain the bound for \(\ell = 0\), we reason like in the proof of Lemma 3.5. We prove that for any fixed \(\gamma > \gamma _c^*\) there exists \(s >0 \) small enough, such that
If this is the case, by (A.7) we obtain the needed bound by noticing that
Changing variables in (A.9) and (A.10), we obtain
and
Next we use the identities
and
to obtain the formulae
and
So that,
with
To prove the bound (A.15), it is enough to show that \(f_0 + f_1 \geqslant 0\). Since \(f_0\) and \(f_1\) are even functions of k, it is enough to consider \(k\geqslant 0\). We have
Noticing that for \(s >0 \) small enough one has \( \sqrt{3} (1-s) \cdot 9^j/2 + 4\cdot 4^j - 4 \sqrt{3} >0\) for all \(j\geqslant 1\), we have the lower bound (it is convenient to keep the first two terms of the series)
with \(\gamma _c^*= \frac{7}{4} \sqrt{3} - 2 \). Similarly, we have
Since \( \gamma \cdot 3^{2j+1}- 2\sqrt{3} \cdot 2^{2j+1} +2>0\) for \(\gamma>\gamma _c^*>1 \) and \(j \geqslant 1\) we obtain a lower by keeping only the term \(j=0\)
Hence, taking into account the fact that we are assuming \(\gamma > \gamma _c^*\), we have
and the r.h.s. is positive for \(s>0\) small enough because \(\gamma - \gamma _c^*>0\) and \(\frac{\sqrt{3}}{12} \pi + \frac{8}{3}\pi +\frac{6\sqrt{3}}{\pi } + 12 \gamma _c^* - 16 \sqrt{3} +8 \simeq 4.8>0\). \(\square \)
Remark A.3
As a consequence, we have
since every term belong to \(L^2\) for \(\xi \in H^1(\mathbb {R}^3)\), that is all four operators are bounded from \(H^1\) to \(L^2\). The claim is obvious for \({\Gamma }^{ \lambda }_{\text {diag}} \), it was proved in [23, Proposition 5] for \({\Gamma }^{ \lambda }_{\text {off}} \), while regarding \( {\Gamma }^{(2)}_{\text {reg}}\) it amounts to Hardy’s inequality. It is trivial for \( {\Gamma }^{(1)}_{\text {reg}}\), since \(a\in L^\infty \).
1.2 A.2 Regularity of the Charge
Now, we study the regularity of the charge associated with \(\psi \in H\), that is the solution of the equation appearing in (2.18).
Proposition A.4
Assume (H2) and \(\gamma >2\), let \(f\in H^{1/2}(\mathbb {R}^3)\) and let \(\xi \in {\mathcal {D}}(\Gamma ^\lambda )\) be the solution of
then \(\xi \in H^{3/2}(\mathbb {R}^3)\) and \( \Vert \xi \Vert _{H^{3/2}}\leqslant c \Vert f\Vert _{H^{1/2}}\).
Proof
We argue as in Proposition A.2 and recast (A.16) as \(T\xi = f^\lambda \) (see (A.1) and (A.4)). We note that \(f^\lambda \in H^{1/2}(\mathbb {R}^3)\). To convince oneself that this is the case, recall that by Proposition A.2, \(\xi \in H^{1}(\mathbb {R}^3)\) then \(a\xi \in H^{1}(\mathbb {R}^3)\) since a and \(a'\) are bounded; moreover, it can be checked that the last term at the r.h.s. of Eq. (A.4) also belongs to \(H^{1/2}\):
introducing polar coordinates, one easily sees that the last integral is finite.
To conclude the proof we are going to show that
Decomposing \(\xi \) and \(f^\lambda \) on the basis of the spherical harmonics, we obtain
and
where \( \hat{f}_{\ell m}^\lambda (p)\) satisfies the condition
To prove the bound (A.17), we will show that
where c does not depend on \(\ell \) and m, and
We remark that for \(\ell = 0\) we have a slightly weaker bound involving both \(\Vert f^\lambda \Vert \) and \(\Vert \Delta ^{1/2} f^\lambda \Vert \).
i) Case \(\ell \geqslant 1\).
Decomposing the equation \(T\xi = f^\lambda \) on the basis of the spherical harmonics, we obtain (A.5). We multiply it by \(p^2\), change variables as \(p=e^x\), and \(q=e^y\), and set \(\zeta _{\ell m}(x) = e^{3 \,x} {\hat{\xi }}_{\ell m}(e^x)\), and \(h_{\ell m}(x)= e^{2 \,x} {\hat{f}}_{\ell m}(e^x)\). In this way, we obtain the equation.
Since
and
the bound (A.18) is equivalent to the inequality
The integral equation (A.20) can be conveniently studied via Fourier transform. To proceed we start by noticing that, taking into account the identity \(\int _{-1}^{1} \! \text {d}\nu \, P_\ell (\nu )=0\), \(\ell \geqslant 1\), we have
The representation formula (A.22) shows that \(S_{\mathrm {off},\ell }(k)\) can be holomorphically extended to the strip \(\{ |{\text {Im}}k| <2\}\). The same argument can be repeated for \(S_{\mathrm {reg},\ell }(k)\). Therefore, in Fourier transform, (A.20) reads
with \(S_\ell \) given as in (3.17), i.e.,
By the unitarity of the Fourier transform to prove the bound (A.21) it is enough to find a lower bound for \(|S_\ell (k+i)|\) (see also the similar argument used in the proof of Proposition A.2). We concentrate on the real part of \(S_\ell (k+i)\).
With straightforward calculations, starting from (3.9), one finds:
Analogously
Let us observe that, using the recurrence formula for the Legendre polynomials \((\ell +1) P_{\ell +1}(\nu ) = (2\ell +1) \nu P_\ell (\nu ) - \ell P_{\ell -1}(\nu )\), we can rewrite
and analogously
Then, using Lemma 3.4, for any \(k \in \mathbb {R}\) we obtain that
and
Hence,
Next, we focus attention on \({\text {Re}}S_{\mathrm {off},\ell } (k+i) \) with \(\ell \) even. Notice that (A.24) and Lemma 3.4 imply
Then, using [8, Lemma 3.5], we obtain
where \( \text {d} \in (0,1)\).
Therefore, we find
The latter bound, together with the one in (A.25), give (A.18) with \(c = (\frac{\sqrt{3}}{2}(1-d))^{-2}\) for all \(\ell \geqslant 1\).
Next, we proceed with the analysis of the case \(\ell =0\).
ii) Case \(\ell =0\)
For \(\ell =0\), the kernels in (A.20) are too singular, in particular the regularization described in (A.22) does not apply; hence, we cannot extend \(S_{\mathrm {off},0} (k)\) and \(S_{\mathrm {reg},0} (k)\) to \({\text {Im}}k =1\) and proceed starting from an equation of the form (A.23).
In order to circumvent this difficulty, we define
for \(t \in (0,1)\). Then, we multiply (A.5) by \(p^{2-t}\) and introduce the change of variables \(p=e^x\), \(q=e^y\) to obtain
where
The integral kernels in (A.28) are regular for \(t\in (0,1)\). Let us rewrite the equation in such a way to isolate the term that becomes singular for \(t \rightarrow 0\).
In the Fourier space equation (A.28) is
where
From equation (A.29), we have
We notice that
Moreover,
by [8, Lemma 3.5], see also (A.26). Similarly,
Hence, we have
where (see (A.30))
By dominated convergence theorem, we find that \(\Vert Q^1_{t} - Q^1_{0}\Vert _{L^{\infty }} \rightarrow 0\) for \(t \rightarrow 0. \;\) Analogously, we have
where (see (A.31))
The last integral in (A.36) can be estimated as follows. For \(|x|<1\), we have
where we have used the inequality \(\cosh x \geqslant 1 + \frac{x^2}{2}\). For \(|x|\geqslant 1\) we have
Using the above estimates, we can apply the dominated convergence theorem in (A.36) and we find that \(\Vert Q^2_{t} - Q_{\mathrm {reg},0}\Vert _{L^{\infty }} \rightarrow 0\) for \(t \rightarrow 0.\;\) Taking into account (A.33), (A.34), (A.35) and considering t sufficiently small, from (A.32) we obtain
where \(\,c\,\) is a constant independent of t. Moreover, by (A.27) we have \(|\zeta _{t}(x)|^2 = |e^{(3-t)x} \, \xi _{00}(e^x)|^2 \rightarrow |e^{3x} \, \xi _{00}(e^x)|^2 \) for \(t \rightarrow 0\) a.e.. Then, applying Fatou’s lemma, we find
this gives the bound (A.19) and concludes the proof of the proposition. \(\square \)
Remark A.5
Notice also that \(\Gamma ^\lambda : H^{s+1}\rightarrow H^s\) for \(s\in (0,1/2)\).The claim is obvious for \({\Gamma }^{ \lambda }_{\text {diag}} \) and it was proved in [23, Proposition 5] for \({\Gamma }^{ \lambda }_{\text {off}} \). Regarding \({\Gamma }^{ (2)}_{\text {reg}}\), due to Hardy’s inequality, it is sufficient to prove that it is a bounded operator from \(\dot{H}^{s+1}\) to \(\dot{H}^{s}\) for \(0<s<1/2\). Let us consider
If we put \(f(\mathbf{p})= p^{-3/2}\), it is straightforward to prove that for \(0<s<1/2\) we have
and then T is the integral kernel of an \(L^2\)-bounded operator by Schur’s test and the claim on \({\Gamma }^{ (2)}_{\text {reg}}\) follows. For \({\Gamma }^{ (2)}_{\text {reg}}\), the claim is trivial since \(\theta \in C^1\).
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Basti, G., Cacciapuoti, C., Finco, D. et al. Three-Body Hamiltonian with Regularized Zero-Range Interactions in Dimension Three. Ann. Henri Poincaré 24, 223–276 (2023). https://doi.org/10.1007/s00023-022-01214-9
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DOI: https://doi.org/10.1007/s00023-022-01214-9