Rigorous derivation of the Efimov effect in a simple model

We consider a system of three identical bosons in R3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^3$$\end{document} with two-body zero-range interactions and a three-body hard-core repulsion of a given radius a>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ a > 0$$\end{document}. Using a quadratic form approach, we prove that the corresponding Hamiltonian is self-adjoint and bounded from below for any value of a. In particular, this means that the hard-core repulsion is sufficient to prevent the fall to the center phenomenon found by Minlos and Faddeev in their seminal work on the three-body problem in 1961. Furthermore, in the case of infinite two-body scattering length, also known as unitary limit, we prove the Efimov effect, i.e., we show that the Hamiltonian has an infinite sequence of negative eigenvalues En\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_n$$\end{document} accumulating at zero and fulfilling the asymptotic geometrical law En+1/En→e-2πs0forn→+∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\;E_{n+1} / E_n \; \rightarrow \; e^{-\frac{2\pi }{s_0}}\,\; \,\text {for} \,\; n\rightarrow +\infty $$\end{document} holds, where s0≈1.00624\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s_0\approx 1.00624$$\end{document}.


INTRODUCTION
The Efimov effect is an interesting physical phenomenon occurring in three-particle quantum systems in dimension three ( [10,11], see also [25]).It consists in the appearing of an infinite sequence of negative eigenvalues E n , with E n → 0 for n → ∞, of the three-body Hamiltonian if the two-particle subsystems do not have bound states and at least two of them exhibit a zero-energy resonance (or, equivalently, an infinite two-body scattering length).A remarkable feature of the effect is that the distribution of eigenvalues satisfies the universal geometrical law where the parameter s > 0 depends only on the mass ratios and, possibly, on the statistics of the particles.
According to an intuitive physical picture, the three-particle bound states (or trimers) associated to the eigenvalues are determined by a long range, attractive effective interaction of kinetic origin, which is produced by the resonance condition and does not depend on the details of the two-body potentials.Roughly speaking, in a trimer the attraction between two particles is mediated by the third one, which is moving back and forth between the two.It should also be stressed that the Efimov effect disappears if the two-body potentials become more attractive causing the destruction of the zero-energy resonance.For interesting experimental evidence of Efimov quantum states see, e.g., [19].
The first mathematical result on the Efimov effect was obtained by Yafaeev in 1974 [33].He studied a symmetrized form of the Faddeev equations for the bound states of the three-particle Hamiltonian and proved the existence of an infinite number of negative eigenvalues.In 1993 Sobolev [29] used a slightly different symmetrization of the equations and proved the asymptotics where N(z) denotes the number of eigenvalues smaller than z < 0. Note that (1.2) is consistent with the law (1.1).In the same year Tamura [31] obtained the same result under more general conditions on the twobody potentials.Other mathematical proofs of the effect were obtained by Ovchinnikov and Sigal in 1979 [27] and Tamura in 1991 [30] using a variational approach based on the Born-Oppenheimer approximation.For more recent results on the subject, see [5] (for the case of two identical fermions and a different particle), [15] (for a two-dimensional variant of the problem) and [16].
We notice that in the above mentioned mathematical results a rigorous derivation of the law (1.1) is lacking.
It is also worth observing that, before the seminal works of Efimov, Minlos and Faddeev [23,24] studied the problem of constructing the Hamiltonian for a system of three bosons with zero-range interactions in dimension three.It was known that such Hamiltonian cannot be defined considering only pairwise zero-range interactions.
Minlos and Faddeev showed that a self-adjoint Hamiltonian can be constructed by imposing suitable two-body boundary conditions at the coincidence hyperplanes, i.e., when the positions of two particles coincide, and also a three-body boundary condition at the triple-coincidence point, when the positions of all the three particles coincide.They also proved that the Hamiltonian is unbounded form below, due to the presence of an infinite sequence of negative eigenvalues diverging to −∞.Such instability property 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 the three particles are very close to each other.A further interesting result of the analysis of Minlos and Faddeev, even if it is not explicitly emphasized, is the proof of the Efimov effect in the case of infinite two-body scattering length (corresponding to the resonant case), with a rigorous derivation of the law (1.1).This in particular shows that the occurrence of the Efimov effect can be obtained also with zero-range interactions, the only crucial condition being the presence of an infinite two-body scattering length.Such a result is somewhat tainted by the fact that the Hamiltonian in unbounded from below and therefore unsatisfactory from the physical point of view.
Our aim is to present a mathematical proof of the Efimov effect and law (1.1) for a bounded from below Hamiltonian obtained by a slight modification of the Minlos and Faddeev Hamiltonian.We mention that the problem of constructing a lower bounded Hamiltonian for a three-body system with zero-range interactions has been recently approached in the literature (see, e.g., [4,13,12,22]).The idea is to introduce an effective three-body force acting only when the three particles are close to each other, preventing the fall to the center phenomenon.
In the present work we consider a Hamiltonian with two-body zero-range interactions and another type of threebody interaction.More precisely, the effective three-body force is replaced by a three-body hard-core repulsion.
We shall prove that such Hamiltonian is self-adjoint and bounded from below and then prove the Efimov effect, i.e., the existence of an infinite sequence of negative eigenvalues satisfying (1.1) when the two-body scattering length is infinite.
Our work can be viewed as an attempt to make rigorous the original physical argument of Efimov.Indeed, Efimov takes into account three identical bosons and his approach is based on the replacement of the twobody potential with a boundary condition, which is essentially equivalent to consider a two-body zero-range interaction.Then, he introduces hyper-spherical coordinates and shows that if the two-body scattering length is infinite then the problem becomes separable and in the equation for the hyper-radius R the long range, attractive effective potential −(s 2 0 + 1/4)/R 2 appears.The behavior for small R of this potential is too singular and an extra boundary condition at short distance must be imposed.After this ad hoc procedure, he obtains the infinite sequence of negative eigenvalues satisfying the law (1.1) as a consequence of the large R behavior of the effective potential.
The self-adjoint and bounded from below Hamiltonian constructed in this paper can be considered as the rigorous counterpart of the ad hoc regularization scheme mentioned above.Furthermore, we show that the eigenvalues and eigenvectors found in a formal way in the physical literature are in fact eigenvalues and eigenvectors of our Hamiltonian in a rigorous sense and, accordingly, we obtain a mathematical proof of (1.1).
Let us introduce some notation.Here and in the sequel: x 1 , x 2 , x 3 ∈ R 3 are the coordinates of the three bosons in a fixed inertial reference frame; the units of measure employed are such that h = m 1 = m 2 = m 3 = 1.It is convenient to introduce the system of Jacobi coordinates r cm , x, y ∈ R 3 defined as Correspondingly, we have Upon factorizing the center of mass coordinate r cm (i.e., adopting the center-of-mass reference frame) the heuristic Hamiltonian describing our three-boson system is expressed by where, at a formal level, V hc a indicates a hard-core potential corresponding to a Dirichlet boundary condition on the hyper-sphere of radius a in R 6 , centered at (x, y) = (0, 0) and the "δ-potentials" represent the zero-range interactions between the pair of particles (1, 2), (2, 3) and (3, 1) respectively.Notice that therefore the hard-core potential V hc a plays the role to prevent the three bosons from reaching the triplecoincidence point x 1 = x 2 = x 3 , avoiding the above mentioned fall to the center phenomenon.
The bosonic Hilbert space of states for our system is where (1.5) Definition (1.4) encodes the symmetry by exchange given by σ 12 and σ 23 which clearly imply also the condition corresponding to the exchange performed by σ 31 , i.e. ψ(x, In the following we shall construct the rigorous counterpart of (1.3) as a self-adjoint and bounded from below operator in L 2 s (Ω a ).The first step is to interpret the formal unperturbed operator −∆ x − ∆ y + V hc a (x, y) as the Dirichlet Laplacian in Ω a , namely dom It is well known that (1.6) is the self-adjoint and positive operator uniquely defined by the positive quadratic (1.7) The second, and more relevant, step is to define a self-adjoint perturbation of the Dirichlet Laplacian H D supported by the coincidence hyperplanes Following the analogy with the one particle case [1], a natural attempt is to construct an operator which, roughly speaking, acts as the Dirichlet Laplacian H D outside the hyperplanes and it is characterized by a (singular) boundary condition on each hyperplane.Specifically, given α ∈ R, we demand that where B c a := R 3 \ B a and B a = y ∈ R 3 |y| < a .The above condition describes the interaction between particles 1 and 2. Due to the bosonic symmetry requirements, it also accounts for the interactions between the other two admissible pairs of particles.We recall that −α −1 has the physical meaning of two-body scattering length.The strategy for the mathematical proof is based on a quadratic form approach, similar to the one adopted for the construction of singular perturbations of a given self-adjoint and positive operator in analogous contexts (see, e.g., [32,8,4]).
More precisely, starting from the formal Hamiltonian (1.3) characterized by the boundary condition (1.8), we construct the corresponding quadratic form Q D,α and we formulate our main results (section 2).
The main technical part of the paper is the proof that Q D,α is closed and bounded from below in L 2 s (Ω a ) (section 3).
Then, we define the Hamiltonian H D,α of our system as the unique self-adjoint and bounded from below operator associated to Q D,α and we give an explicit characterization of the domain and action of H D,α , providing also an expression for the associated resolvent operator (section 4).
Finally we show that for α = 0 the Efimov effect occurs and the law (1.1)holds (section 5).
In Appendix A we collect some useful representation formulas for the integral kernel of the resolvent of the free Laplacian in R 6 and for the integral kernel of the resolvent of the Dirichlet Laplacian in Ω a .
In Appendix B we recall the solution of the eigenvalue problem in the case α = 0 following the treatment usually given in the physical literature.

FORMULATION OF THE MAIN RESULTS
In this section we first give a heuristic argument to derive the quadratic form associated to the formal Hamiltonian (1.3) and then we formulate our main results.
Let us introduce the potentials G λ ij ξ ij (λ > 0) produced by a suitable charge ξ ij with support concentrated on the hyperplane π ij : Here, for the sake of brevity, we have introduced the notation and we have denoted by R λ D (X, X ′ ) the integral kernel associated to the resolvent operator R λ D := (H D + λ) −1 .For later convenience, we write the kernel R λ D X, X ′ as where R λ 0 X, X ′ is the integral kernel associated to the resolvent operator of the free Laplacian in R 6 and the function g λ X, X ′ is a reminder term, solving the following elliptic problem for any fixed X ′ ∈ Ω a and λ > 0: In Appendix A we give explicit expressions for R λ 0 X, X ′ , g λ X, X ′ and then for R λ D X, X ′ .Furthermore, the potentials G λ ij ξ ij fulfills the following equation in distributional sense By a slight abuse of notation, we set In order to ensure that G λ ξ actually meets the bosonic symmetries encoded in L 2 s (Ω a ), we must require Taking this into account and noting that from Eq. (2.1) we infer the following: Notice that, due to the singularity for |x| → 0, the potential G λ 12 ξ 12 does not belong to H 1 (Ω a ).Of course, the same is true for G λ 23 ξ 23 and G λ 31 ξ 31 , due to the same kind of singularities for |x − √ 3y| → 0 and for |x + √ 3y| → 0, respectively.Moreover, such a singular behavior is exactly of the same form appearing in (1.8).This fact suggests to write a generic element of the operator domain as In view of the previous arguments, Eq. (1.8) is equivalent to (2.7) We now proceed to compute the quadratic form associated to the formal Hamiltonian H introduced in (1.3).
For functions of the form ψ = ϕ λ + G λ ξ, taking into account that (H D + λ)G λ ξ = 0 outside the two-particle coincidence hyperplanes, a heuristic computation yields where By means of Eqs.(2.4), (2.5), (2.6) and of the bosonic symmetry of ϕ λ one has a dy ξ(y) ϕ λ (0, y) . (2.9) Moreover, using the boundary condition (2.7) we find (2.10) By (2.8), (2.9) and (2.10) we obtain the expression for the quadratic form associated to H.In order to give a precise mathematical definition, we first consider the quadratic form in L 2 (B c a ) appearing in (2.10): with In (2.12) we have introduced the weighted L 2 space where for any b > a, the continuous function w is given by and H 1/2 (B c a ) denotes the Sobolev-Slobodeckiȋ space of fractional order 1/2 given by Notice that the choice of the parameter b > a is irrelevant and, since w 1, we have We remark that the reason for the choice of (2.12) as the form domain of Φ λ α will be clear in the course of the proofs reported in section 3. We also point out that the form domain (2.12) is isomorphic (as a Hilbert space) to the Lions-Magenes space H We are now in position to define the quadratic form in L 2 s (Ω a ) where Q D is the Dirichlet quadratic form defined in (1.7).
We stress that the definition of the quadratic forms (2.11), (2.12) and (2.20), (2.21) is the starting point of our rigorous analysis.
Before proceeding, let us mention an equivalent characterization of the potential G λ ξ.This will play a role in some of the proofs presented in the following sections.Let Sobolev trace operator defined as the unique bounded extension of the evaluation map It is crucial to notice that the range of τ actually keeps track of the bosonic symmetry encoded in dom . Taking this into account, noting that R λ D : L 2 s (Ω a ) → dom H D and using the natural embedding where, in compliance with (2.6) and (2.12), we put Accordingly, with a slight abuse of notation we can rephrase Eq. (2.5) as In the sequel we shall refer especially to the bounded operator In the rest of this section we formulate the main results of the paper.(i) The quadratic form Φ λ α in L 2 (B c a ) defined by (2.11), (2.12) is closed and bounded from below for any λ > 0.More precisely, there exists a constant B > 0 such that for any λ > 0 . (2.24) Furthermore, there exist λ 0 > 0, A 0 > 0 and A(λ) > 0, with A(λ) = o(1) for λ → +∞, such that for any λ > λ 0 . (2.25) ) is independent of λ, closed and lower-bounded.
Let us define Γ λ α , λ > 0, as the unique self-adjoint and lower-bounded operator in L 2 (B c a ) associated with the quadratic form Φ λ α .Notice that Γ λ α is positive and has a bounded inverse whenever λ > λ 0 (with λ 0 as in Theorem 2.1).We also recall that τ is the Sobolev trace operator on the coincidence hyperplane π 12 , see (2.23).

Theorem 2.2 (Characterization of the Hamiltonian).
The self-adjoint and bounded from below operator H D,α in L 2 s (Ω a ) uniquely associated to the quadratic form Q D,α is characterized as follows: For any λ > λ 0 (with λ 0 as in Theorem 2.1), the associated resolvent operator R λ D,α := (H D,α + λ) −1 is given by the Krein formula (2.27) Remark 2.3.As a consequence of the previous Theorem, one immediately sees that Ψ µ ∈ dom H D,α is an eigenvector of H D,α associated to the negative eigenvalue −µ, with µ > 0, if and only if The last result concerns the proof of the Efimov effect in the case of infinite two-body scattering length, i.e., when α = 0, also known as the unitary limit.

Theorem 2.4 (Efimov effect).
The Hamiltonian H D,0 has an infinite sequence of negative eigenvalues E n accumulating at zero and fulfilling where θ = arg Γ (1 + is 0 ) and s 0 ≈ 1.00624 is the unique positive solution of the equation In particular, the geometrical law (1.1)holds.Furthermore, the eigenvector associated to E n is given by where imaginary order and t n is the n-th positive simple root of the equation K is 0 (t) = 0.

ANALYSIS OF THE QUADRATIC FORM
As a first step we derive upper and lower bounds for the quadratic form Φ λ α [ξ] defined in (2.11), (2.12).The estimates reported in the forthcoming Lemma 3.1 ultimately account for the main results stated in Theorem 2.1.Lemma 3.1.For any λ > 0 there holds Moreover, there exist positive constants A i (λ) (i = 1, 2) and B i (i = 1, 2, 3, 4) such that: In particular, the constants A 1 (λ) and A 2 (λ) fulfill Proof.We discuss separately the terms Making reference to the definition (2.13), we first consider the decomposition Taking into account that the integral kernel R λ 0 (X, X ′ ) can be written explicitly in terms of the modified Bessel function of second kind K 2 (a.k.a.Macdonald function), see (A.2) in Appendix A, by direct evaluation we get Eq. 10.29.4], we deduce that t ∈ R + → t ν K ν (t) is a continuous and decreasing function.In particular, we have inf t∈[0,c] t 2 K 2 (t) = c 2 K 2 (c) for any c > 0 and sup t∈R+ t 2 K 2 (t) = lim t → 0 + t 2 K 2 (t) = 2, see [26, Eq. 10.30.2].
On one side, these arguments suffice to infer that For any y ∈ B c a , a direct computation yields It is easy to check that the expression between square brackets appearing above is a positive and decreasing In view of the previous considerations, this implies On the other side, noting that by computation similar to those outlined above (recall, in particular, that t 2 K 2 (t) 2 for all t 0), we infer Summing up, we obtain From here we readily deduce (3.2), recalling the basic relations (2.17) (2.19) and noting that The claim in (3.6) regarding the constant A 1 (λ) follows by elementary considerations, noting that the map t → t 2 K 2 (t) vanishes with exponential rate in the limit t → +∞ [26, Eq. 10.40.2].
2) Estimates for Φ λ 2 [ξ].Recall the explicit expression (2.14).By arguments similar to those described before, it is easy to see that Φ λ 2 [ξ] 0. Next, let us fix arbitrarily ε > 0 and consider the set We re-write the definition (2.14) accordingly as On one side, considerations analogous to those reported in part 1) of this proof yield for all (y, y ′ ) ∈ ∆ a,ε , which implies, in turn, On the other side, since we readily get The above arguments imply This in turn accounts for (3.3), recalling the definition of the regional Gagliardo-Slobodeckiȋ semi-norm for the Sobolev space H 1/2 B c a , see (2.18), and noting that ∆a,ε dy dy ′ ξ(y) − ξ(y ′ ) Also in this case, the statement about A 2 (λ) in (3.6) follows from the exponential vanishing of the function t → t 2 K 2 (t) in the limit t → +∞.
3) Estimates for ℓ) be the associated family of normalized spherical harmonics (see, e.g., [2]).In the sequel, for any ξ ∈ L 2 (B c a ) we consider the representation where the coefficients ξ ℓ,m are determined by It is worth noting that To proceed, we refer to the explicit expression (2.15) for Φ λ 3 [ξ] and consider the series expansion for g λ X, X ′ reported in (A.8) of Appendix A. Let us mention that the Gegenbauer polynomials C 2 ℓ appearing therein enjoy the following identity, for any pair of angular coordinates ω, ω ′ ∈ S 2 and any ℓ ∈ {0, 1, 2, . ..} [2, Eq. (66)]: On account of the above considerations and of the orthogonality of the spherical harmonics Y ℓ,m [2, Eq. ( 56)], by direct calculations we deduce From here and from the positivity of the Bessel functions K ν [26, §10.37], it readily follows that Φ λ 3 [ξ] 0. On the other hand, by Cauchy-Schwarz inequality and the already mentioned monotonicity of the map t ∈ R + → t ν K ν (t) for any fixed ν > 0, we get Taking also into account that, for any fixed ν > 0, the map t ∈ R + → I ν (t) K ν (t) is continuous, positive and strictly decreasing with lim t → 0 + I ν (t) K ν (t) = 1 2ν (see [3] and [26, Eq. 10.29.2 and §10.37, together with §10.30(i)]), we finally obtain which proves the upper bound in Eq. (3.4).

4) Estimates for Φ λ 4 [ξ].
Let us refer to the definition (2.16) and recall that R λ D (X, X ′ ) is the integral kernel associated to the resolvent operator of the Dirichlet Laplacian H D on Ω a ⊂ R 6 .The corresponding heat kernel K D (t; X, X ′ ) is known to fulfill the following Gaussian upper bound, for all t > 0, X, X ′ ∈ Ω a and some suitable c 1 , c 2 > 0 (see, e.g., [7, p. 89, Corollary 3.2.8]and [17,34]): Taking this into account and using a well-known integral representation for the Bessel function K 2 [26, Eq. 10.32.10], by classical arguments [7, p. 101, Lemma 3.4.3]we deduce Then, using the elementary inequality y•y ′ − |y| 2 +|y ′ | 2 /2 and recalling that the map On account of the above arguments, by Cauchy-Schwarz inequality and basic symmetry considerations, from (2.16) we infer This in turn implies the thesis (3.4), in view of the fact that Building on the estimates derived in Lemma 3.1, we can now proceed to prove Theorem 2.1.

Proof of Theorem
L 2 is a norm on H 1 (Ω a ) equivalent to the standard one.Then, using the upper bound (2.24) for Φ λ α [ξ] , by elementary estimates we deduce that for any λ > 0 there exists a positive constant C 1 > 0 such that This suffices to prove the well-posedness of Q D,α [ψ] on the domain (2.21) for any λ > 0.
Let us now show that the form does not depend on λ.To this purpose, we fix λ 1 > λ 2 > 0 and consider the alternative representations ψ = ϕ λ 1 + G λ 1 ξ and ψ = ϕ λ 2 + G λ 2 ξ.From the first resolvent identity and from Eq. (2.22), we deduce (2.12)).In particular, we can write . Taking this into account and using the definition (2.20) of Q D,α [ψ], by a few integration by parts we get Finally, from the lower bound (2.25) for Φ λ α [ξ] and from the asymptotic relations reported in (3.6), for any λ > λ 0 we infer This shows that, for any fixed λ > 0 large enough there exist two positive constants γ 2 , C 2 > 0 such that which proves that the form Q D,α is coercive.From here, closedness follows as well by standard arguments [32].
To say more, since the right-hand side of (3.9) is clearly positive, we readily get that Q D,α is lower-bounded.

THE HAMILTONIAN
Proof of Theorem 2.2.Let us first introduce the sesquilinear form defined by the polarization identity, starting from Eq. (2.20).With respect to the decompositions ψ 1 = ϕ λ 1 + G λ ξ 1 and ψ 2 = ϕ λ 2 + G λ ξ 2 , this is given by where we have set and Since we have already proved that the form Q D,α is closed and lower bounded, there exists a unique associated self-adjoint and lower bounded operator H D,α .Moreover, if ψ 2 ∈ dom H D,α then there exists an element For Thus, ϕ λ 2 ∈ dom H D and w = H D ϕ λ 2 − λ G λ ξ 2 , which entails the identity For On account of the bosonic exchange symmetries (see (2.4) and (2.6)), from the basic identity (2.22) we readily deduce the following, for all ξ 1 ∈ dom Φ λ α and ϕ λ 2 ∈ dom H D : where τ is the Sobolev trace operator introduced in (2.23).Let us also remark that, since R λ 0 0, y ′ ; 0, y = R λ 0 0, y; 0, y ′ , the definition (??) can be conveniently rephrased as Summing up, we obtain Then the thesis follows by the arbitrariness of ξ 1 , recalling the definition of Γ λ α (see the comments reported after Theorem 2.1).Finally, the representation (2.27) for the resolvent operator R λ α follows by general resolvent identities, noting that Γ λ α has a bounded inverse for λ > 0 large enough (see, e.g., [6,28]).
In view of the arguments reported in the above proof, it appears that the action of the operator Γ λ α on any ξ ∈ dom Γ λ α is given by Lemma 4.1.For any λ > 0, there holds dom Proof.We refer to the explicit expression (4.2) for Γ λ α ξ, regarding it as a sum of five distinct terms.The first linear addendum is trivially defined on the whole space L 2 (B c a ).Hereafter we discuss separately the remaining terms.

Recalling that, for any
[20, p. 74, Example 9.12], the above estimate proves that the second term in (4.2) is a bounded operator from H 1 0 (B c a ) into L 2 (B c a ).2) On the third term in (4.2).Let us consider the decomposition where we have set On one side, for any given ξ ∈ H 1 0 (B c a ) we consider the extension ξ ∈ H 1 (R 3 ) such that ξ = ξ a.e. in B c a and ξ = 0 a.e. in B a [21,Thm. 11.4].Then, by an explicit calculation reported in the proof of Lemma 3.1, for a.e.y ∈ B c a we obtain where ) is the square root of the Laplacian [9, §3].Keeping in mind that the function between square brackets is upper and lower bounded, and recalling again that . On the other side, noting that the function h(t) := 1 − 1 2 t 2 K 2 (t) (t > 0) is strictly increasing with h ′ (t) = 1 2 t 2 K 1 (t), lim t → 0 + h(t)/t = 0 and lim t → +∞ h(t) = 0 [26, Eqs.10.29.4,10.30.2 and 10.25.3], we infer by direct inspection that H λ ∈ L 1 (R 3 ). 1 As a consequence, by elementary estimates and Young's convolution inequality [18,Thm. 4.5.1]we infer Summing up, the previous results allow us to infer that the third term in (4.2) defines a bounded operator from 3) On the fourth term in (4.2).Retracing the arguments described in step 3) of the proof of Lemma 3.1, it can be shown that the term under analysis is a bounded operator in L 2 (B c a ).More precisely, decomposing ξ into spherical harmonics Y ℓ,m and exploiting the explicit representation (A.8) for g λ (X, X ′ ), from the summation formula (3.7) and the orthonormality of the spherical harmonics we deduce Then, recalling that the function t → t ν K ν (t) (t ∈ R + , ν > 0) is decreasing and noting that the map t → I ν (t) K ν (t) (t ∈ R + , ν > 0) is decreasing as well with lim t → 0 + I ν (t) K ν (t) = 1 2ν (see [3] and [26, Eq.
1 More precisely, integrating by parts and using a known integral identity for the Bessel function [14, p. 676, Eq. 6.561.16],we obtain 10.29.2 and §10.37, together with §10.30(i)]), we obtain 4) On the fifth term in (4.2).This term identifies a bounded operator in L 2 (B c a ).We proceed to justify this claim using arguments analogous to those reported in step 4) of the proof of Lemma 3. where ψ µ = G µ 12 ξ µ , for some suitable ξ µ ∈ L 2 (B c a ), so that Ψ µ is decomposed in terms of the so-called Faddeev components.To simplify the notation, from now on we drop the dependence on µ.We look for ψ depending only on the radial variables r = |x| and ρ = |y|, so with an abuse of notation we set ψ = ψ(r, ρ).Hence we have where (5.3) Moreover, we impose the Dirichlet boundary condition and the (singular) boundary condition Ψ = ξ(ρ) 4π r + o(1) for r → 0, see (1.8).Taking into account that ξ(ρ) = 4π (r ψ)(0, ρ), in view of (5.1) the boundary condition reads lim for ρ > a . (5.5) It turns out that the function ψ can be explicitly determined as a solution in L 2 (R 6 ) to the boundary value problem (5.2), (5.4), (5.5).We outline the construction in appendix B for convenience of the reader.Here we simply state the result.Let K is 0 : R + → R be the modified Bessel function of the second kind with imaginary order, where s 0 is the unique positive solution of (2.30).
Let {t n } n∈N be the sequence of positive simple roots of the equation K is 0 (t) = 0, where t n → 0 for n → +∞.
Taking into account that the asymptotic expansion of K is 0 (t) for t → 0 is given by [26, Eq. 10.45.7] one also has the following asymptotic behavior Then we have that, for each the boundary value problem (5.2), (5.4), (5.5) has a solution in L 2 (R 6 ) given by (2.32), i.e., where C n is an arbitrary constant.We define the charge distribution associated to ψ n as (5.9) Let us stress that ξ n actually keeps track of the Dirichlet boundary condition (5.4) for ψ n ; in fact, we have ξ n (a) = Cn a K is 0 (t n ) = 0.With a slight abuse of notation, in the sequel we refer to the function The next step is to show that the function ψ n can be written as the potential generated by the charge (5.9) distributed on the hyperplane π 12 , i.e. to prove the following lemma.
one hand, using the explicit expression (2.32) for ψ n together with the identity (5.11), we get By direct computations and simple estimates,2 for all r > 0 and ρ > a we deduce Finally, we remark that the last term in (5.13) coincides with ξ n | τϕ for any smooth ϕ.
The above results prove (5.12) for all ϕ ∈ D(Ω a ).Now the thesis follows by plain density arguments.In fact, for any given ϕ ∈ dom H D = H 1 0 (Ω a ) ∩ H 2 (Ω a ) there exists an approximating sequence {ϕ j } j∈N ⊂ D(Ω a ), converging to ϕ in the natural topology on dom H D induced by the graph norm, such that We are now ready to prove Theorem 2.4.
Proof of Theorem 2.4.Recall that, for all n ∈ N, the function Ψ n given by (2.31), (2.32) is a formal eigenfunction of H D,0 by construction.Lemma 5.10 further ensures that Ψ n = G µn ξ n , where µ n is fixed according to (5.8) and we are employing the slightly abusive notation (2.5).Since ξ n ∈ dom Γ λ 0 , this suffices to infer that Ψ n ∈ dom H D,0 .Moreover, making reference to Remark 2.3, let us stress that the boundary condition for Ψ n = 0 + G µn ξ n encoded in dom H D,0 reduces to Γ λ 0 ξ n = 0.

APPENDIX A. ON THE INTEGRAL KERNEL FOR THE DIRICHLET RESOLVENT
In this appendix we collect some results regarding the integral kernel R λ D X, X ′ associated to the Dirichlet resolvent R λ D := (H D + λ) −1 (λ > 0).We recall that throughout the paper we refer to the decomposition (2.2), namely, R λ D X, X ′ = R λ 0 X, X ′ + g λ X, X ′ , where R λ 0 X, X ′ is the resolvent kernel associated to the free Laplacian in R 6 and, for fixed We first remark that by elementary spectral arguments it follows that which entails, in turn, g λ X, X ′ = g λ X ′ , X .
Regarding R λ 0 X, X ′ , a well known computation yields where K 2 is the modified Bessel function of second kind, a.k.a.Macdonald function.Then, using a noteworthy summation theorem for Bessel functions [14, p. 940, Eq. 8.532 1], for where C 2 ℓ are the Gegenbauer (ultraspherical) polynomials defined by the identity [14, §8.930] In the last part of this appendix we derive a series representation for the remainder function g λ (X, X ′ ).
Correspondingly, a complete orthonormal set of eigenfunctions is given by the hyper-spherical harmonics Y ℓ,m , 3 The identity (A. for some constants α ℓ,m = 0 to fulfill the condition g λ → 0 for r → +∞.Furthermore, let us point out that the solution g λ has to be invariant under rotations around the fixed vector X ′ .Keeping in mind that we chose X ′ to lie on the 6 th axis, this means that we have to fix α (K) ℓ,m = 0 for all m = (ℓ, 0, 0, 0, 0) (ℓ ∈ N 0 ).The previous arguments, together with a sum rule for the hyper-spherical harmonics Y ℓ,m [2, p. 1372, Eq. 66], entail or, equivalently, Here, C 2 ℓ are the Gegenbauer polynomials defined by (A.4) and (α ℓ ) ℓ = 0,1,2,... ⊂ R are suitable coefficients.We now fix these coefficients so as to fulfill the non-homogeneous Dirichlet boundary condition in the second line of (2.3).In view of the identity (A.3), the said boundary condition for |X| = a becomes Upon varying X ∈ ∂Ω a , this implies which, together with Eq. (A.7), ultimately yields Taking these into account it is easy to see that, for any fixed X, X ′ ∈ Ω a , the series in Eq. (A.8) behaves as
(4.2).By arguments analogous to those described in the proof of Lemma 3