From Short-Range to Contact Interactions in the 1d Bose Gas

For a system of N bosons in one space dimension with two-body δ-interactions the Hamiltonian can be defined in terms of the usual closed semi-bounded quadratic form. We approximate this Hamiltonian in norm resolvent sense by Schrödinger operators with rescaled two-body potentials, and we estimate the rate of this convergence.


Introduction
Short-range interactions with large scattering length in quantum mechanical systems of bosons or distinguishable particles are conveniently described by δ-potentials, unless the space dimension is three and the number of particles exceeds two [1,2,6]. This has a long tradition in physics and rigorous formulation in mathematics [1,11,12]. Yet, a mathematical justification of such idealized models based on manyparticle Schrödinger operators with suitably rescaled two-body potentials is still at the beginning [3,14,20]. In the present paper we address this problem for the system of N bosons in one space dimension. If a trapping potential were included, this would be the Lieb-Liniger model [15]. We show that the Hamiltonian is the limit, in norm resolvent sense, of rescaled Schrödinger operators and we estimate the rate of convergence.
The Hilbert space of the system to be considered is the N-fold symmetric tensor product H := ⊗ N sym L 2 (R) (1.1) and the Hamiltonian is formally given by (1.2) where H 0 = − describes the kinetic energy of the bosons, α ∈ R determines the interaction strength, and x i ∈ R denotes the position of the ith boson. It is well known that H may be self-adjointly realized in terms of a closed semi-bounded quadratic form (see Appendix C) and that vectors from the domain can be characterized by a jump condition in the first partial derivatives at the collision planes [3, Theorem 2], [15]. We are interested in the approximation of H in terms of Schrödinger operators of the form where V ∈ L 1 ∩ L 2 (R), V (r) = V (−r), and V ε (r) := ε −1 V (r/ε), ε > 0. (1.4) Of course, the coupling constant g ε ∈ R will be chosen in such a way that In the case N = 2 it is well known and in the case N = 3 it was recently shown that H ε → H in the norm resolvent sense [1,3]. For general N ≥ 2 we will see that convergence in the strong resolvent sense is easily established with the help of -convergence. In fact, it is not hard to see that H ε + C ≥ 0 uniformly in ε > 0 for some C > 0. Let q ε and q denote the quadratic forms associated with H ε + C and H + C, respectively. Then q ε → q in the sense of weak and strong -convergence, and, by an abstract theorem, this is equivalent to convergence H ε → H in the strong resolvent sense [9]. The main result of the present paper is that H ε → H in the norm resolvent sense with estimates on the rate of convergence in terms of the decay of V . Norm resolvent convergence, unlike the weaker strong resolvent convergence, implies convergence of the spectra [19] and convergence of the unitary groups in a (weighted) operator norm (see remark below).
We set (1.5) where L 2 ev (R) ⊂ L 2 (R) denotes the subspace of even functions. Here, r and R correspond to the relative and center of mass coordinates |V (r)| 1/2 (R − εr 2 , R + εr 2 , x 3 , ..., x N ), (1.7) which is nothing but the operator of multiplication by √ (N − 1)N/2 |V ε (x 2 − x 1 )| 1/2 written in the new coordinates (1.6) and followed by the (unitary) rescaling r → εr. Let J denote multiplication by sgn(V ) in L 2 ev (R, dr) and let B ε = J A ε . Then (1.10) The formula (1.9) is our starting point for proving resolvent convergence. It allows us to generalize the methods familiar from the case N = 2 [1]. It is not hard to see that the limit exists for some, and hence for all z ∈ ρ(H 0 ). This is independent of the space dimension d ≤ 3. The subtle point in two and three space dimensions, even for N = 2, is the convergence of (g −1 ε − φ ε (z)) −1 , which involves the cancellation of divergencies [1,7]. For d = 3 and N ≥ 3 there is, in addition, a partly open problem known as Thomas effect [4,16,17,21]. In the present paper we avoid these complications by considering d = 1 only. In this case the limit (1.12) exists in the norm of L ( H ). In combination with (1.11), this allows us to take the limit ε → 0 in (1.9) and leads to the following theorem: suppose that g = lim ε→0 g ε exists, and let α = g V (r)dr. Then H ε → H in the norm resolvent sense as ε → 0, and

The norm convergence established by this theorem implies that
uniformly on compact (or growing, if s > 0) time intervals. In contrast, strong resolvent convergence implies a similar result in the strong operator topology [18]. 2. The operators φ(z) and S(z) depend on V and so the left hand side of (1. 13) seems to depend on V as well. This apparent dependence may be removed by integrating out the potential in the second term of (1.13), see Section 5. In particular, this term vanishes if V (r) dr = 0.
Our proofs of (1.11) and (1.12), and hence of Theorem 1.1, rely on explicit expressions for the integral kernels of A ε (H 0 + z) −1 and φ ε (z) in terms of the Green's function G n z of − + z in R n . This procedure is fairly involved in the case of φ ε (z) = i<j φ ij ε (z) because the kernel of φ ij ε (z) depends on the pair (i, j ) of particles. The bosonic symmetry is lost, in part, because of the symmetry breaking choice (1.6) of coordinates. Once we have shown convergence of the resolvent (H ε + z) −1 , to conclude the proof of the first statement of the theorem, it suffices to show that H ε → H in the strong resolvent sense. By a general theorem [9], this is equivalent to strong and weak -convergence of the associated quadratic forms q ε and q, which we prove in Appendix C.
Important elements of our approach, such as the representation (1.8) and the Konno-Kuroda formula (1.9) are independent of the space dimension and the statistics of the particles. A result similar to Theorem 1.1 for (distinguishable) particles with shortrange interactions in two dimensions is in preparation (see also [14]). This is related to, yet distinct from work described in [11][12][13], where two-dimensional systems with contact interactions are approximated by systems with ultraviolet cutoff.
A result similar to Theorem 1.1 for three distinct particles in one dimension was previously established in [3]. The proof in [3], however, relies on Fadeev equations, which do not generalize to N > 3. In another closely related work, the Lieb-Liniger model with repulsive δ-interactions is derived from a trapped 3d Bose gas with nonnegative two-body potentials [20].
The proof of Theorem 1.1 is given in Section 5. Sections 2, 3 and 4 provide all preparations apart from generalities, which we collect in the appendix. In Appendix A we collect the basic properties of the Green's function G n z along with some nonstandard inequalities. In Appendix B the Konno-Kuroda formula (1.9) is established in an abstract framework, and in Appendix C we prove the -convergence q ε → q.
Notations In this paper the resolvent set ρ(H ) of a closed operator H is defined as the set of z ∈ C for which H + z : D(H ) ⊂ H → H is a bijection. This differs by a minus sign from the conventional definition. The L 2 -norm will be denoted by · , without index, while all other norms carry the space as an index, as e.g. in V L 1 .

Auxiliary Operators
This section defines auxiliary operators that will be helpful in the proofs of (1.11) and (1.12).
The change of coordinates (1.6) is implemented by the coordinate transformation K : H → H defined by It follows that K * : H → H is given by 2) where the hat inx i indicates omission of this variable. Since terms arising from distinct pairs (i, j ) in (2.2) will be treated separately later on, we further introduce for 3) The asterisk in K * ij is part of the notation, which reminds us of the decomposition It does not have the meaning of adjoint. Let now V ∈ L 1 ∩ L 2 (R) be a given even potential, let v = |V | 1/2 and let u = sgn(V )v so that V = vu. Let U ε denote the unitary scaling in H defined by U ε (r, R, x 3 , ..., x N ) := ε 1/2 (εr, R, x 3 , ..., x N ). (2.5) The closed operator A ε : D(A ε ) ⊆ H → H defined by agrees with (1.7). With the help of (2.2), it is straightforward to verify that on D(H 0 ), which proves (1.8). It follows, in particular, that A * ε B ε and A * ε A ε are infinitesimally H 0 -bounded. Hence Theorem B.1 applies to (1.8), which justifies (1.9).

The Limit of A ε (H 0 + z) −1
In this section the limit of A ε (H 0 + z) −1 : H → H as ε → 0 is computed assuming V ∈ L 1 (R) only. The rate of convergence is estimated in terms of the decay of V at r = ∞. While A ε will be an unbounded operator in general, the operator A ε (H 0 +z) −1 is bounded as will be seen. In this section the restriction to d = 1 space dimension would not be necessary, all arguments go through for general d ≤ 3.
The Laplacian H 0 expressed in the relative and center of mass coordinates (1.6) reads In terms of the coordinate transformation K from (2.1) this means that Hence, (2.6) implies that with an operator T ε (z) in H defined by It remains to prove existence of lim ε→0 T ε (z). Upon a Fourier transform in (R, x 3 , ..., x N ), the operator (3.4) acts pointwise in the associated momentum variable P = (P , P 3 , ..., P N ) by an operator T ε (z, P ) that is given by The integral kernel associated with T ε (z, P ) is where G λ := G 1 λ denotes the Green's function of − + λ : H 2 (R) → L 2 (R), which is explicitly given by is a Hilbert-Schmidt operator. Let T 0 (z, P ), and thus T 0 (z), be defined by (3.7) with ε = 0. We expect that T ε (z) converges to T 0 (z) as ε → 0. Lemma 3.1 and Proposition 3.2 below are concerned with this convergence. The first step is to show that it suffices to consider potentials V with compact support. For this purpose, we introduce for any k ≥ 0 the cutoff potential and we set v k (r) := |V k (r)| 1/2 . By T ε,k (z) and T 0,k (z) we denote the operators T ε (z) and T 0 (z) with v replaced by v k , respectively. The kernels of T ε,k (z, P ) and T 0,k (z, P ) are given by (3.7) with v k instead of v. The next lemma shows that T ε,k (z) is close to T ε (z) uniformly in ε ≥ 0 for large k > 0.
Hence, the Hilbert-Schmidt norm of T ε (z, P ) − T ε,k (z, P ) is bounded by the right side of (3.10), which is independent of P . This proves the lemma.
Proof We first assume that dr |r| 2s |V (r)| < ∞ for some s ∈ (0, 1]. For fixed P , it follows from (3.7) that the difference T ε (z, P ) − T 0 (z, P ) is associated with the kernel where Q ≥ 0 is defined by (3.6). Using Lemma A.2, the L 2 -norm thereof can be bounded by where the right side is independent of P . To prove that T ε (z) − T 0 (z) = O(ε s ) as ε → 0, it is thus sufficient to show that for every λ > 0 there exists a constant C(λ) > 0 such that This proves (3.16) and hence the second part of the lemma.
In the case of general V ∈ L 1 (R), we use an approximation argument together with Lemma 3.1. This reduces the proof to showing that for every k > 0. But this is clear from the above because |r| 2s |V k (r)| dr < ∞.
Remark Proposition 3.2 implies that the limit exists for every z > 0.

Convergence of φ ε (z)
In this section, as in the previous one, the assumption V ∈ L 1 (R) will be sufficient. It ensures that A ε is a densely defined, closed operator from H to H and that A ε (H 0 + z) −1 is bounded. In the following S(R N ) denotes the Schwartz space and all operators are introduced on the subspace S(R N ) ∩ H ⊆ D(A * ε ), which is dense in H . We will see, however, that some of them have bounded extensions.
Let z > 0 be fixed. In view of the identities (1.10), (2.4), and (2.6), we have the decomposition For the further analysis of these operators, we fix a pair (i, j ) with 1 ≤ i < j ≤ N and we compute the kernel of φ ij ε (z) in terms of the Green's function G N z of − + z : H 2 (R N ) → L 2 (R N ). Inserting the defining relations (2.5), (2.1), and (2.3) for U ε , K and K * ij , respectively, we obtain that 2) Here, the second equation results from the substitution where two more integrations, which are compensated by the two δ-distributions, were introduced. A priori the operators φ ij ε (z) (1 ≤ i < j ≤ N) are all unbounded, but we will see in the subsequent lemmata that they all extend to bounded operators given by the same integral kernels. Hence, we expect that they converge to the (formal) limit operators φ ij 0 (z), which are defined in terms of the corresponding kernels with ε = 0. In the following, we prove that this is indeed the case. For this purpose, we divide these operators into the three groups (i, j ) = (1, 2), i ∈ {1, 2} and j ≥ 3, and {i, j } ⊂ {3, . . . , N}, and we analyze these groups separately.

The Limit of φ 12 ε (z)
For the kernel of φ 12 ε (z) we shall not use (4.2) but instead we derive a simpler expression as follows. By the defining expression in (4.1) and by (3.2), It follows from (4.3), after a Fourier transform in (R, x 3 , ..., x N ), that the operator φ 12 ε (z) acts pointwise in the associated momentum variable P = (P , P 3 , ..., P N ) by the operator where Q ≥ 0 is defined by (3.6). This operator has the integral kernel where G λ = G 1 λ denotes the one-dimensional Green's function, which is explicitly given by (3.8). Due to the facts that u, v ∈ L 2 (R) and G 1 2 (z+Q) is bounded, we see that φ 12 ε (z, P ) is a Hilbert-Schmidt operator and we expect, and prove below, that lim ε→0 φ 12 ε (z) = φ 12 0 (z), where φ 12 0 (z, P ) is defined in terms of the kernel (4.5) with ε = 0, which is As in the previous section, the first step in the analysis of the limit ε → 0 is to reduce the problem to the case of compactly supported potentials. Let V k (k > 0) denote the cutoff potential introduced in (3.9) and let φ 12 ε,k (z) and φ 12 0,k (z) denote the operators φ 12 ε (z) and φ 12 0 (z) with V replaced by V k . The corresponding kernels are given by (4.5) and (4.6), respectively, with the substitutions u → u k and v → v k , where v k (r) = |V k (r)| 1/2 and u k (r) = sgn(V k (r))v k (r). The next lemma shows that φ 12 ε (z) and φ 12 0 (z) define bounded operators and φ 12 ε,k (z) is close to φ 12 ε (z) uniformly in ε ≥ 0 for large k > 0.
Furthermore, for any k > 0, we have the estimate Proof For fixed z > 0, ε ≥ 0 and P , it follows from (4.5) and the L ∞ -bound that the Hilbert-Schmidt norm of φ 12 ε (z, P ) is bounded by the right side of (4.7), which is independent of P . This proves the first part of the lemma.
For the second part of the lemma, we note that φ 12 ε (z, P ) − φ 12 ε,k (z, P ) has the kernel (4.10) With the help of (4.9) and the relation we see that the L 2 -norm of (4.10) can be bounded by This shows that the Hilbert-Schmidt norm of φ 12 ε (z, P ) − φ 12 ε,k (z, P ) is bounded by the right side of (4.8), which is independent of P . Hence, (4.8) is established. Now, we can prove that φ 12 0 (z), which is defined by the kernel (4.6), is the limit of φ 12 ε (z): Proposition 4.2 Let z > 0. Then φ 12 ε (z) converges in operator norm to φ 12 0 (z) as ε → 0. If dr |r| 2s |V (r)| < ∞ for some s ∈ (0, 1], then φ 12 Proof Let us first assume that dr |r| 2s |V (r)| < ∞ for some s ∈ (0, 1]. Then we note that the kernel of φ 12 ε (z, P ) − φ 12 0 (z, P ), for z > 0 and P ∈ R N−1 fixed, is given by 1 (4.12) With the help of the elementary inequality 1 − exp(−x) ≤ x s , which is valid for x ≥ 0, and the explicit formula (3.8) for G z , we find that Using this to estimate the Hilbert-Schmidt norm of φ 12 ε (z, P ) − φ 12 0 (z, P ), we find that which proves the second part of the lemma.
In the case of general V ∈ L 1 (R), by Lemma 4.1, it suffices to prove that for every fixed k > 0. This is clear from the above because |r| 2s |V k (r)| dr < ∞.

The Limits of φ
1j ε (z) and φ 2j ε (z) for j ≥ 3 Next, we discuss the operators φ 1j ε (z) and φ 2j ε (z) with j ∈ {3, ..., N}. Their kernels are derived from (4.2): after the evaluation of the δ-distributions in x 1 and x j followed by the substitution x 2 → x j , we see that the kernel of φ 1j ε (z) reads where Hence, φ 1j ε (z) simply acts by convolution in the variables (x 3 , ... x j ..., x N ). Consequently, it follows from Lemma A.1 (vi) that φ 1j ε (z) acts pointwise in the associated momentum variables P j = (P 3 , ... P j ..., P N ) by an operator φ 1j ε (z, P j ) with kernel This kernel depends on the three-dimensional Green's function, which is explicitly given by Similarly, the operator φ 2j ε (z) acts pointwise in P j by the operator φ 2j where A comparison of (4.18) and (4.20) shows that the kernels of the operators φ 1j ε (z) and φ 2j ε (z) only differ by the reflection r → −r. Hence, it suffices to consider the operators φ 1j ε (z) henceforth. In Lemma 4.4, we will see that φ 1j ε (z) and φ 2j ε (z) extend to bounded operators. Hence we expect that they converge to the formal limit operators φ 1j 0 (z) and φ 2j 0 (z), respectively, which are defined by the corresponding kernels with ε = 0. The next lemma explains the norm bounds in Lemma 4.4: Proof Applying the Schur test yields In the second inequality we first made a substitution and then used that G 3 z (x) is decreasing as a function of |x|. The integral can be evaluated directly or with the help of (A.3).
The first step in proving that φ ij 0 (z) = lim ε→0 φ ij ε (z) for i ∈ {1, 2} and j ∈ {3, ..., N} is again a reduction to compactly supported potentials. For this purpose, let φ ij ε,k (z) (ε ≥ 0) be the variant of φ ij ε (z) with the potential V replaced by the cutoff potential V k from (3.9). The next lemma is the analog of Lemma 4.1: Furthermore, for any k > 0, we have the estimate Proof Without loss of generality, we may assume i = 1. The proofs of (4.23) and (4.24) are similar: in both cases we have to estimate the norm of an operator that, for fixed P j = (P 3 , ... P j ..., P N ), is given by a kernel of the form Explicitly, we have W (r, r ) = u(r)v(r ) in the case of (4.23) and W (r, r ) = u(r)v(r ) − u k (r)v k (r ) in the case of (4.24). Therefore, we only demonstrate the desired estimate in the case of (4.23). For fixed , the Cauchy-Schwarz inequality in the r -integration yields In the case of (4.24), the identity (4.11) implies that The rest of the proof of (4.24) is the same as for (4.23). We continue estimating the right side of (4.25). For fixed r, r ∈ R and P j , the with the integral operator F z+Q j ∈ L (L 2 (R 2 )) from Lemma 4.3 and ∈ L 2 (R 3 ) defined by where the right side is independent of P j . Hence φ 1j ε (z, P j ) extends to a bounded operator in L 2 (R 3 ) and in view of (4.26) its norm is bounded by the right side of (4.23). This completes the proof of (4.23). The proof of (4.24) is similar with the only exception that (4.26) has to be replaced by (4.27).
After these preparations, we are in position to prove: Proof Again, it suffices to consider the case i = 1. Let us first assume that dr |r| 2s |V (r)| < ∞ for some s ∈ (0, 1) and let where X ε := X 1j ε for short. Using the Cauchy-Schwarz inequality in the rintegration, we obtain for fixed ∈ L 2 (R 3 where X := (r , R , x j ) for brevity. For a further estimate of (4.31), we consider for fixed r, r ∈ R, Q j ≥ 0 and ε > 0 the integral operator F r,r ,Q j ,ε in L 2 R 2 , d(R, x j ) that is defined in terms of the kernel We are going to estimate F r,r ,Q j ,ε with the help of a Schur test. To this end, we introduce for ε ≥ 0 the intermediate point Using the properties of the Green's function (see Lemma A.2), From (4.32) and a similar estimate with the roles of (R, x j ) and (R , x j ) interchanged, we conclude, using the Schur test, that Hence, Lemma A.3 implies that F r,r ,Q j ,ε 2 ≤ Cε 2s (|r| 2s +|r | 2s ) for some constant C = C(s, z) > 0, which does not depend on r, r , Q j and ε. Using this in (4.31) results in where I (V , s) is defined by (4.29). As the right side is independent of P j , this proves the second part of the lemma. If the assumption dr |r| 2s |V (r)| < ∞ is not satisfied, then the proposition follows -by an approximation argument -from Lemma 4.4 and from the fact that I (V k , s) < ∞ for finite k.
where we made use of the shorthand notation After a Fourier transform in (x 3 , ... x i ... x j ..., x N ), property (vi) of Lemma A.1 implies that φ ij ε (z) acts pointwise in the associated momentum variable P ij = (P 3 , ... P i ... P j ..., P N ) by an operator φ ij ε (z, P ij ) with kernel In Lemma 4.7, we will see that φ ij ε (z) extends to a bounded operator from H to L 2 (R N ). Moreover, this is still true for the (formal) limit operator φ ij 0 (z) that, for fixed P ij ∈ R N−4 , is defined by the kernel (4.36) with ε = 0. The norm bound in Lemma 4.7 will be a consequence of the following lemma: Proof The Schur test and the substitutions w − As G 4 z (x) is decreasing as a function of |x| (see Lemma A.1 (v)), we can continue estimating where we used (A.3) to evaluate the integral.
As in the previous sections, the first step in proving convergence is a reduction to potentials with compact support. For this reason, we introduce for k > 0 and ε ≥ 0 the operator φ ij ε,k (z) that is defined as the operator φ ij ε (z) with the potential V replaced by the cutoff potential V k from (3.9). Hence, for fixed P ij , the kernel of φ ij ε,k (z, P ij ) is given by (4.36) with u k instead of u and v k instead of v. The next lemma is the analog of Lemma 4.4: Lemma 4.7 Let z > 0, ε ≥ 0 and 3 ≤ i < j ≤ N. Then φ ij ε (z) extends to a bounded operator from H to L 2 (R N ), which satisfies the norm estimate Furthermore, for any k > 0, we have the estimate Proof The proof follows the line of arguments in the proof of Lemma 4.4 with very few adjustments. The role of F z in the proof of Lemma 4.4 is now played by B z ∈ L (L 2 (R 3 )), for which we use the estimate from Lemma 4.6.
As the last step before the proof of Theorem 1.1, we show convergence of φ ij ε (z): Proof As in the proof of Proposition 4.5, we first assume that |r| 2s |V (r)| dr < ∞ for some s ∈ (0, 1). In the following, we use the shorthand notations X ε := X ij ε for ε ≥ 0 and q := Q ij . Notice that (4.36) implies that φ ij ε (z) − φ ij 0 (z) acts pointwise in P ij by an operator with kernel Hence, similar to (4.31), by the Cauchy-Schwarz inequality, where X := (r , R , x i , x j ) for short. For further estimates of (4.42), we analyze for fixed r, r ∈ R, q ≥ 0 and ε > 0 the integral operator B ε,r,r ,q in L 2 (R 3 , d(R, x i , x j )) with kernel
Finally, we show that the potential V may be integrated out in the expression (5.4) for the resolvent of H . To this end, we introduce an auxiliary Hilbert space H red by (1.5) and H = L 2 ev (R, dr) ⊗ H red . By inspection of the defining expressions (3.7), (4.6), (4.18), and (4.36), the operators S(z) ∈ L (H , H ) and φ(z) ∈ L ( H ) factor as follows: where S(z) ∈ L (H , H red ) and φ(z) ∈ L ( H red ) are bounded operators, which do not depend on V . Explicitly, it follows from (3.20) that where the action of T 0 (z) is pointwise in P = (P , P 3 , ..., P N ) and given by  In view of (5.9), we have that φ(z) = φ(z) V L 1 , so Lemma 4.1, Lemma 4.4 and Lemma 4.7 show that φ(z) ≤ K √ z (5.12) for some constant K > 0. Therefore, 1 − α φ(z) is invertible for large enough z > 0 and with the help of (5.9) and α = g V (r) dr = g v|u it is straightforward to verify that Step 3. If z ∈ ρ(H 0 ) ∩ ρ(H ), then (1 − φ(z)) is invertible and 1 + (z) = (1 − φ(z)) −1 .
Since RanBR 0 (i) ⊂ D(A * ), it follows that (1 − φ(z)) −1 leaves D(A * ) invariant as well, and that is well defined. Now it is a matter of straightforward computations to show that (H + z)R(z) = 1 on H and that R(z)(H + z) = 1 on D(H ).
Let V ∈ L 1 (R) with V (−r) = V (r), let g = lim ε→0 g ε , and let α = g V (r)dr. Let q and q ε denote the quadratic forms on H 1 (R N ) defined by for ψ ∈ C ∞ 0 (R N ). It is well known that this operator extends to a bounded operator from H 1 (R N ) to H 1/2 (R N−1 ). By Lemma C.1, the quadratic forms q and q ε are bounded below and closed. More precisely, we may choose C so large, that q ≥ 0 and q ε ≥ 0 for all ε > 0.
We are going to prove weak and strong -convergence q ε → q as ε → 0. To this end, it is convenient to extend all quadratic forms to L 2 (R N ) by setting q = q ε = +∞ in L 2 (R N )\H 1 (R N ). The main ingredients of this section are the inequalities