On the Weakness of Short-Range Interactions in Fermi Gases

Ultracold quantum gases of equal spin fermions with short range interactions are often considered free even in the presence of strongly binding spin-up-spin-down pairs. We describe a large class of many-particle Schr\"odinger operators with short-range pair interactions, where this approximation can be justified rigorously.


Introduction
Short-range interactions among equal spin fermions in ultracold quantum gases are often neglected, while at the same time the interaction between particles of opposite spin is modeled by zero-range (i.e. contact) interactions [6,10,20]. This can be justified by the fact that zero-range interactions among spinless (or equal spin) fermions are prohibited by the Pauli principle (see Theorem 5.1,below), and by the recent approximation results for zero-range interactions in terms of short-range potentials [2,11,12]. In the present paper we give a more direct analysis of the weakness of short-range interactions among spinless fermions in terms of estimates for the resolvent difference of free and interacting Hamiltonians. Our main results hold for all space dimensions d ≤ 3.
We consider fermionic N -particle systems in the Hilbert space

described by Schrödinger operators
where V ε (r) = ε −2 V (r/ε), V (−r) = V (r), λ ε > 0 and d ∈ {1, 2, 3}. We are primarily interested in the case, where the interaction strength of two distinct particles, in their center-of-mass frame described by is independent of ε in the sense that (2) has a ground state energy E ε that is fixed or convergent E ε → E with limit E < 0. It is well-known in the spectral theory of Schrödinger operators what this means for λ ε [17,22]. In fact, ε → λ ε and V can be chosen, depending on d, in such a way that the resolvent of (2) -and hence the spectrum of (2) -has a limit as ε → 0. Since E < 0, the limit of (2) describes a non-trivial point interaction at the origin [1]. Our main result, Theorem 3.1, can be described in simplified form as follows. Assuming V ∈ L 1 ∩ L 2 (R d ) with V ≥ 0, C V = sup r∈R d V (r)|r| 2 < ∞, some further decay of V in the case d = 1, and we show that in norm resolvent sense. The rate of convergence depends on the size of λ ε and -to some extent -on the decay of V (x) as |x| → ∞. Given sufficient decay of V , the regularity of H 2 (R d )-functions, and hence the dimension d, begins to play a role. If we choose λ ε as described above, where E ε → E < 0, then condition (3) is satisfied for all N if d ≤ 2, and for some N ≥ 3 if d = 3. Surprisingly, this particular choice of λ ε conspires with the regularity of Sobolev functions in such a way that with the bound O(ε 2 ) independent of the space dimension d. Another choice for the coupling constant, consistent with (3) as well, is the one where λ ε is a positive constant smaller than d 2 /C V N . Then operator (2), upon a rescaling, is proportional to ε −2 and hence the (negative) binding energy E ε diverges. In this case, the limit ε → 0 amounts to a combined short-range and strong interaction limit, which is interesting and relevant physically [20]. In summary we can say that fully spin polarized Fermi gases in d ≤ 2 with short-range interactions -the spin-up-spin-down interaction strength being fixed -are asymptotically free in the limit of zero-range interaction. This is true in dimension d = 3 as well for suitable V and small N ≥ 3, depending on V . The result remains correct even in a suitable combined limit of short-range and strong interaction.
We conclude with some remarks on the literature: For single particles the approximation of point-like disturbances by short-range potentials is discussed at length and in rich detail in [1], see also [5]. For systems of N ≥ 3 particles in d ≤ 2 dimensions, it was recently shown in a series of papers, that contact interactions (of TMS-type) can be approximated by rescaled two-body potentials in the norm resolvent sense of N -particle Hamiltonians [2,[11][12][13][14]. From these results the mere convergence (4) can be derived by reduction of the Hilbert space to antisymmetric wave functions. This works for the very special choice of λ ε needed for the approximation of contact interactions, and for d ≤ 2, only. For d = 3, systems of N ≥ 3 distinct (or bosonic) particles with two-body short-range interactions are prone to suffer collapse, a phenomenon known as Thomas effect. See Proposition 4.1, below. In order to avoid this effect, suitable many-body forces are required [3,8,9]. This work is organized as follows. In Section 2 we present an explicit estimate of the norm of (H ε=1 + z) −1 − (−∆ + z) −1 in terms of the pair potential V . This estimate is then used in Section 3 to prove (4) and (5). The proofs benefit from the methods and tools developed in [11,12]. Section 4 gives examples illustrating our results, in particular for d = 3 and N = 3. Finally, in Section 5, we prove the impossibility of fermionic contact interaction in d ≥ 2. This improves, for fermions, a well known result about the impossibility of contact interactions in dimensions d ≥ 4 [23].

The resolvent difference
is real-valued and even. There is no scaling parameter and no coupling constant. We assume d ≤ 3 and hence H is self-adjoint on D(H) = D(H 0 ) [21]. The result of this section, Proposition 2.1, is based on a suitable factorization W = A * B and an iterated (second) resolvent identity related to the Konno-Kuroda formula [18]. We start by constructing the factorization. Let where L 2 odd denotes the subspace of odd functions from L 2 . The integration variables r and R in (6) correspond to the relative and center of mass coordinates of the fermion positions x 1 and x 2 . This change of coordinates is implemented isometrically by the operator K : H f → X f given by From ϕ, V ij ψ = ϕ, V 12 ψ and from ϕ, We therefore write V = vu with Recall that V ∈ L 1 (R d ) and hence u, v ∈ L 2 (R d ). The domain D(A) of A : D(A) ⊂ H f → X f is determined by the domain of the multiplication operator v ⊗ 1, so it follows that A and B are densely defined and closed on D(A) ⊃ D(H 0 ). Hence, H can be rewritten in the form By an iteration of the resolvent identity, we find Upon setting W = A * B, Identity (12) can be written in the form The following proposition is our tool for proving norm resolvent convergence in the next section.
Then H ≥ 0 and, for all z > 0, where v = |V | 1/2 , and C δ is a function of C and δ. Here · odd denotes the operator norm in L 2 odd (R d , dr).
Proof. Suppose, temporarily, that H ≥ 0 and let z > 0. Then, from (13), the definition of A, and from (−∆ r ⊗ 1 + z)K (H 0 + z) −1 ≤ 1, it follows that It remains to prove H ≥ 0 and S(z) ≤ C δ under the various assumptions on V .
In the case V ≤ 0 we have J = −1, B = −A, and hence, In the case V ≥ 0 we have J = 1, B = A, and hence the assumption (14). It remains to consider the case where V changes sign. The assumption H 0 −(1+δ)W ≥ 0 implies that

By Lemma 2.2 it follows that
Combining this with the assumption on |V |, that is, This, by Lemma 2.2, implies A(H + µ) −1 A * ≤ C(1 + δ)/δ and the desired bound on S(z) follows from (14). Proof.

The resolvent convergence
We now apply the results from the previous section to Schrödinger operators with rescaled two-body potentials, that is where We therefore define, Note that S λ ≥ 0 for all λ ≤ λ max . This follows from the fact that E λ := inf σ(S λ ) as a function of λ is concave (hence continuous) and E 0 = 0. For Theorem 3.1 to be non-void, we need that λ max > 0. This can be achieved, e.g., by assuming that Then Statement as well as proof of Theorem 3.1 depend on the regularity of H 2 -Sobolev functions. Explicitly we use the embedding 1) and I 3 = [0, 1/2], and we use Lemma 3.2, below, which improves the embedding in the case d = 2. Here C 0,s (R d ) denotes the space of continuous functions that are uniformly Hölder continuous of exponent s.
If λ 0 := lim sup ε→0 λ ε < λ max , then H ε ≥ 0 for ε small enough, and for all z > 0, Moreover, the following is true: (a) If |V (r)||r| 2s dr < ∞ for some s ∈ I d , then In the situation described in the introduction, where inf σ(−2∆ − λ ε V ε ) has a limit E < 0, the bounds (a) and (b) reveal a surprising interplay between λ ε and the (optimal) regularity of provided V decays fast enough, e.g., as in the hypothesis of (b), and N is small if d = 3. Remarks: 1. Part (a) of the theorem shows that, for a large class of potentials, in the norm resolvent sense, provided that λ ε ε d−2+2s → 0 as ε → 0. This is not true for the HamiltoniansH ε defined by (15) on the enlarged Hilbert space L 2 (R N d ), as shown in the next section.
2. For the convergence (19) in norm resolvent sense to hold, it is necessary that inf σ(H ε ) → inf σ(H 0 ) = 0. Therefore the assumption lim sup ε→0 λ ε < λ max in Theorem 3.1 cannot be relaxed significantly. Strong resolvent convergence, by contrast, has much weaker spectral implications and hence -given some decay of V -much less is needed of λ ε , see Proposition 3.3 below.
3. A weaker result, similar to Theorem 3.1, could be derived from [11,12]. Indeed, for suitable λ ε we know from [11,12] that H ε → H in norm resolvent sense, where H = −∆ on H f . Information on the rate of convergence can also be found in these papers.
Proof. We are going to apply Proposition 2.1 to the Hamiltonian (15), and we assume that V changes sign, the other cases being easier. Due to the unitary equivalence of ε 2 H ε and S λε , the hypotheses of Proposition 2.1 are equivalent to for some δ, C > 0. Both (20) and (21) are true for ε small enough. This follows from λ 0 = lim sup ε→0 λ ε < λ max , from sup x∈R d |V (x)||x| 2 < ∞, and from the Hardy inequality for fermions (17). Hence, by Proposition 2.1, for ε > 0 small enough, where v ε (x) := ε −d/2 |V (x/ε)| 1/2 . By the Sobolev embedding H 2 (R d ) ֒→ C 0,s (R d ), valid for s ∈ I d , the elements ψ ∈ H 2 (R d ) ∩ L 2 odd (R d ) are Hölder continuous (of exponent s) odd functions. It follows that ψ(0) = 0 and that Therefore, for all This is true for all s ∈ I d and, combined with (22), it proves statement (a) of the theorem.
To prove (b), we use Lemma 3.2 in (23) (rather than H 2 ֒→ C 0,s ) and then (24) becomes where the integral is finite by the assumptions on V . It remains to prove (18). Eq. (24) with s = 0 implies that v ε (−∆ + z) −1 odd = O(1), which can be improved as follows: let χ k denote the characteristic function of the ball |x| ≤ k in R d and let (vχ k ) ε = v ε χ εk . Then, by (24), for any s > 0 in I d . On the other hand, uniformly in ε > 0, where HS refers to Hilbert-Schmidt norm. The combination of (22), (25) and (26) proves (18) and concludes the proof of the theorem.
In the proof of Theorem 3.1 we have used the following lemma, which can probably be found in the literature, but we are not aware of suitable reference. Lemma 3.2. For all u ∈ H 2 (R 2 ) and all x, y ∈ R 2 , y = 0, we have Remark: By our method of proof, this inequality can be generalized to derivatives ∂ α u, |α| ≤ k, of functions u ∈ H s (R n ) with s − (n/2) = k + 1, s ∈ R and k ∈ N 0 .
We conclude this section with the proposition announced in Remark 2, above. It is a consequence of Theorem 5.1 concerning the essential self-adjointness of the Laplacian. In all the following Ω = R N d \ Γ, where Suppose there exists s ≥ 0 such that V (r) 2 |r| 2s dr < ∞ and lim sup ε→0 λ ε ε s+d/2−2 < ∞. Then, H ε → H 0 in the strong resolvent sense as ε → 0.

Examples and discussion
To put Theorem 3.1 into a broader perspective and to demonstrate its dependence on the Pauli principle, we now view H ε as the restriction whereH ε denotes the Schrödinger operator defined by expression (15) on the enlarged Hilbert space L 2 (R N d ). We shall give choices for λ ε and V , whereH ε , in contrast to H ε , has a limitH describing non-trivial contact interactions or no limit at all. In the cases d = 1 and d = 2 we choose, for simplicity, a two-body potential V ∈ L ∞ (R d ) with compact support and V (r) dr = 1. Suppose further that Then λ max > 0 and λ 0 = lim sup ε→0 λ ε = 0. So, the hypotheses of Theorem 3.1 are satisfied and hence H ε → H 0 in norm resolvent sense. On the other hand, by [11,12],H ε →H, whereH describes non-trivial contact interactions. That is,H is a self-adjoint extension of −∆|C ∞ 0 (R N d \Γ) distinct from −∆. See (30) for the definition of Γ. We now turn to the more interesting case of N particles in d = 3 dimensions. In the following N is exhibited in the notation: we write H N,ε for H ε andH N,ε forH ε . For the coupling constant and the two-body potential we choose λ ε = 2 and Then 0 ≤ V (r) ≤ |r| −2 and hence C V = sup V (r)|r| 2 ≤ 1. It follows that, Thus, for N ≤ 4 we have λ 0 < λ max and hence, by Theorem 3.1, H N,ε → H 0 in norm resolvent sense. On the other hand, concerningH N,ε the following can be said: With the above notations, in the case d = 3 we have (a) For N = 2,H 2,ε →H 2 in norm resolvent sense, whereH 2 is a non-trivial self-adjoint extension of −∆|C ∞ 0 (R 6 \Γ).
Remark. The divergence of the ground state energy established in Part (b) is known as Thomas effect [24].
In the case (b), we use that the Schrödinger operatorH N,ε is unitarily equivalent to ε −2H N,ε=1 . For N = 3 the presence of a zero-energy resonance in the two-body Hamiltonian leads to non-empty (in fact, infinite) discrete spectrum in the three-particle Hamiltonian with center of mass motion removed [7,16,19]. This is the Efimov effect. It means, in particular, that C 3 := inf σ(H 3,ε=1 ) < 0. By the HVZ-theorem, C N := inf σ(H N,ε=1 ) ≤ C 3 for all N ≥ 3.

Absence of contact interactions for d ≥ 2
In space dimensions d ≥ 2 zero-range interactions among equal-spin fermions are prohibited by the Pauli principle. This is true in the very strong form of Theorem 5.1, below. For a related result in the physics literature concerning two fermions in d = 2, see [4]. Let Remarks.
1. The main point of Theorem 5.1 is that elements of C ∞ 0 (Ω) vanish in an entire neighborhood of the collision set Γ. The elements of is true for all d ≥ 1 and it easily follows from the fact that C ∞ 0 (R N d ) is dense in H 2 (R N d ).
2. For d = 1 the assertion of the theorem is false. To see this, consider a sequence (ψ n ) in C ∞ 0 (Ω) with ψ n → ψ in the norm of H 2 . Then ∇ψ n → ∇ψ in the norm of H 1 . Since the trace operators T ij : H 1 (R N ) → L 2 (Γ ij ) are continuous, and since, clearly, ∇ψ n = 0 on all hyperplanes Γ ij , it follows that or, more precisely, T ij ∇ψ = 0 in L 2 (Γ ij ). We now give an example of an antisymmetric wave function ψ ∈ H 2 (R N ) without property (33), which proves that Apart from the Gaussian, this is a Vandermonde determinant. This shows that ψ is antisymmetric. On the hyperplane Γ 12 we have which shows that ∇ψ does not vanish on Γ 12 .
The proof of Theorem 5.1 is based on the following lemmas, Lemma 5.2 being the heart of it.