Stability of the 2+2 fermionic system with point interactions

We give a lower bound on the ground state energy of a system of two fermions of one species interacting with two fermions of another species via point interactions. We show that there is a critical mass ratio m_c \approx 0.58 such that the system is stable, i.e., the energy is bounded from below, for m \in [m_c, m_c^{-1}]. So far it was not known whether this 2+2 system exhibits a stable region at all or whether the formation of four-body bound states causes an unbounded spectrum for all mass ratios, similar to the Thomas effect. Our result gives further evidence for the stability of the more general N+M system.


Introduction
Systems of particles interacting via point interactions are frequently used in physics to model short range forces. In these models the shape of the interaction potential enters only via the scattering length. Originally point interactions were introduced in the 1930s to model nuclear interactions [4,5,12,23,24], and later they were also successfully applied to other areas of physics like polarons (see [15] and references there) or cold atomic gases [25].
Given N ě 1 fermions of one type with mass 1{2 and M ě 1 fermions of another type with mass m{2 ą 0, point interaction models give a meaning to the formal expressioń for γ P R. Because of the existence of discontinuous functions in H 1 pR n q for n ě 2, this expression is ill-defined in dimensions larger than one. In the following we restrict our attention to the three-dimensional case but we note that the system also exhibits interesting behavior in two dimensions [9,10,14]. A mathematically precise version of (1.1) in three dimensions was constructed in [9,13] and we will work here with the model introduced there. We note that even though these models are mathematically well-defined it is not established whether they can be obtained as a limit of genuine Schrödinger operators with interaction potentials of shrinking support. (See, however, [1] for the case N " M " 1, and [2] for models in one dimension.) It was already known to Thomas [23] that systems with point interactions are inherently unstable for bosons, in the sense that the energy is not bounded from below, if there are at least three particles involved. It turns out that in the case that the particles are fermions the question of stability is more delicate as it depends on the mass ratio of the two species, in general.
The case N " M " 1 is completely understood as it reduces to a one particle problem [1]. In this case there exists a one-parameter family of Hamiltonians describing point interactions parameterized by the inverse scattering length, and they are bounded from below for all masses.
Beside this trivial case also the 2`1 case (i.e., N " 2 and M " 1), where the two particles of the same species are fermions, is well understood [3, 6-9, 17-20, 22]. There is a critical mass ratio m˚« 0.0735 such that the system is unstable for m ă m˚and stable otherwise. It is remarkable that this critical mass ratio does not depend on the strength of the interaction, i.e., the scattering length. Recently in [3] the spectrum of the 2`1 system was discussed in more detail. Moreover, it was shown in [7,19] that in a certain mass range other models describing point interactions can be constructed.
For larger systems of fermions even the question of stability is generally open. In [21] the stability result for the 2`1 case was recently extended to the general N`1 problem (N ě 2 and M " 1). In particular it was shown that there exists a critical mass m 1 « 0.36 such that the system's energy is bounded from below, uniformly in N , for m ě m 1 . As a consequence of the 2`1 case this N`1 system is unstable for m ă m˚, but the behavior for m P rm˚, m 1 q is unknown.
By separating particles one can obtain an upper bound on the ground state energy of the general N`M problem using the bounds for the N`1 or the 1`M problem. We note that the latter is, up to an overall factor, equivalent to the M`1 problem with m replaced by its inverse. Hence the fact that m 1 ă 1 gives hope that there exists a mass region where the general N`M system is stable for all N and M . The simplest problem of this kind is the 2`2 case. So far there are only numerical results on its stability available [11,16]. In particular, the analysis in [11] suggests that the critical mass for the 2`2 case should be equal to m˚, i.e., the one for the 2`1 case.
In this paper we give a rigorous proof of stability for the 2`2 system in a certain window of mass ratios. We find a critical mass m 2 « 0.58 such that the system is stable if m P rm 2 , m´1 2 s « r0.58, 1.73s. We note that the critical mass m 2 is not optimal and we cannot make any further statements about the mass range rm˚, m 2 s Y rm´1 2 , m˚´1s. The behavior for these masses, and in particular the question whether m 2 " m˚, still represents an open problem.

The model
For p 1 , p 2 , k 1 , k 2 P R 3 and m ą 0, let (2.1) We will work with the quadratic form F α introduced in [13] for 2`2 particles. Its form domain is given by where, for some (arbitrary) µ ą 0, G µ ξ is the function with Fourier transform and we used the notation thatp 1 " p 2 ,p 2 " p 1 and analogously for k. The space H 1 as pR 6 q denotes antisymmetric functions in H 1 pR 3 q b H 1 pR 3 q. Note that because of the requirement ϕ P H 1 pR 12 q the decomposition ψ " ϕ`G µ ξ is unique. Note also that the Hilbert space under consideration consists of functions that are antisymmetric in the first two and last two variables, i.e., under both the exchange p 1 Ø p 2 and k 1 Ø k 2 .
For α P R, the quadratic form we consider is given by and T µ pξq " We note that F α is independent of the choice of µ ą 0. The parameter α corresponds to the inverse scattering length; more precisely, α "´2π 2 {a, with a P p´8, 0q Y p0, 8s the scattering length. It was shown in [13] that T µ pξq is well-defined on H 1{2 pR 9 q. To show stability, we need to prove that it is in fact positive. If, on the contrary, there exists a µ ą 0 and a ξ P H 1{2 pR 9 q such that T µ pξq ă 0, a simple scaling argument (choosing ϕ " 0 and using the scale invariance of F 0 ) can be used to deduce that F α is unbounded from below for all α P R.
The functionals φ 0 and φ 1 also appear in a similar form in the discussion of the 2`1 problem, and φ 2 can be seen as the analogous 1`2 term. The term φ 3 has no analogue in the 2`1 or 1`2 systems. Note that none of the φ i for 1 ď i ď 3 has a sign, and we expect that cancellations occur between them that are important for stability. In our proof below, we will first bound φ 0`φ3 from below by a positive quantity, which we then use to compensate separately the negative parts of φ 1 and φ 2 . Since we shall neglect some positive terms, we cannot expect to obtain a sharp bound. In particular, whether m 2 " m˚, as suggested in [11], cannot be determined using this method.

Main result
for any ξ P H 1{2 pR 9 q and any µ ą 0.
This bound readily implies stability for F α , as the following corollary shows.
Corollary 1. For m such that Λpmq`Λp1{mq ă 1, we have for any ψ P DpF α q.
Proof. Without loss of generality we can assume that }ψ} 2 " 1. Using Theorem 1 and Hpϕq ě 0, we get In case α ě 0 we obtain F α pψq ě´µ , which shows the result as µ ą 0 was arbitrary. If α ă 0, we choose which yields the desired result.
We thus proved stability as long as Λpmq`Λp1{mq ă 1. To investigate the implication on m, let us first check what happens for a " 0 and b " 0. An explicit calculation following [6] shows that which satisfiesΛpmq`Λp1{mq ă 1 for 0.139 À m À 7.189. This range of masses is the largest possible for which our approach can show stability. While we do not know whether Λpmq "Λpmq, we shall give in Section 5 a rough upper bound on Λpmq which shows that Λpmq`Λp1{mq ă 1 for 0.58 À m À 1.73.

Proof of Theorem 1
We shall split the proof into several steps.

4.2.
Bound on φ 1 . For the term φ 1 in (2.7), we shall switch to center-of-mass and relative coordinates for the particles pp 1 , p 2 , k 1 q. With P " p 1`p2`k1 , q 1 " 1`m 2`m p 11 2`m pp 2`k1 q and q 2 " 1`m 2`m p 2´1 2`m pp 1`k1 q, as well as k " k 2 for short, we have Defining λ pq, P, kq " L λ p 1`m 2`m P´q, P 2`m`q`k , mq our aim is to obtain a lower bound on the operator on L 2 pR 3 q with integral kernel λ pq 1 , P, kq´1 {2 λ pq 2 , P, kq´1 {2 1 q 2 1`q 2 2`2 1`m q 1¨q2`m p1`mqp2`mq P 2`1 1`m k 2`m 1`m µ for suitable λ, uniformly in the fixed parameters P and k.
Let us take λ 2 " 1{2 for simplicity, in which case we have 1{ ?
To summarize, what we have shown here is that 2 pq, P, kq dP dq dk . Using (4.10), a simple change of variables shows that this is equivalent to ? 2 pP 1 , P 2 , qq dP 1 dP 2 dq . 4.3. Bound on φ 2 . In exactly the same way we proceed with φ 2 in (2.8), which we rewrite as If we now definẽ λ pq, P, pq " L λ p 1`m 1`2m P´q, p`q`m P 1`2m , mp we need a lower bound on the operator on L 2 pR 3 q with integral kernel for fixed P and p. By proceeding as in the previous subsection, one readily checks that, for λ 2 " 1{2, its best lower bound is´2π 2 Λp1{mq, with Λ defined in (3.2). In particular, we have ? 2 pP 1 , P 2 , qq dP 1 dP 2 dq . (4.20) 4.4. Combining above bounds. By combining the bounds (4.7), (4.16) and (4.20) from the previous three subsections, we obtain ? 2 pP 1 , P 2 , qq dP 1 dP 2 dq (4.21) with L 1{ ? 2 defined in (4.8). In the case Λpmq`Λp1{mq ď 1, we can further use L 1{ ? 2 pP 1 , P 2 , qq ě a mµ{p2p1`mqq for a lower bound. This completes the proof of Theorem 1.