Quadratic Forms for the Fermionic Unitary Gas Model

Vol. 69 (2012)

REPORTS ON MATHEMATICAL PHYSICS

No. 2

QUADRATIC FORMS FOR THE FERMIONIC UNITARY GAS MODEL D OMENICO F INCO Facolt`a di Ingegneria, Universit`a Telematica Internazionale Uninettuno Corso Vittorio Emanuele II, 39 - 00186 Roma, Italy (e-mails: d.fi[email protected])

and A LESSANDRO T ETA Dipartimento di Matematica Pura ed Applicata, Universit`a di L’Aquila Via Vetoio - Loc. Coppito - 67010 L’Aquila, Italy (e-mail: [email protected]) (Received June 25, 2011 — Revised November 16, 2011) We consider a quantum system in dimension three composed by a group of N identical fermions, with mass 1/2, interacting via zero-range interaction with a group of M identical fermions of a different type, with mass m/2. Exploiting a renormalization procedure, we construct the corresponding quadratic form and define the so-called Skornyakov–Ter-Martirosyan extension Hα , which is the natural candidate as a possible Hamiltonian of the system. It is shown that if the form is unbounded from below then Hα is not a self-adjoint and bounded from below operator, and this in particular suggests that the so-called Thomas effect could occur. In the special case N = 2, M = 1 we prove that this is in fact the case when a suitable condition on the parameter m is satisfied. Keywords: zero-range interactions, unitary gas, Skornyakov–Ter-Martirosyan extension.

1.

Introduction In many models in condensed matter physics and statistical mechanics, a gas of n quantum particles in R3 is described through the formal Hamiltonian n n   1 H =− xi + μ δ(xi − xj ) (1.1) 2mi i=1 i,j =1 i
where xi ∈ R3 , mi is the mass of the i-th particle, xi is the free Laplacian relative to the coordinate xi and μ ∈ R is the strength of the δ, or zero-range, interaction acting between each pair of particles of the gas. To simplify the notation we fix h¯ = 1. One reason of interest for the Hamiltonian (1.1) is that it is a simple but nontrivial modification of the free Hamiltonian and then it can be used for concrete computations [131]

132

D. FINCO and A. TETA

of relevant physical properties of the quantum gas. It is worth mentioning that in recent years these models have been widely used in the physical literature. They are studied for systems of bosons or fermions, possibly with harmonic confining potential, in the unitary limit, i.e. for infinite two-body scattering length (see the review [3] and also [5, 19, 20]). From the mathematical point of view a Hamiltonian of the type (1.1) in the appropriate Hilbert space is defined as a self-adjoint extension of the free Hamiltonian restricted to a domain of smooth functions vanishing on each hyperplane xi = xj . The most used techniques for the concrete construction of such extensions are Krein’s theory of self-adjoint extensions and limiting procedure of smooth approximating Hamiltonians (in the sense of the resolvent or the quadratic form). The problem is completely understood in the case n = 2, where one is reduced to study a fixed δ interaction in the relative coordinate and all the self-adjoint extensions can be explicitly constructed (we refer to [1] for a complete mathematical analysis of this case). A direct generalization of the same construction to the case n > 2 naturally leads to the definition of the so-called Skornyakov–Ter-Martirosyan extension Hα . Roughly speaking, such extension is a symmetric operator acting on a set of functions ψ which are smooth outside the hyperplanes xi = xj , i, j = 1, . . . , n, while on each hyperplane they exhibit the following singular behaviour ψ

fij + α fij + o(1) |xi − xj |

for |xi − xj | → 0,

(1.2)

where fij is a function defined on the hyperplane xi = xj and α is a real parameter. One can see that α −1 is proportional to the two-body scattering length and therefore the unitary limit is obtained for α = 0. As a matter of fact the operator Hα is not self-adjoint and all its self-adjoint extensions are unbounded from below due to the presence of an infinite sequence of energy levels Ek going to −∞ for k → ∞. This result was first rigorously proved for a system of (at least) three identical bosons in [16] (see also [13] for the case of three different particles) using the theory of self-adjoint extensions. This effect, known as Thomas effect, prevents Hα from being a good physical Hamiltonian. Notice that the effect is absent in dimension two [7]. It is expected that the Thomas effect could be absent if the Hilbert space of states is appropriately restricted, e.g. introducing suitable symmetry constraints on the wave function. A typical example is the antisymmetry requirement. If the wave function is antisymmetric under the exchange of the coordinates of two particles of the system then such two particles cannot “see” the zero-range interaction and therefore the whole interaction term in the Hamiltonian is less singular. In this paper we shall consider the case of a system in dimension three made of two subsystems A and B , where A consists of N identical fermions of one kind and B of M identical fermions of another kind. We assume that no interaction is present between particles of the same species while each particle of A interacts

QUADRATIC FORMS FOR THE FERMIONIC UNITARY GAS MODEL

133

with each particle of B through a zero-range potential. Without loss of generality, we fix the mass of a particle in A equal to 1/2 and the mass of a particle in B equal to m/2. The first mathematical problem is to construct the corresponding Skornyakov–TerMartirosyan extension and to show that it is self-adjoint and bounded from below or, on the contrary, it is only symmetric and, possibly, there is Thomas effect. In this generality the problem is open and one can only stress that the answer seems to be strongly dependent on the physical parameters m, N, M. Some partial results are available in the simpler case M = 1. In [15] it is proved that for M = 1, N ≤ 4 and m sufficiently large the Skornyakov–Ter-Martirosyan extension is self-adjoint and bounded from below. The case M = 1, N = 3 has been approached in [5] where, exploiting analytical and numerical arguments, it is shown that there is Thomas effect if m−1 > 13.384. For M = 1, N = 2 it is known in the physical literature (see e.g. [3] and references therein) that for m−1 > 13.607 the Thomas effect is present while for m−1 < 13.607 the Hamiltonian is expected to be bounded from below. As rigorous results in this case, we mention [14,17] for m = 1 and [18], where it is proved that for m−1 > 13.607 the Skornyakov–Ter-Martirosyan extension is not self-adjoint and any self-adjoint extension is unbounded from below. These results are not suprprising. For small m the dynamics can be described through the Born–Oppenheimer approximation, at least on a variational level. Therefore the light particle undergoes the influence of two attractive delta potentials whose centers parametrically depend on the heavy particle position. It is known, see [1], that the ground state energy of such reduced system diverges to −∞ when the distance between the two centers goes to 0. This provide a simple heuristic explanation for the physical behaviour of the three body problem. Our aim here is to propose a quadratic form approach for the study of the Hamiltonian of the system composed by the subsystems A and B . More precisely, following the line of [7], we construct a renormalized quadratic form Fα which is naturally associated to the formal Hamiltonian of the system for any N and M (Section 2). We also introduce an independent definition of the Skornyakov–Ter-Martirosyan extension Hα for the same system and we show that (u, Hα u) = Fα (u) for any u ∈ D(Hα ) (Section 3). The relation between Fα and Hα is briefly discussed exploiting the Birman–Krein extension theory of a positive symmetric operator [2, 4, 12]. It is proved that if Fα is unbounded from below then Hα is not a self-adjoint and bounded from below operator (Section 4). We apply the above general result to the simplest case M = 1, N = 2, studying in particular the restriction of Fα to the subspaces of fixed angular momentum l. For any fixed l, we find that if an explicit condition on m (see (5.12)) is satisfied then the quadratic form is unbounded from below. It is conjectured that for l even the condition is never satisfied while for l odd it generalizes the result obtained for l = 1 in [18] (Section 5).

134

D. FINCO and A. TETA

In the appendix the renormalization procedure for the derivation of the quadratic form Fα is briefly described and some results from potential theory are recalled. We finally collect here some notation that will be used in the paper. • With an abuse of notation, the scalar product and the norm of various L2 -spaces introduced throughout the paper will be all denoted by the same symbols (·, ·),  · . • L2a (R3(N +M) ) is the space of L2 -functions in R3(N +M) which are antisymmetric in the first N coordinates and in the second M coordinates. • H s (Rd ) is the standard Sobolev space of order s ∈ R in dimension d ∈ N. • Has (R3(N +M) ) = H s (R3(N +M) ) ∩ L2a (R3(N +M) ). • fˆ is the Fourier transform of f . • x N = (x1 , . . . , xN ) ∈ R3N , xi ∈ R3 , y M = (y1 , . . . , yM ) ∈ R3M , yj ∈ R3 . • pˆ i= (p1 , . . .  , pi−1 , pi+1 , . . . , pN ) ∈ R3(N −1) .   • f pˆ i , r, kˆ j , s = f p1 , . . . , pi−1 , r, pi+1 , . . . , pN , k1 , . . . , kj −1 , s, kj +1 , . . . , kM . •

 (i,j )

2.

=

M N   i=1 j =1

,

 (i,j ) =(l,h)

=

M N   ,



i=1 j =1

(i,j ) =(l,h)

i =l j =h

=

M N  

.

i,l=1 j,h=1 (i,j ) =(l,h)

The quadratic form and its wellposedness

In this section we introduce and study the quadratic form naturally associated to the Hamiltonian of a system composed by two species A and B of identical fermions described in the introduction. In the Hilbert space of the system L2a (R3(N +M) ) the formal Hamiltonian describing the dynamics is  δ(yj − xi )u(x N , y M ), (2.1) (H u)(x N , y M ) = (H0 u) (x N , y M ) − μ (i,j )

H0 = −x N −

1 y . m M

(2.2)

In order to give a precise meaning as an operator in L2a (R3(N +M) ) to the formal expression (2.1) we exploit a renormalization procedure for the corresponding quadratic form, following the line of [7]. The first step is to introduce an ultraviolet cut-off and to define a regularized Hamiltonian HR , R > 0; then we consider the corresponding quadratic form which is re-written in a suitable form; finally we renormalize the coupling constant μ and we explicitly compute the limit for R → ∞. The procedure is described with some details in the appendix. We underline that our aim is only to identify the limit quadratic form and therefore no attempt is made to give a rigorous proof of the convergence. The limit quadratic form is

135

QUADRATIC FORMS FOR THE FERMIONIC UNITARY GAS MODEL

denoted by Fα , D(Fα ), α ∈ R, and it is explicitly given by  D(Fα ) = u ∈ L2a (R3(N +M) ) | u = wλ + Gλ ξ, w λ ∈ Ha1 (R3(N+M) ),



ξ ∈ H 1/2 (R3(N +M−1) ) Fα (u) = F (u) + λ

where

λα (ξ )

(2.3) (2.4)



 λ ξ )(p , k )|2 − λ|u(p dpN dk M (h0 (p N , k M ) + λ)|(uˆ − G ˆ N , k M )|2 , (2.5) M N

λα (ξ ) = N M αξ 2 + λ0 (ξ ) + λ1 (ξ ) + λ2 (ξ ) + λ3 (ξ ) , (2.6)

F (u) = λ

and

3/2

 dq d pˆ 1 d kˆ 1 h1 (q, pˆ 1 , kˆ 1 ) + λ |ξˆ (q, pˆ 1 , kˆ 1 )|2 , (2.7)     p2 +k1 1 ˆ ˆ ˆ √ ˆ ˆ ξ , p , k , p , k ξˆ p1√+k 1 1 1 2 2 2 , (2.8) λ1 (ξ ) = (N − 1) dp N dk M h0 (pN , k M ) + λ     p1 +k2 1 ˆ ˆ ˆ √ ,p ˆ ˆ ξ , p , k , k ξˆ p1√+k 1 2 1 1 2 2 , (2.9) λ2 (ξ ) = (M − 1) dpN dk M h0 (pN , k M ) + λ     p2 +k2 1 ˆ ˆ ˆ √ ,p ˆ ˆ ξ , p , k , k ξˆ p1√+k 1 2 1 2 2 2 . (2.10) λ3 (ξ ) = −(N − 1)(M − 1) dp N dk M h0 (pN , k M ) + λ

λ0 (ξ ) = 2π 2

2m m+1

In the above definition we have denoted h0 (pN , k M ) = p2N

k 2M + , m

2 kˆ j 2 2 2 h1 (q, pˆ i , kˆ j ) = q + pˆ i + , m+1 m   p +k i j  (−1)i+j ξˆ √2 , pˆ i , kˆ j  λ ξ (p , k ) = G . M N h0 (p N , k M ) + λ (i,j )

(2.11) (2.12)

(2.13)

In the following we shall often refer to Gλ ξ as the potential while ξ will be called the charge. The condition Gλ ξ ∈ L2a (R3(N+M) ) implies that ξˆ must be antisymmetric in the variables pˆ i and in the variables kˆ j . We also notice that if A and B would consist of distinguishable particles the definition of the quadratic form would be the same with

136

D. FINCO and A. TETA

 λ ξ (p , k ) = G M N



 λ ξ (p , k ) ≡ G ij M N

(i,j )

 ξˆij (i,j )

 p +k i

√ j 2

, pˆ i , kˆ j



h0 (pN , k M ) + λ

.

(2.14)

Since we want to consider fermions, the final step of the construction is to take into account the requirement of antisymmetry by particle index exchange. This requires (see more details in Appendix) that ξˆij (q, pˆ i , kˆ j ) = (−1)i+j ξˆ11 (q, pˆ i , kˆ j ) ≡ (−1)i+j ξˆ (q, pˆ i , kˆ j ).

(2.15)

Let us show well-posedness of our definition of the quadratic form. It is easy to see that for ξ ∈ H 1/2 (R3(N +M−1) ) the potential Gλ ξ belongs to L2 (R3(N +M) ) but it does not belong to H 1 (R3(N +M) ) and therefore the decomposition u = wλ + Gλ ξ for the generic element of the form domain (2.3) is unique. It remains to show that the quadratic form on the charges (2.6) is well defined for ξ ∈ H 1/2 (R3(N +M−1) ). For the proof we shall exploit the following lemma. LEMMA 2.1. Let us consider the following integral operator in L2 (R3 ) u(x ) (Qu)(x) = dx √ √ . |x|(x 2 + x 2 ) |x |

(2.16)

Then Q is a bounded, positive operator with Q = 2π 2 .

(2.17)

Proof : Introducing spherical coordinates x = (r, z) and denoting the standard measure on S 2 by dz, we have ∞ ∞ (r r )3/2 (u, Qv) = dz dz

dr dr 2 u(r, ¯ z)v(r , z ). (2.18)

2 r + r 0 0 ˜ in L2 (R+ , r 2 dr) with integral kernel The operator Q

3/2 ˜ r ) = (r r ) (2.19) Q(r, r 2 + r 2 can be explicitly diagonalized. It is sufficient to introduce the unitary operator

D : L2 (R+ , r 2 dr) → L2 (R), and to observe that 1 (DQ˜ D−1 g)(y) = 2

(Df )(y) = e3y/2 f (ey ),

dy

g(y ) . ch(y − y )

(2.20)

(2.21)

Taking the Fourier transform, the above operator is reduced to the multiplication operator (see e.g. [8]) π  ˜ D−1 g)(k) = (D Q g(k) ˆ (2.22) 2 ch π2 k

QUADRATIC FORMS FOR THE FERMIONIC UNITARY GAS MODEL

137

˜ is positive and its norm is π . Using this fact in (2.18) we have and therefore Q 2 π  v(k, z ). (2.23) ¯ z) dz D dz Du(k, (u, Qv) = dk 2 ch π2 k From (2.23) it easily follows that Q is positive and its norm is 2π 2 .

2

In the next proposition we show that λα (ξ ) is well defined for ξ ∈ H 1/2 (R3(N +M−1) ). PROPOSITION 2.2. There exist positive constants Ci = Ci (N, M, m, λ) such that |λi (ξ )| ≤ Ci λ0 (ξ ),

i = 1, 2, 3,

(2.24)

for any ξ ∈ H 1/2 (R3(N +M−1) ). Proof : Let us first consider λ1 defined in (2.8). We introduce the change of the integration variables (p1 , p2 , k1 ) → (x, y, z) given by ⎧

1/2   m+1 ⎪ ⎪ ⎪ x= k1 + p1 − (m + 1)p2 , ⎪ 2 ⎪ m(m + 2) ⎪ ⎪ ⎪ 1/2

⎨   m+1 (2.25) k1 + p2 − (m + 1)p1 , y= 2 ⎪ m(m + 2) ⎪ ⎪ ⎪ 1/2

⎪ ⎪ 1 ⎪ ⎪ ⎩z = (k1 + p1 + p2 ) m(m + 2) with inverse given by ⎧

1/2 1/2 m m ⎪ ⎪ ⎪p1 = z− y, ⎪ ⎪ m+2 m+1 ⎪ ⎪ ⎪

1/2 1/2 ⎨ m m z− x, p2 = ⎪ m+2 m+1 ⎪ ⎪ ⎪

1/2 1/2 ⎪ ⎪ m m3 ⎪ ⎪ ⎩ k1 = (x + y) + z. m+1 m+2

(2.26)

Moreover we define η(x, pˆ 1 , kˆ 1 ) =   

  m(m + 1)2 m m m ˆ ˆξ p2 + x, p2 − x, p3 , . . . , pN , k 1 . 2(m + 2) 2(m + 1) m+2 m+1 Then we have λ1 (ξ ) = (N − 1)|J1 |



d pˆ 1 d kˆ 1

dx dy

(2.27)

η(x, ¯ pˆ 1 , kˆ 1 )η(y, pˆ 1 , kˆ 1 ) , 2 1 ˆ2 2 2 2 x +y + x · y + pˆ 1 + k 1 + λ m+1 m (2.28)

138

D. FINCO and A. TETA

where

3   3/2 √   ∂(p1 , p2 , k1 )  m + 2 m = |J1 | =  ∂(x, y, z)  m+1

(2.29)

is the Jacobian of the transformation of coordinates (2.25). Taking into account 2

2 kˆ m (2.30) x +y + x · y + pˆ 21 + 1 ≥ (x 2 + y 2 ) m+1 m m+1 we have the following estimate: √ √ m+1 |x||η(x, ¯ pˆ 1 , kˆ 1 )| |y||η(y, pˆ 1 , kˆ 1 )| λ ˆ |1 (ξ )| ≤ (N − 1)|J1 | . d pˆ 1 d k 1 dx dy √ √ m |x|(x 2 + y 2 ) |y| (2.31) Using the estimate (2.17) we obtain m+1 λ 2 ˆ |1 (ξ )| ≤ 2π (N − 1)|J1 | (2.32) d pˆ 1 d k 1 dx |x||η(x, pˆ 1 , kˆ 1 )|2 . m 2

2

Let us rewrite also λ0 (ξ ) in terms of η defined by (2.27). It is convenient to introduce a further change of coordinates (q, q2 ) → (x, p2 ) given by  ⎧  m + 1 √ ⎪ ⎪x = 2 q − (m + 1)q2 , ⎨ 2 m(m + 2) (2.33) √  ⎪ 1 ⎪ ⎩p2 = √ 2 q + q2 , m(m + 2) with inverse given by ⎧  ⎪ ⎪ ⎪ q = ⎨

 m m(m + 1)2 x+ p2 , 2(m + 1) 2(m + 2)   ⎪ ⎪ m m ⎪ ⎩q2 = − x+ p2 . m+1 m+2

The diagonal term now reads 

m λ 2 |J2 | 0 (ξ ) = 2π m+1  2 kˆ 1 m(m + 2) 2 2 2 ˆ · d pˆ 1 d k 1 dx x + (m − 1)p2 + pˆ 1 + + λ |η(x, pˆ 1 , kˆ 1 )|2 , (m + 1)2 m where

|J2 | = m

3

m+2 2(m + 1)

(2.34)

(2.35)

3/2 (2.36)

QUADRATIC FORMS FOR THE FERMIONIC UNITARY GAS MODEL

139

is the Jacobian of the transformation of coordinates (2.33). By the trivial estimate   2 ˆ k m(m + 2) 2 m(m + 2) x + (m − 1)p22 + pˆ 21 + 1 + λ ≥ |x| (2.37) 2 (m + 1) m (m + 1)2 and (2.32), (2.35) we conclude |λ1 (ξ )| ≤ C1 λ0 (ξ ),

  23/2 (m + 1)2 C1 = max 1, (N − 1) . √ m3/2 m2

Proceeding exactly in the same way we also have   23/2 (m + 1)2 λ λ C2 = max 1, (N − 1) |2 (ξ )| ≤ C2 0 (ξ ), . √ m3/2 m2

(2.38)

(2.39)

Let us consider λ3 defined in (2.10). Introducing the coordinates v=

p1 + k1 √ , 2

x=

p1 − k1 √ , 2

z=

p2 + k2 √ , 2

y=

p2 − k2 √ , 2

(2.40)

and exploiting the fact that h0 (pN , k M ) ≥ m−1 (x 2 + y 2 ), we have λ |3 (ξ )| ≤ (N − 1)(M − 1)m dp3 . . . dpN dk3 . . . dkM dvdz       ˆ   z+y z−y √ , p3 , . . . , pN , v−x √ , k3 , . . . , kM  ξ v, √2 , p3 , . . . , pN , √2 , k3 , . . . , kM  ξˆ z, v+x 2 2 . · dx dy x2 + y2 (2.41) Using the estimate (2.17) we also obtain |λ3 (ξ )| ≤ (N − 1)(M − 1) 2π 2 m dp3 . . . dpN dk3 . . . dkM dvdz  2    v−x v+x · dx x 2 + v 2 ξˆ z, √ , p3 , . . . , pN , √ , k3 , . . . , kM  2 2  = (N − 1)(M − 1) 2π 2 m dz d pˆ 1 d kˆ 1 p22 + k22 |ξˆ (z, pˆ 1 , kˆ 1 )|2

(m + 1)3/2 λ 0 (ξ ) ≡ C3 λ0 (ξ ) 3/2 1/2 2 m and the proof of (2.24) is concluded. ≤ (N − 1)(M − 1)

(2.42)

2

Since the norm induced by λ0 is clearly equivalent to the H 1/2 norm, it immediately follows from Proposition 2.2 that λα (ξ ) < +∞ if ξ ∈ H 1/2 (R3(N +M−1) ) and therefore the well-posedness of the definition of Fα is proved.

140

D. FINCO and A. TETA

3.

The Skornyakov–Ter-Martirosyan extension In this section we introduce the Skornyakov–Ter-Martirosyan extension Hα , i.e. the symmetric operator which is usually considered as a possible candidate for the description of the dynamics of our system. Then we show that the energy form associated with it coincides with the quadratic form defined in the previous sections. We define the operator Hα as follows. Let us introduce the 3(N +M −1)-dimensional hyperplanes in R3(N +M) ,   (3.1) ij = (x N , y M ) ∈ R3(N +M) | xi = yj  and the open domain ij . (3.2)

= R3(N +M) \ (i,j )

Then  D(Hα ) = u ∈ L2a (R3(N +M) ) | u = wλ + Gλ ξ, ξ ∈ H 3/2 (R3(N +M−1) ), w λ ∈ Ha2 (R3(N +M) ),     √ λ ˆ  λ | ( 2q, p ˆ i , kˆ j ) = α ξˆij + 8π 3/2 w Tij,lh ξlh (q, pˆ i , kˆ j ) , ij

(3.3)

(l,h)

(Hα + λ)u = (H0 + λ)w λ ,

(3.4)

λ acting on the surface charges ξˆlh is defined in the following where the operator Tij,lh way λ ˆ Tij,lh ξlh (q, pˆ i , kˆ j ) ⎧  

2m 3/2 ⎪ ⎨2π 2 h1 (q, pˆ i , kˆ j ) + λ ξˆij (q, pˆ i , kˆ j ) (i, j ) = (l, h), m+1 (3.5) = ⎪ √ ⎩ 3/2  λ ˆ (i, j ) = (l, h). −8π G ξlh | ij ( 2q, pˆ i , k j )

It is useful to give explicit expressions for the nondiagonal terms in (3.5). We distinguish the three possible cases: l = i and h = j , l = i and h = j , l = i and h = j. (1) l = i and h = j : Gλ ξlh | ij (xi , xˆ i , yˆ j )     ˆlh pl√+kh , pˆ l , kˆ h ξ ˆ 1 2 i pˆ ·xˆ +k ·yˆ d pˆ i d kˆ j e i i j j dpi dkj ei(pi +kj )xi = 3 (N+M) h (p 0 N , kM ) + λ (2π ) 2   √ (−1)l+h i 2q·xi +pˆ i ·xˆ i +kˆ j ·yˆ j ˆ ˆ = d k e dq d p j i 3   (2π ) 2 (N+M) ˆ h | q−s , k ξˆ pl√+k2 h , pˆ l |p = q+s kj = √ i √2 2 · ds , (3.6) ˆ h˜ 0 (q, s, pˆ i , k j ) + λ

141

QUADRATIC FORMS FOR THE FERMIONIC UNITARY GAS MODEL

where 1 2 m+1 2 m+1 2 m−1 h˜ 0 (q, s, pˆ i , kˆ j ) = q + s + q · s + pˆ 2i + kˆ j . 2m 2m m m Then the Fourier transform reads   pl +kh ˆ ˆ √ ˆ q+s q−s , p | , k | ξ h k =√ l pi = √ √ 2 j (−1)l+h 2 2 λ ξ | ( 2q, p ˆ i , kˆ j ) = G ds . lh ij 3/2 8π h˜ 0 (q, s, pˆ i , kˆ j ) + λ A similar computation can be done for the other two cases. (2) l = i and h = j ;   kh q+s ˆ ˆ √ ˆ q−s + , p , k | ξ h i kj = √ 2 √ 2 (−1)i+h 2 λ ξ | ( 2q, p ˆ i , kˆ j ) = ds . G ih ij ˜h0 (q, s, pˆ i , kˆ j ) + λ 8π 3/2 (3) l = i and h = j ; √ (−1) λ ξ | ( 2q, p ˆ i , kˆ j ) = G lj ij 8π 3/2

l+j

ξˆ

ds



p √l 2

+

q−s , pˆ l |p = q+s , kˆ j 2 i √ 2

h˜ 0 (q, s, pˆ i , kˆ j ) + λ

(3.7)

(3.8)

(3.9)

 .

(3.10)

The last equality in (3.3) should be considered as the boundary condition satisfied by u on ij and it connects the regular and the singular part of an element of D(Hα ). It is easy to verify that the operator Hα ,D(Hα ) is independent of the choice of λ > 0 and it is symmetric. In the next proposition we show that our definition of Hα ,D(Hα ) coincides with the standard definition usually found in the literature, except for an irrelevant modification of the coupling constant α. PROPOSITION 3.1. Let u ∈ D(Hα ). Then Hα u| = H0 u| , lim |xi − yj |u(x N , y M ) = fij (xi , xˆ i , yˆ j ), |xi −yj |→0

lim

|xi −yj |→0

u(x N , y M ) −

fij (xi , xˆ i , yˆ j ) |xi − yj |

where

(3.11) (3.12)

 = α0 fij (xi , xˆ i , yˆ j ),

√ 2 πm √ ξ( 2xi , xˆ i , yˆ j ), m+1

(3.13)

(3.14) fij (xi , xˆ i , yˆ j ) = (−1) √ 2(m + 1) α. (3.15) α0 = 8π 2 m Proof : Taking into account (3.4) and the fact that (H0 + λ)Gλ ξij | = 0 (see (6.23)), we have i+j

142

D. FINCO and A. TETA

Hα u| = (Hα + λ)u| − λu| = (H0 + λ)w λ | − λu|    = (H0 + λ) u − Gλ ξij | − λu| = H0 u| .

(3.16)

(i,j )

Let us characterize the singularity of an element of (3.3) at the hyperplane ij . Exploiting (6.24), for |xi − yj | → 0 we have  u(x N , y M ) = w λ (x N , y M ) + Gλ ξlh (x N , y M ) + Gλ ξij (x N , y M ) (l,h) =(i,j )

√ √ 1 2 πm G ξlh (x N , y M ) + ξij ( 2xi , xˆ i , yˆ j ) = w (x N , y M ) + |xi − yj | m + 1 (l,h) =(i,j ) 

λ

2π 2 (2m)3/2

−

3

·

λ

(2π ) 2 (N+M) (m + 1)3/2 i dqd pˆ i d kˆ j e

√

2xi q+xˆ i ·pˆ i +yˆ j ·kˆ j



h1 (q, pˆ i , kˆ j ) + λ ξˆij (q, pˆ i , kˆ j ) + o(1). (3.17)

We notice that  lim |xi − yj | w λ (x N , y M ) + |xi −yj |→0



 Gλ ξlh (x N , y M ) = 0.

Therefore from (3.17) we obtain (3.12). Moreover 

fij (xi , xˆ i , yˆ j ) u(x N , y M ) − lim |xi −yj |→0 |xi − yj |  = = wλ (x N , y M )| ij + − ·

(3.18)

(l,h) =(i,j )

2π

2

3

(2π ) 2 (N+M)

u(x N , y M ) −

|xi − yj |(m + 1)

Gλ ξlh (x N , y M )| ij

(l,h) =(i,j )

2m m+1

i dq d pˆ i d kˆ j e

≡ f (xi , xˆ i , yˆ j ).



lim

|xi −yj |→0

 √ √ 2 π m ξij ( 2xi , xˆ i , yˆ j )

3/2

 √ 2xi q+xˆ i ·pˆ i +yˆ j ·kˆ j

h1 (q, pˆ i , kˆ j ) + λ ξˆij (q, pˆ i , kˆ j ) (3.19)

The computation of f is more easily done in the Fourier space. Exploiting (3.5) we have √ f ( 2q, pˆ i , kˆ j )  √ √ 1 λ ˆ λ ξ | ( 2q, p  ˆ i , kˆ j ) + ˆ i , kˆ j ) − G (Tij,ij = w| ξij )(q, pˆ i , kˆ j ) ij ( 2q, p lh ij 3/2 8π (l,h) =(i,j )

QUADRATIC FORMS FOR THE FERMIONIC UNITARY GAS MODEL

√  ˆ i , kˆ j ) − = w| ij ( 2q, p

143

 1  λ ˆlh (q, pˆ i , kˆ j ) T ξ ij,lh 8π 3/2 (l,h)

α (−1)i+j ξˆ (q, pˆ i , kˆ j ), (3.20) 8π 3/2 where, in the last line, we have used the boundary condition in (3.3). Taking the inverse Fourier transform and using (3.14), (3.15), we find =

f (xi , xˆ i , yˆ j ) = α0 fij (xi , xˆ i , yˆ j )

(3.21)

2

concluding the proof of the proposition.

The next step is to verify that the mean value of the operator Hα coincides with our quadratic form Fα restricted to D(Hα ). PROPOSITION 3.2. If u ∈ D(Hα ) then (u, Hα u) = Fα (u). Proof : Let us introduce the tubular neighbourhood ijε , for ε > 0, of the hyperplane ij ,   ijε = (x N , y M ) ∈ R3(N +M) | |xi − yj | ≤ ε (3.22) and the open domain

ε = R3(N +M) \



ijε .

(3.23)

(i,j )

Taking into account (3.11), for any u ∈ D(Hα ) we can write dx N dy M u¯ H0 u. (u, Hα u) = lim ε→0 ε

(3.24)

The r.h.s. of (3.24) can be computed using Definition (3.3) and Eq. (6.23) proved in Proposition 6.1. In fact we have (u, Hα u)       λ¯ λ λ λ dx N dy M w + G ξij (H0 + λ) w + G ξij − λ dx N dy M |u|2 = lim ε→0 zε

(i,j )

=

dx N dy M

w λ (H

= F λ (u) + 8π 3/2 = F λ (u) +

 (i,j )

0 + λ)w



λ

(i,j )

−λ

dx N dy M |u| + 2



dx N dy M Gλ ξ¯ij (H0 + λ)wλ

(i,j )

√  λ | ( 2q, p ˆ i , kˆ j ) dq d pˆ i d kˆ j ξ¯ˆij (q, pˆ i , kˆ j )w ij

(i,j )

  λ ˆ Tij,lh ξlh (q, pˆ i , kˆ j ) dq d pˆ i d kˆ j ξ¯ˆij (q, pˆ i , kˆ j ) α ξˆij +

(3.25)

(l,h)

where in the last line we have used the boundary condition satisfied by u on ij (see (3.3)). Now we closely look at the last term appearing in r.h.s. of (3.25) and

144

D. FINCO and A. TETA

show that they reconstruct λα (ξ ). First we have  α dq d pˆ i d kˆ j ξ¯ˆij (q, pˆ i , kˆ j )ξij (q, pˆ i , kˆ j ) = αN Mξ 2 .

(3.26)

(i,j )

Using (3.5), the diagonal terms can be written as  λ ˆ ξij (q, pˆ i , kˆ j ) dq d pˆ i d kˆ j ξ¯ˆij (q, pˆ i , kˆ j )Tij,ij (i,j )

= 2π

2

2m m+1

3/2 

dq d pˆ i d kˆ j

 h1 (q, pˆ i , kˆ j ) + λ |ξ(q, pˆ i , kˆ j )|2 = N Mλ0 (ξ ).

(i,j )

(3.27) λξ | Concerning the nondiagonal terms, we use the explicit expression of G lh ij and we find  λ ˆ ξlh (q, pˆ i , kˆ j ) = N M(λ1 (ξ ) + λ2 (ξ ) + λ3 (ξ )). dq d pˆ i d kˆ j ξ¯ˆij (q, pˆ i , kˆ j )Tij,lh (i,j ) =(l,h)

The proof of the proposition is concluded. 4.

(3.28) 2

On the relation between the form Fα and the operator Hα

For any physical application the crucial point is to show that the symmetric operator Hα is a good Hamiltonian for our system, i.e. to prove that it is self-adjoint and bounded from below (stability condition). As we already remarked, in general the problem is open and we expect that the answer can be positive only under appropriate conditions on the physical parameters of the system N, M, m. Exploiting the representation theorem of self-adjoint operators (see e.g. [11]), the result could be obtained proving that the associated quadratic form Fα is closed and bounded from below. On the other hand one can also prove a “negative” result. More precisely, we show that if the form Fα is unbounded from below then Hα is not a self-adjoint and bounded from below operator in L2a (R3(N +M) ). First we recall some facts from Birman–Krein theory of positive extensions of a given symmetric and positive operator on a Hilbert space [4, 9, 12]. Proofs can also be found in [2] where a detailed discussion of the original Russian literature is given. Let S0 be a symmetric and positive operator on a Hilbert space H and let N be the kernel of S0∗ . We shall denote the Friedrichs extension of S0 by SF . Notice that since S0 is positive then SF is positive and has a bounded inverse. The main result of the Birman–Krein theory is that the positive self-adjoint extensions of S0 are in a one-to-one correspondence with positive operators on N .

QUADRATIC FORMS FOR THE FERMIONIC UNITARY GAS MODEL

145

More precisely, if S is a positive self-adjoint extension of S0 then there exists a positive operator B : D(B) ⊆ N → N such that D(S) = {u ∈ H | u = φ + SF−1 (Bf + g) + f, φ ∈ D(S0 ), f ∈ D(B), g ∈ N ∩ D(B)⊥ } (4.1) and (4.2) Su = S0∗ |D(S) u = S0 φ + Bf + g. Notice that the closure of D(B) may be a proper subspace of N . Let us specialize this general result to our concrete case. We have H = 2 La (R3(N +M) ) and S0 = H˜ 0 + λ, λ > 0, where H˜ 0 is the free Hamiltonian restricted to  D(H˜ 0 ) = u ∈ L2a (R3(N +M) ) | u ∈ H 2 (R3(N +M) ),  u| ij = 0, i = 1, . . . , N, j = 1, . . . , M . (4.3) Moreover (see e.g. [9], [15])   N = u ∈ L2a (R3(N +M) ) | u = Gλ μ, μ ∈ H −1/2 (R3(N +M−1) ) .

(4.4)

The Friedrichs extension of H˜ 0 + λ is HF + λ, where HF is the free Laplacian with domain   D(HF ) = u ∈ L2a (R3(N +M) ) | u ∈ H 2 (R3(N +M) ) . (4.5) We shall denote G λ = (HF + λ)−1 . Notice that G λ : L2 (R3(N +M) ) → H 2 (R3(N +M) ) while Gλ : H −1/2 (R3(N +M−1) ) → L2 (R3(N +M) ) even if they act in the same way as multiplication operators in Fourier space. By the Birman–Krein theory we have that any self-adjoint positive extension of H˜ 0 + λ is given by HB + λ, where  D(HB ) = u ∈ L2a (R3(N +M) ) | u = ϕ λ + G λ (BGλ μ + Gλ ν) + Gλ μ, ϕ λ ∈ D(H˜ 0 ),  μ, ν ∈ H −1/2 (R3(N +M−1) ), Gλ μ ∈ D(B), Gλ ν ∈ N ∩ D(B)⊥ , (4.6) (HB + λ)u = (H0 + λ)ϕ λ + BGλ μ + Gλ ν,

(4.7)

where B : D(B) ⊆ N → N is a positive operator. Exploiting this fact we can prove the following result. PROPOSITION 4.1. If the form Fα is unbounded from below then the Skornyakov– Ter-Martirosyan extension Hα , defined by (3.3) and (3.4), is not a self-adjoint and bounded from below operator in L2a (R3(N +M) ). Proof : We shall prove the proposition by contradiction. Let us assume that Hα +λ is a positive, self-adjoint extension of H˜ 0 + λ for a sufficiently large λ > 0. Then there exists a positive operator B in N such that D(Hα ) = D(HB ). In particular, for any u = wλ + Gλ ξ ∈ D(Hα ) there exist ϕ λ ∈ D(H˜ 0 ) and μ, ν ∈ H −1/2 (R3(N +M−1) ), with Gλ μ ∈ D(B) and Gλ ν ∈ N ∩ D(B)⊥ , such that the following identity holds u ≡ w λ + Gλ ξ = ϕ λ + G λ (BGλ μ + Gλ ν) + Gλ μ.

(4.8)

146

D. FINCO and A. TETA

From (4.8) we conclude that Gλ ξ = Gλ μ and therefore, by (6.25), it follows ξ = μ and μ ∈ H 3/2 (R3(N +M−1) ). Moreover, from Propositions 3.1 and 6.1 we obtain √

 √ 2 πm u(x N , y M ) − lim ξij ( 2xi , xˆ i , yˆ j ) |xi −yj |→0 (m + 1)|xi − yj | √ α ξij ( 2xi , xˆ i , yˆ j ) = 3/2 (2π )    1 3/2  λ λ λ −1 λ ˆ ˆ ˆ = 8π G (BG ξ + G ν) (x , y )| − F T , x , y ) ξ (x N i i M ij j , ij,lh lh (2π )3/2 (l,h) (4.9) where F −1 denotes the inverse Fourier transform. Formula (4.9) holds in particular for ξ = 0 and this means that ν = 0. Then in the Fourier space formula (4.9) reads       λ λξ | ˆ i , kˆ j ). Tij,lh (4.10) ξˆlh (q, pˆ i , kˆ j ) = 8π 3/2 G λ BG α ξˆij + ij (q, p (l,h)

From (4.10) we obtain      λ  λξ | Tij,lh ξˆlh = 8π 3/2 λα (ξ ) ≡ ξˆij , α ξˆij + ξˆij , G λ BG ij . (i,j )

(l,h)

(4.11)

(i,j )

By a direct computation one sees that the r.h.s. of (4.11) equals 8π 3/2 (Gλ ξ, B Gλ ξ ) and therefore we conclude λα (ξ ) = 8π 3/2 (Gλ ξ, B Gλ ξ ) ≥ 0.

(4.12)

On the other hand if (4.12) holds for any λ > 0 sufficiently large then the form 2 Fα is bounded from below and we obtain a contradiction. 5.

The case M = 1 and N = 2

From now on we shall limit ourselves to the special case M = 1 and N = 2. The quadratic form λα (see (2.6)) reads λα (ξ ) = 2αξ 2 + 2λ0 (ξ ) + 2λ1 (ξ ), 

 2m 3/2 λ 2 0 (ξ ) = 2π dq dp h1 (q, p) + λ |ξˆ (q, p)|2 , m+1

(5.1)

h1 (q, p) =

2 q 2 + p2 , (5.2) m+1

λ1 (ξ ) =

ξˆ dp1 dp2 dk



p1 +k √ , p2 2

   2 +k ξˆ p√ , p1 2

h0 (p1 , p2 , k) + λ

,

h0 (p1 , p2 , k) = p12 + p22 +

1 2 k . m (5.3)

QUADRATIC FORMS FOR THE FERMIONIC UNITARY GAS MODEL

147

The regular part F of the quadratic form is written as in (2.5) where the potential Gλ ξ is now given by  1 +k   2 +k  ˆ p√ ξˆ p√ , p , p − ξ 2 1 2 2  λ ξ (p , p , k) = . (5.4) G 1 2 h0 (p1 , p2 , k) + λ Notice that no symmetry condition is required on ξ to ensure that Gλ ξ ∈ L2a (R9 ). With the above notation we have Fα (u) = F λ (u) + λα (ξ ).

(5.5)

In the next proposition we give a detailed analysis of the subspaces where the form (5.5) is unbounded from below. It is well known, see [18], that Hα can be unbounded from below in the subspace corresponding to p-wave for the charges. Here we shall show that, depending on the value of m, this is not the only case. For this pourpose it is more convenient to write λα using the η function introduced in (2.27), and use the coordinates (2.25) also on the total Hilbert space L2a (R9 ). With an abuse of notation, we write 

m + 2 3/2 λ 3 η2 + 2λ0 (η) + 2λ1 (η), (5.6) α (η) = 2αm 2(m + 1)  2 9/2 3/2 2π m (m + 2) m(m + 2) 2 λ0 (η) = x + mp2 + λ |η(x, p)|2 , (5.7) dx dp 3 (m + 1) (m + 1)2 m9/2 (m + 2)3/2 η(x, p)η(y, p) λ1 (η) = dp dx dy , (5.8) 2 2 2 (m + 1)3 x + y + m+1 x · y + p2 + λ Gλ η(x, y, p) =

x2

η(x, p) − η(y, p) . 2 + y 2 + m+1 x · y + p2 + λ

(5.9)

First we notice that in these coordinates the quadratic form Fα has some simple decomposition properties. In fact, there is an orthogonal decomposition L2a (R9 ) =

∞ !

Hl

(5.10)

l=0

such that if u ∈ Hl and u ∈ Hl then Fα (u, u ) = 0

if l = l .

(5.11)

The charge space is L2 (R6 ) = L2 (R3 ) ⊗ L2 (R3 ). We shall decompose the first L2 (R3 ), corresponding to the x variable, into subspaces of fixed angular momentum "l . With this choice we have H L2 (R6 ) =

∞ ! l=0

"l ⊗ L2 (R3 ). H

148

D. FINCO and A. TETA

"l , that The subspaces Hl is defined as the subspace generated by the charges in H is   "l ⊗ L2 (R3 ) . Hl = Gλ η s.t. η ∈ H If we take u = wλ + Gλ η ∈ Hl then, due to the uniqueness of the decomposition, we have that both w λ and Gλ η belong to Hl . Morover it is straightforward to verify that the Laplacian is invariant by rotations in these coordinates and that (5.11) holds true. PROPOSITION 5.1. Let m and l be such that 1 arc cos (y/(m + 1)) (m + 1) dy Pl (y) < 0, 1+ √ sin arc cos (y/(m + 1)) π m(m + 2) −1

(5.12)

where Pl is the Legendre polynomial of order l. Then inf Fα (u) = −∞.

u∈Hl

(5.13)

Proof : We fix λ > 0 and consider a trial function of the form u = Gλ η,

"l ⊗ L2 (R3 ). η∈H

(5.14)

Therefore we have Fα (u) = λα (η) − λu2 . Let us take η(x, p) = f (x)g(p), with "l and g ∈ S (R3 ), and define the sequence un by the following rescaling of η, f ∈H 1 x  (5.15) g(p), un = G λ η n . ηn (x, p) = f n n Exploiting the estimate (6.25), one can easily check that the sequence un satisfies the condition infn un  > 0. Using (5.6) and (5.8) we have  2 9/2 (m + 2)3/2 m(m + 2) 2 m p 2 + λ λ 2 2π m 0 (ηn ) = n x + |f (x)|2 |g(p)|2 , dx d p ˆ (m + 1)3 (m + 1)2 n2 (5.16) 9/2 3/2 2 m (m + 2) f (x)f (y)|g(p)| λ1 (ηn ) = n2 . dp dx dy 2 3 2 (m + 1) x + y + 2x/(m + 1)x · y + (p2 + λ)/n2 (5.17) Let us compute the leading terms of λ0 (ηn ) and λ1 (ηn ) for n → ∞. Using the inequality  √ m(m + 2) 2 mp 2 + λ m(m + 2) 1 2 x + − mp + λ |x| ≤ (m + 1)2 n2 m+1 n it follows λ0 (ηn )

=n

2 2π

m5 (m + 2)2 (m + 1)4

2

dx |x| |f (x)|2 + O(n).

(5.18)

QUADRATIC FORMS FOR THE FERMIONIC UNITARY GAS MODEL

149

Moreover we prove that λ1 (ηn ) = n2

m9/2 (m + 2)3/2 (m + 1)3

dx dy

f (x)f (y) + O(n). x 2 + y 2 + 2x · y/(m + 1)

(5.19)

In fact we have λ1 (ηn )

−n

2m

= n2

m

(m + 2)3/2 (m + 1)3

9/2

(m + 2) (m + 1)3

9/2

3/2

dx dy

x2

+

y2

f (x)f (y) + 2x · y/(m + 1)

dx dy f (x)f (y) Tn (x, y; p2 + λ),

dp|g(p)|2

(5.20)

where we have defined Tn (x, y; p2 + λ) =−

1 p2 + λ    2 2 2 2 n x + y + 2x · y/(m + 1) x + y 2 + 2x · y/(m + 1) + (p2 + λ)/n2 (5.21)

For any n, p, λ, the integral kernel (5.21) defines a Hilbert–Schmidt operator and therefore its norm can be estimated as 2 2 Tn (p + λ)  dx dy |Tn (x, y; p2 + λ)|2 (p2 + λ)2 (m + 1)4  n 4 m4

dx dy (x 2

+

y 2 )2

1



x2

+

y2

2

+ (m + 1)(p2 + λ)/n2 m

c (p2 + λ)(m + 1)3 , (5.22) n2 m3 where c is a numerical constant. Using this estimate in (5.20), we obtain (5.19). Moreover we notice that (5.23) αηn 2 = O(n). =

Therefore, from (5.18), (5.19), (5.23), we have Fα (un ) = λα (ηn ) − λun 2 = n2 where



4π 2 m5 (m + 2)2 " (f ) + O(n), (m + 1)4

1 m+1 √ 2 2π m(m + 2)



(5.24)

f (x)f (y) . + + 2x · y/(m + 1) (5.25) " ) < 0. We introduce polar Thus the problem is reduced to find f such that (f coordinates (ρ, θ, ϕ) in R3 and denote the standard measure on S 2 by dz. We " )= (f

dx|x||f (x)|2 +

dx dy

x2

y2

150

D. FINCO and A. TETA

further specialize our choice of the trial function by f (ρ, θ, ϕ) =

1 a(ρ) Pl (cos θ ), Pl 2

(5.26)

where the radial part a will be specified later. Then we have " ) = 2π (f



+∞ 0

+∞ 1 m+1 1 dρ ρ |a(ρ)| + dρ1 dρ2 ρ12 ρ22 a(ρ1 )a(ρ2 ) √ 2π 2 m(m + 2) Pl 22 0 Pl (cos θ1 ) Pl (cos θ2 ) dz1 dz2 2 (5.27) × 2 ρ1 + ρ22 + m+1 ρ1 ρ2 cos θ12 S2 3

2

where θ12 is the angle between x and y. With the change of variable ex = ρ we arrive at 1 m+1 1 " ) = 2π dx |e2x a(ex )|2 + dx1 dx2 e2x1 a(ex1 )e2x2 a(ex2 ) (f √ 2 2 4π P  m(m + 2) R l 2 R Pl (cos θ1 ) Pl (cos θ2 ) dz1 dz2 , (5.28) × 1 ch(x1 − x2 ) + m+1 cos θ12 S2 Both terms appearing in (5.28) can be diagonalized by Fourier transform, see e.g. [8], and we get 1 " ) = dk |d(k)|2 Sl (k), (f dx e−ixk e2x a(ex ), d(k) = √ (5.29) 2π where Sl (k) is the smooth and even function given by 1 sh γ k m+1 1 dz1 dz2 Pl (cos θ1 ) Pl (cos θ2 ) Sl (k) = 2π + (5.30) √ 2 2π m(m + 2) Pl 2 S 2 sin γ sh π k   θ12 . Now we study the sign of Sl (0) and, in particular, we and γ = arc cos cos m+1 show that if (5.12) holds then S(0) < 0. We have 1 γ m+1 1 Sl (0) = 2π + dz1 dz2 Pl (cos θ1 ) Pl (cos θ2 ) . (5.31) √ 2π 2 m(m + 2) Pl 22 S 2 sin γ Let us briefly focus on the angular integral appearing in (5.31). Introducing ϕ12 = ϕ1 − ϕ2 and performing the change of variables (ϕ1 , θ1 , ϕ2 , θ2 ) → (ϕ1 , θ1 , ϕ12 , θ12 ), the integral takes the form γ (θ12 ) dz12 dz1 Pl (cos θ1 )Pl (cos θ1 cos θ12 + sin θ1 sin θ12 cos(ϕ1 − ϕ12 )). sin γ (θ12 ) S 2 S2 (5.32)

QUADRATIC FORMS FOR THE FERMIONIC UNITARY GAS MODEL

151

Now we exploit the additional properties of Legendre polynomials, see [10], Pl (cos θ1 cos θ12 + sin θ1 sin θ12 cos(ϕ12 − ϕ1 )) ∞  (l − k + 1) k = Pl (cos θ1 )Pl (cos θ12 ) + 2 Pl (cos θ1 )Plk (cos θ12 ) cos(k(ϕ12 − ϕ1 )). (l + k + 1) k=1 (5.33) The integral over ϕ12 kills all the terms in the series in (5.33) and we obtain 1 γ arc cos (y/(m + 1)) 2 2 dz1 dz2 Pl (cos θ1 ) Pl (cos θ2 ) dy Pl (y) = 4π Pl 2 . 2 sin γ sin arc cos (y/(m + 1)) S −1 (5.34) Subsituting in (5.31), we arrive at # $ 1 m+1 arc cos (y/(m + 1)) dy Pl (y) Sl (0) = 2π 1 + √ , (5.35) sin arc cos (y/(m + 1)) π m(m + 2) −1 and Sl (0) < 0 by (5.12). Then it is sufficient to fix the radial function a such that d(k) is, roughly speaking, supported around k = 0. We choose √ β 1 , β > 0. (5.36) a(ρ) = c 2 β ρ ch 2 (ρ + ρ −β ) A straightforward calculation gives

 c k 1 d(k) = √ h , h(x) = . (5.37) β ch(ch x) β We fix c such that d = h = 1. For β sufficiently small, possibly depending " ) < 0 and the proof is complete. 2 on l, (f We conclude this section with a discussion of more detailed discussion condition (5.12) for l = 0, 1, 2, 3. We shall make use of the identities arc cos z = π/2−arc sin z, cos arc sin z = sin arc cos z and of the explicit expression of the first Legendre polynomials. For l = 0, a straightforward computation shows that the l.h.s. of (5.12) reads 1 1 m+1 dy 1+ √ cos arc sin (y/(m + 1)) m(m + 2) 0 and this means that condition (5.12) is never satisfied for any m > 0. For l = 1, condition (5.12) reads arc sin 1 m+1 2(m + 1)3 x sin x dx < 0 1− √ π m(m + 2) 0 and therefore it reduces to the unboundedness conditions appearead in [18] which is satisfied for m−1 > 13.607. For l = 2, condition (5.12) cannot be satisfied. It is convenient to decompose the integral into a positive and a negative part according to the sign of the Legendre

152

D. FINCO and A. TETA

polynomial and estimate each term separately taking into account the monotonicity of the other the other part of the integrand. In facts, we have 1 m+1 1 1+ √ dy (3y 2 − 1) cos arc sin (y/(m + 1)) π m(m + 2) 0 √1 1 m+1 3 dy (3y 2 − 1) =1+ √ cos arc sin (y/(m + 1)) π m(m + 2) 0 1 1 m+1 dy (3y 2 − 1) + √ 1 cos arc sin (y/(m + 1)) π m(m + 2) √ 3

m+1 ≥1+ √ π m(m + 2) m+1 + √ π m(m + 2)





√1 3

cos arc sin

0 1 √1 3

1 

dy (3y 2 − 1) 1 

dy (3y 2 − 1) cos arc sin

√1 (m 3

√1 (m 3

+ 1)



 = 1.

+ 1)

For l = 3, condition (5.12) is satisfied at least for sufficiently small m. This can be easily seen estimating the integral as in the previous case. 1 arc sin (y/(m + 1)) 2(m + 1) dy (5y 3 − 3y) 1− √ cos arc sin (y/(m + 1)) π m(m + 2) 0   √ 3 5 arc sin 3/5/(m + 1) 2(m + 1) 3  1 √  dy (5y − 3y) ≤1− √ π m(m + 2) 0 cos arc sin m+1 3/5  √ 1 arc sin 3/5/(m + 1) 2(m + 1) 3  1 √  +− √  dy (5y − 3y) 3 π m(m + 2) cos arc sin 3/5 m+1 5  √ arc sin 3/5/(m + 1) m+1  1 √ . =1− √ 2π m(m + 2) cos arc sin m+1 3/5 A numerical analysis of condition (5.12) shows that for l = 3 it is satisfied for m−1 > 10.659. The structure exhibited by these examples can be extended to the general case. It can be proved, see [6], that condition (5.12) is never satisfied for even l and it is satisfied for m less than a critical value m∗ (l) for odd l. 6.

Appendix: Renormalization procedure and properties of the potential In order to obtain a well-defined operator corresponding to the formal Hamiltonian (2.1), the first step is to introduce a regularization and this is more conveniently done in the Fourier space. Using the representation

153

QUADRATIC FORMS FOR THE FERMIONIC UNITARY GAS MODEL

δ(yj − xi ) =

1 (2π )3



dw eiw(yj −xi ) ,

(6.1)

a direct computation yields

 μ (Hˆ u)(p ˆ N , k M ) = h0 (p N , k M )u(p ˆ N , kM ) − dz u( ˆ pˆ i , pi + z, kˆ j , kj − z) (2π )3 (i,j ) = h0 (p N , k M )u(p ˆ N , kM )

 3/2  pi + kj z ˆ kj − z z 2 μ dz uˆ pˆ i , + √ , kj , −√ . − (2π )3 (i,j ) 2 2 2 2

(6.2)

A natural regularization of (6.2) is the following Hamiltonian depending on the cut-off R > 0 ˆ N , k M ) = h0 (p N , k M )u(p ˆ N , kM ) (Hˆ R u)(p

   pi − kj  pi + kj z ˆ kj − z z dz 1R (z)uˆ pˆ i , 1R − μR + √ , kj , −√ , √ 2 2 2 2 2 (i,j )

(6.3)

where μR is a new coupling constant explicitly dependent on the cut-off and 1R is the characteristic function of the ball in R3 of radius R and center in the origin. It is obviously true that (6.3) defines a lower bounded self-adjoint operator for any R > 0 with the same domain of the free Hamiltonian. The next step is to compute the quadratic form associated to (6.3) and then to take the limit R → ∞ for a suitably chosen μR . The identification of the limit is easier if one introduces the following “volume charges” for i = 1, . . . , N , j = 1, . . . , M, ρˆijR (p N , k M )

= μR 1 R

pi − kj √ 2



 pi + kj z ˆ kj − z z + √ , kj , −√ , dz 1R (z)uˆ pˆ i , 2 2 2 2

(6.4)

and the corresponding “potentials” produced by ρˆijR ,  λ ρ R (p , k ) = G M N



 λ ρ R (p , k ) = G M N ij

(i,j )



ρˆijR (p N , k M )

(i,j )

h0 (p N , k M ) + λ

,

(6.5)

where λ > 0. Hence a direct computation yields  2 ˆ ˆ = dp N dk M h0 |u| ˆ − dpN dk M uˆ ρˆijR (u, ˆ HR u) =

(i,j )

  λ ρ R |2 − λ|u| λ ρ R |2 dpN dk M (h0 + λ)|uˆ − G ˆ 2 − dpN dk M (h0 + λ)|G   λ R +2 Re dpN dk M uˆ (h0 + λ)G ρ − dpN dk M uˆ ρˆijR





(i,j )

154

D. FINCO and A. TETA



 λ ρ R |2 − λ|u| dp N dk M (h0 + λ)|uˆ − G ˆ2

= −



dpN k M

(i,j ) =(l,h)

+



R ρˆijR ρˆlh

h0 + λ

−

 (i,j )

dpN k M

|ρˆijR |2 h0 + λ

dpN dk M uˆ ρˆijR ,

(6.6)

(i,j )

where we have used (6.3), (6.4), (6.5) and the fact that Im dpN dk M uˆ ρˆijR = 0.

(6.7)

Let us define the following “surface charges” for i = 1, . . . , N , j = 1, . . . , M,

 q +z q −z ˆξijR (q, pˆ i , kˆ j ) = μR dz 1R (z)uˆ pˆ i , √ , kˆ j , √ . (6.8) 2 2 We notice that 



pi + kj pi + kj z ˆ pi + kj z R ˆ + √ , kj , −√ , ξˆij √ , pˆ i , k j = μR dz 1R (z)uˆ pˆ i , 2 2 2 2 2 (6.9) and therefore

  pi − kj R pi + kj R ˆ ˆ ρˆij (pN , k M ) = 1R (6.10) ξij √ √ , pˆ i , k j . 2 2 Let us rewrite the last two integrals in (6.6). We have dpN dk M uˆ ρˆijR = μ−1 dq d pˆ i d kˆ j |ξˆijR (q, pˆ i , kˆ j )|2 R and dpN dk M

|ρˆijR |2 h0 + λ

=

dp N dk M 1R

=

(6.11)

  √  |ξˆ R (pi + kj )/ 2, pˆ , kˆ j |2 i ij pi − kj √ h0 + λ 2

dq d pˆ i d kˆ j |ξˆijR (q, pˆ i , kˆ j )|2 IR (q, pˆ i , kˆ j )

(6.12)

√ √ where we have introduced the integration variables q = (pi +kj )/ 2, z = (pi −kj )/ 2 and we have defined 1R (z) IR (q, pˆ i , kˆ j ) = dz , (m + 1)z2 /2m + m−1 q ·z+γ m (6.13) 1 ˆ2 m+1 2 2 q + pˆ i + k j + λ. γ = 2m m

QUADRATIC FORMS FOR THE FERMIONIC UNITARY GAS MODEL

155

Using (6.11), (6.12) in (6.6), we have # $ 2 2  λ R ˆ ˆ ˆ = dpN dk M (h0 + λ)|uˆ − G ρ | − λ|u| (u, ˆ HR u)    ˆ i , kˆ j ) + dq d pˆ i d kˆ j |ξˆijR (q, pˆ i , kˆ j )|2 μ−1 R − IR (q, p (i,j )

−



dpN k M

(i,j ) =(l,h)

For R → ∞ one has

IR (q, pˆ i , kˆ j ) =

2m m+1

dz

R ρˆijR ρˆlh

h0 + λ

.

(6.14)

1R (z) z2

m−1 q ·z+γ m  + o(1) dz m−1 2 2 z (m + 1)z /2m + q ·z+γ m  

2m 3/2 8π m 1 2 2 2 = R − 2π q 2 + pˆ 2i + kˆ j + λ + o(1), (6.15) m+1 m+1 m+1 m 2m − m+1



where in the last line we have used the explicit integration  δ·z+γ δ 2 < 4γ . dz 2 2 = π 2 4γ − δ 2 , z (z + δ · z + γ )

(6.16)

Therefore, in order to obtain a nontrivial limit for R → ∞, we fix 8π m μ−1 R+α (6.17) R = m+1 where α ∈ R is a new coupling constant. At least formally, with this choice we can remove the cut-off and define the renormalized quadratic form as the limit of (6.14) for R → ∞. More precisely, we are lead to the following definition of quadratic form # $    2 2  λ G ξij (p N , k M )| − λ|u(p ˆ N , k M )| Gα (u) = dpN dk M (h0 (p N , k M ) + λ)| uˆ − (i,j )

   2m 3/2 h1 (q, pˆ i , kˆ j ) + λ |ξˆij (q, pˆ i , kˆ j )|2 + dq d pˆ i d kˆ j α + 2π 2 m+1 (i,j )     √ √ ξˆij (pi + kj )/ 2, pˆ i , kˆ j ξˆlh (pl + kh )/ 2, pˆ l , kˆ h  − dpN dk M , h0 (p N , k M ) + λ (i,j ) =(l,h) 



(6.18)

156

D. FINCO and A. TETA

where ξˆij (q, pˆ i , kˆ j ) = lim ξˆijR (q, pˆ i , kˆ j ) R→∞

(6.19)

and the potential produced by the surface charges ξˆij was introduced in (2.13). In the quadratic form (6.18) the particles in the two groups A and B are still considered distinguishable. Since we want to describe fermions, the final step of the construction is to take into account the requirement of antisymmetry. From (6.19), (6.8) it is easily seen that ξˆij (q, pˆ i , kˆ j ) = (−1)i+j ξˆ11 (q, pˆ i , kˆ j ) ≡ (−1)i+j ξˆ (q, pˆ i , kˆ j )

(6.20)

which in particular means that the interaction is completely described by the unique surface charge ξˆ11 ≡ ξˆ . We also denote   √ i+j ˆ ˆ ˆ (−1) + k )/ 2, p , k ξ (p  i j j i  λ ξ (p , k ) = (6.21) G M N h0 (p N , k M ) + λ (i,j ) Moreover for the products of surface charges in (6.18) we have     √ √ ξˆij (pi + kj )/ 2, pˆ i , kˆ j ξˆlh (pl + kh )/ 2, pˆ l , kˆ h     √ √ if i = l, j = h. = ξˆ (p1 + k1 )/ 2, pˆ 1 , kˆ 1 ξˆ (p2 + k2 )/ 2, pˆ 2 , kˆ 2     √ √ ξˆij (pi + kj )/ 2, pˆ i , kˆ j ξˆih (pi + kh )/ 2, pˆ i , kˆ h     √ √ if i = l, j = h. = −ξˆ (p1 + k1 )/ 2, pˆ 1 , kˆ 1 ξˆ (p1 + k2 )/ 2, pˆ 1 , kˆ 2     √ √ ξˆij (pi + kj )/ 2, pˆ i , kˆ j ξˆlj (pl + kj )/ 2, pˆ l , kˆ j     √ √ if i = l, j = h. = −ξˆ (p1 + k1 )/ 2, pˆ 1 , kˆ 1 ξˆ (p2 + k1 )/ 2, pˆ 2 , kˆ 1 (6.22) Taking into account the above symmetry constraints in (6.18), we finally arrive at the quadratic form (2.4), (2.5) and (2.6). In the next proposition we collect some useful properties of the potential produced by the surface charges ξij . PROPOSITION 6.1. For ξij ∈ L2 (R3(N +M−1) ) the corresponding potential G ξij (x N , y M ) satisfies √ % & (6.23) (H0 + λ) Gλ ξij (x N , y M ) = 8π 3/2 ξij ( 2 xi , xˆ i , yˆ j ) δ(xi − yj ) λ

in the distributional sense. For ξij ∈ H 1 (R3(N +M−1) ) the singularity for |xi − yj | → 0 is characterized as follows

QUADRATIC FORMS FOR THE FERMIONIC UNITARY GAS MODEL

157

√ √ 1 2 πm λ (G ξij )(x N , y M ) = ξij ( 2xi , xˆ i , yˆ j ) |xi − yj | m + 1 2π 2 (2m)3/2

−

3

(2π ) 2 (N+M) (m + 1)3/2  √ i 2xi q+xˆ i ·pˆ i +yˆ j ·kˆ j · dq d pˆ i d kˆ j e h1 (q, pˆ i , kˆ j ) + λ ξˆij (q, pˆ i , kˆ j ) + o(1). Moreover for ξij ∈ H −1/2 (R3(N +M−1) ) one has   |ξˆ q, pˆ i , kˆ j |2 ˆ c1 dq d pˆ i d k j  ≤ Gλ ξij 2 2 2 2 q + pˆ + kˆ /m + λ j

i

≤ c2

  |ξˆ q, pˆ i , kˆ j |2

, dq d pˆ i d kˆ j  2 2 2 ˆ q + pˆ i + k j /m + λ

(6.24)

(6.25)

where c1 = π 2 min{m, 1}, c2 = π 2 max{m, 1}. Proof : Let us fix a test function φ and let us consider the definition (2.14) of  λ ξ . Then, using Fourier transform, we have G ij % & dx N dy M φ(x N , y M ) (H0 + λ) Gλ ξij (x N , y M )   √ ˆ N , k M )ξˆij (p1 + kj )/ 2, pˆ i , kˆ j = dpN dk M φ(p =



8π 3/2 3 (2π ) 2 (N+M)

√ dxi d xˆ i d yˆ j ξij ( 2xi , xˆ i , yˆ j )



ˆ N , k M ) e−i[(pi +kj )·xi +pˆ i ·xˆ i +kˆ j ·yˆ j )] dpN dk M φ(p √ 3/2 = 8π dxi d xˆ i d yˆ j ξij ( 2xi , xˆ i , yˆ j )φ(xˆ i , xi , yˆ j , xi ) ·

(6.26)

and this proves (6.23). From (2.14) we also have Gλ ξij (x N , y M ) = =

1 3

(2π ) 2 (N+M)



1



3 (2π ) 2 (N+M)

dq d pˆ i d kˆ j e

dpN dk M ei(x N ·pN +y M ·kM ) #

i

xi +yj √ ·q+xˆ i ·p ˆ i +yˆ j ·kˆ j 2

$

  √ ξˆij (p1 + kj )/ 2, pˆ i , kˆ j h0 (pN , k M ) + λ

ξˆij (q, pˆ i , kˆ j ) L(xi − yj , q, pˆ i , kˆ j ), (6.27)

158

D. FINCO and A. TETA

where

L(xi − yj , q, pˆ i , kˆ j ) =



√

ei(xi −yj )z/ 2 dz , m+1 2 m−1 z + q ·z+γ 2m m

(6.28)

m+1 2 1 2 q + pˆ 2i + kˆ j + λ. 2m m For |xi − yj | → 0 the last integral is given by γ =

L(xi − yj , q, pˆ i , kˆ j ) m−1 q ·z+γ m  + o(1) dz m+1 2 m−1 z2 z + q ·z+γ 2m m √ 2 3/2 

4 2π m 1 2m = h1 (q, pˆ i , kˆ j ) + λ + o(1) − 2π 2 m + 1 |xi − yj | m+1 (6.29)

2m = m+1



√

ei(xi −yj )/ dz z2

2·z

2m − m+1



where we have used the explicit integration (6.16). Using (6.29) in (6.27) we obtain (6.24). Finally for the proof of (6.25) we observe that   √  ξˆ (pi + kj )/ 2, pˆ i , kˆ j 2 λ 2 . (6.30) G ξij  = dpN dk M 2  h0 (p N , k M ) + λ √ √ Introducing the coordinates q = (pi + kj )/ 2, v = (pi − kj )/ 2 and using the elementary inequality − 12 (v 2 + q 2 ) ≤ v · q ≤ 12 (v 2 + q 2 ) we have c1 dq d pˆ i d kˆ j |ξˆ (q, pˆ i , kˆ j )|2 M ≤ Gλ ξij 2 ≤ c2 dq d pˆ i d kˆ j |ξˆ (q, pˆ i , kˆ j )|2 M, (6.31) where

M=

1 π2



1 dv  2 v 2 + q 2 + pˆ 2i + kˆj2 /m + λ

By an explicit computation of the above integral we obtain (6.25).

(6.32)

2

REFERENCES [1] S. Albeverio, F. Gesztesy, R. Hoegh-Krohn and H. Holden: Solvable Models in Quantum Mechanics, Springer, New York 1988. [2] A. Alonso and B. Simon: The Birman–Krein–Vishik theory of self-adjoint extensions of semibounded operators, J. Operator Theory 4 (1980), 251–270.

QUADRATIC FORMS FOR THE FERMIONIC UNITARY GAS MODEL

159

[3] E. Braaten and H. W. Hammer: Universality in few-body systems with large scattering length, Phys. Rep. 428 (2006), 259–390. [4] M. S. Birman: On the self-adjoint extensions of positive definite operators (in Russian), Math. Sb. 38 (1956), 431–450. [5] Y. Castin, C. Mora and L. Pricoupenko: Four-body Efimov effect, arXiv:1006.4720v 1. [6] M. Correggi, G. Dell’Antonio, D. Finco A. Michelangeli and A. Teta: Zero-range interactions for n identical fermions plus a different particle, in preparation. [7] G. Dell’Antonio, R. Figari and A. Teta: Hamiltonians for systems of N particles interacting through point interactions, Ann. Inst. Henri Poincare, Phys. Theo. 60 (1994), 253–290. [8] A. Erdely et al,: Tables of Integrals Transforms, Mc Graw-Hill, New York 1954. [9] G. Flamand: Mathematical theory of non-relativistic two- and three-particle systems with point interactions, in Cargese Lectures in Theoretical Physics, Lurcat F. ed., 1967, 247–287. [10] I. S. Gradshteyn and I. M. Ryzhik: Table of Integrals, Series, and Products, Academiv Press 2007. [11] T. Kato: Perturbation Theory for Linear Operators, Springer, Berlin 1980. [12] M. Krein: The theory of self-adjoint extensions of semibounded hermitian transformations and its applications, Rec. Math. (Math. Sb.) 20 (1947), 431–495. [13] A. M. Melnikov and R. A. Minlos: On the pointlike interaction of three different particles, Adv. Soviet Math. 5 (1991), 99–112. [14] R. A. Minlos: On the point interaction of three particles, Lect. Notes in Physics 324, Springer, 1989. [15] R. A. Minlos: On point-like interaction between n fermions and another particle, Moscow Math. Journal 11 (2011), 113–127. [16] R. A. Minlos and L. Faddeev: On the point interaction for a three-particle system in Quantum Mechanics, Soviet Phys. Dokl. 6 (1962), 1072–1074. [17] R. A. Minlos and M. S. Shermatov: On pointlike interaction of three quantum particles, Vestnik Mosk, Univ. Ser. I Math. Mekh. 6 (1989), 7–14. [18] M. K. Shermatov: Point Interaction between two fermions and a particle of a different nature, Theo. Math. Phys. 136 (2003), 1119–1130. [19] F. Werner and Y. Castin: Unitary gas in an isotropic harmonic trap: symmetry properties and applications, Phys. Rev. A 74 (2006), 053–604. [20] F. Werner and Y. Castin: Unitary quantum three-body problem in a harmonic trap, Phys. Rev. Lett. 97 (2006), 150–401.