Table Of ContentN
Stability for a System of Fermions Plus a Different Particle with
Zero-Range Interactions
M. Correggi1, G. Dell’Antonio2, D. Finco3, A. Michelangeli4, A. Teta5
2
1 1. Dipartimento di Matematica, Universit`a di Roma Tre
0 Largo San Leonardo Murialdo 1, 00146 Roma, Italy, [email protected]
2
2. Dipartimento di Matematica, ”Sapienza” Universit`a di Roma
n
P.le A. Moro 5, 00185 Roma, Italy, and
a
J Scuola Internazionale di Studi Superiori Avanzati
7 Via Bonomea 265, 34136 Trieste, Italy, [email protected]
2
3. Facolta` di Ingegneria, Universit`a Telematica Internazionale Uninettuno
] Corso V. Emanuele II 39, 00186 Roma, Italy, d.fi[email protected]
h
4. Institute of Mathematics, LMU Munich
p
- Theresienstr. 39, 80333 Munich, Germany, [email protected]
h
t 5. Dipartimento di Matematica Pura ed Applicata, Universit`a di L’Aquila
a
Via Vetoio - Loc. Coppito - 67010 L’Aquila, Italy, [email protected]
m
[
January 30, 2012
1
v
0
4 Abstract
7
5 Westudythestabilityproblemforanon-relativisticquantumsystemindimensionthreecom-
. posed by N 2 identical fermions, with unit mass, interacting with a different particle, with
1 mass m, via a≥zero-range interaction of strength α R. We construct the corresponding renor-
0
∈
2 malised quadratic (or energy) form α and the so-called Skornyakov-Ter-Martirosyansymmetric
F
1 extension Hα, which is the natural candidate as Hamiltonian of the system. We find a value of
v: the mass m∗(N) such that for m > m∗(N) the form α is closed and bounded from below. As
F
i a consequence, α defines a unique self-adjoint and bounded from below extension of Hα and
X F
therefore the system is stable. On the other hand, we also show that the form α is unbounded
r from below for m < m∗(2). In analogy with the well-known bosonic case, this sFuggests that the
a
∗
system is unstable for m<m (2) and the so-called Thomas effect occurs.
1
1 Introduction
The dynamics of a quantum system composed by N particles in Rd, d = 1,2,3, interacting via a
zero-range, two-body interaction is described by the formal Hamiltonian
N N
1
= ∆ + µ δ(x x ), (1.1)
H − 2m xi ij i − j
i
i=1 i,j=1
X X
i<j
where x Rd, i = 1,...,N, is the coordinate of the i-th particle, m is the correspondingmass, ∆
i ∈ i xi
is the Laplacian relative to x , and µ R is the strength of the interaction between particles i and
i ij
∈
j. To simplify the notation we set ~ = 1. Formal Hamiltonians of the type (1.1) are widely used in
physical applications. In particular they are relevant in the study of ultra-cold quantum gases, both
in the bosonic and in the fermionic case, in the so-called unitary limit, i.e., for infinite two-body
scattering length (see [BH],[CW1],[CW2],[CMP] and references therein).
Thefirststeptowardsarigorousapproachtotheanalysisofthemodelistogivethemathematical
definition of such a Hamiltonian as a self-adjoint operator on the appropriate L2-space. One first
notices that the interaction term in (1.1) is effective only on the hyperplanes x = x . This
i<j i j
∪ { }
suggeststoconsidertheoperator ˙ definedasthefreeHamiltonianrestrictedtoadomainofsmooth
0
H
functions vanishing in the neighbourhood of each hyperplane x = x . It is easily seen that ˙
i j 0
{ } H
is symmetric but not self-adjoint and a trivial self-adjoint extension is the free Hamiltonian on its
natural domain. Then, by definition, any non trivial self-adjoint extension of the operator ˙ is a
0
H
Hamiltonian for a system of N quantum particles in Rd with a two-body, zero-range interaction.
The second and more relevant problem is the concrete construction of such self-adjoint exten-
sions. It turns out that each extension is characterized by a specific generalized boundary condition
satisfied by the wave function at the hyperplanes. The two most frequently used techniques for
the construction are Krein’s theory of self-adjoint extensions and the approximation by regularized
Hamiltonians, in the sense of the limit of the resolvent or of the quadratic form. The difficulty of
the analysis depends on the dimension d. For d = 1 the problem is greatly simplified by the fact
that the interaction term is a small perturbation of the free Hamiltonian in the sense of quadratic
forms. For d = 2 a natural class of Hamiltonians with local zero-range interactions was constructed
in [DFT] exploiting renormalised quadratic forms and it was also shown that such Hamiltonians are
allboundedfrombelow. Ford= 3theproblemismoredelicate. Inordertoillustratethemainpoint,
let us consider the special case n = 2 where, in the center of mass reference frame, one is reduced
to study a one-body problem in the relative coordinate x with a fixed δ-interaction placed at the
origin. In this case the problem is completely understood (see, e.g., [AGH-KH]) and the entire class
of Hamiltonians can be explicitly constructed. It turns out that the domain of each Hamiltonian
consists of functions ψ L2(R3) H2(R3 0 ) which exhibit the following singular behaviour for
∈ ∩ \{ }
x 0
| | → q
ψ(x) = +r+o(1), with r = αq, (1.2)
x
| |
where q C and α R is a parameter proportional to the inverse of the scattering length. We un-
∈ ∈
derline that the relation r = αq in (1.2) should be understood as the generalized boundarycondition
satisfied at the origin by all the elements of the domain. In the general case N > 2 the characteri-
zation of all possible self-adjoint extensions of ˙ is more involved. However, a class of extensions
0
H
based on the analogy with the case N = 2 can be explicitly constructed. More precisely, one con-
siders the so-called Skornyakov-Ter-Martirosyan (STM) extension of ˙ which, roughly speaking, is
0
H
a symmetric operator acting on functions ψ L2(R3N) H2(R3N x = x ) satisfying the
i<j i j
∈ ∩ \∪ { }
2
following condition for x x 0:
i j
| − | →
q
ij
ψ(x ,...,x ) = +r +o(1), with r = α q , (1.3)
1 n ij ij ij ij
x x
i j
| − |
where q is a suitable function defined on the hyperplane x = x and α is a collection
ij i j ij
{ } { }
of real parameters labelling the extension. Noticeably, the boundary condition (1.3) defining the
STM extension of ˙ is a natural generalization to the case N > 2 of the condition (1.2) that
0
H
characterizes the two-body case. Unfortunately, unlike (1.2), (1.3) does not necessarily define a self-
adjoint operator. Indeed, for a system of three identical bosons it was shown in [FM] that the STM
extension is not self-adjoint and all its self-adjoint extensions are unbounded from below owing to
the presence of an infinite sequence of energy levels E going to for k . In [MM] this
k
−∞ → ∞
result was generalized to the case of three distinguishable particles with different masses. This kind
of instability is known in the literature as the Thomas effect. It should be stressed that the Thomas
effect is strongly related to the well-known Efimov effect (see, e.g., [BH], [ADFGL], [AH-KW]) even
if, to our knowledge, arigorous mathematical investigation of this connection is still lacking. We also
mentionthatif, insteadof (1.3),oneintroducesa“non-local” boundaryconditiononthehyperplanes
then it is possible to construct a positive Hamiltonian and to study its stability properties for N
large (see, e.g., [FS]). In this paper we do not consider this kind of Hamiltonians.
It is reasonable to expect that the Thomas effect does not occur if the Hilbert space of states
is suitably restricted, e.g., introducing symmetry constraints on the wave function. A remarkably
importantconstraint isantisymmetry. Infact, awave functionthat isantisymmetric underexchange
of coordinates of two particles necessarily vanishes at the coincidence points of such two particles,
thus making their mutual zero-range interaction ineffective. Analogously, in a mixture of fermions
of different species subject to pairwise zero-range interaction, fermions of the same species cannot
“feel” mutual zero-range interaction and therefore the interaction term in the Hamiltonian is less
singular.
In this paper we consider the simplified model consisting of N identical fermions, with unit mass,
and a different particle with mass m, interacting with the fermions through a zero-range potential.
For such a model only partial results are available and it is remarkable that they strongly depend on
the parameters N and m.
Concerning the physical literature, we mention that for N = 2 it is known (see, e.g., [BH]
and references therein) that for m < 0.0735 = (13.607) 1 the Thomas effect is present while for
−
m > 0.0735 theSTMextensions are expected tobeboundedfrombelow. Morerecently (see [CMP]),
it was shown by means of analytical and numerical arguments that in the case N = 3 the Thomas
effectoccursform < 0.0747 = (13.384) 1. This,inparticular,indicatesthatfor0.0735 < m <0.0747
−
there is a sequence of genuine four-body bound states with energy going to .
−∞
From the rigorous point of view, the case N = 2 was investigated first in [M1] and [MS], where
it was proved that the STM extension is self-adjoint if m = 1. In [S], the existence of a critical
mass m (2) 0.0735 was shown such that for m < m (2) any STM extension – more precisely
∗ ∗
≃
its restriction to the subspace of angular momentum l = 1 – is not self-adjoint and its self-adjoint
extensions are unbounded from below. Therefore, the system is unstable and the Thomas effect
occurs. This result was extended in [FT], where it was shown that the quadratic form associated
with the STM extension, restricted to the subspace of angular momentum l, is unbounded from
below if an explicit condition on m and l is satisfied. Finally, in the case N 4 and m sufficiently
≤
large it was shown in [M3] that the STM extension is self-adjoint.
In the present work we study the problem for generic N and m and following the line of [DFT]
we construct a renormalised quadratic form , α R, which is naturally associated with the STM
α
F ∈
symmetricextension H . Here αis thevaluethatall theparameters α labellingtheSTMextension
α ij
3
must be equal to as a consequence of the fermionic symmetry. Note that in the bosonic case the
quadratic form would differ in a sign in front the non-diagonal term (see the remark after (2.15)).
The first question we address is a non-trivial sufficient condition for the stability of the model.
As a first main result (Theorem 2.2) we prove that for any N there is a value of the mass m (N) > 0
∗
such that is closed and boundedfrom below if m > m (N). This implies that is the quadratic
α ∗ α
F F
form of a unique self-adjoint extension of the STM operator H , and this extension is bounded from
α
below. It therefore describes a stable system, where the Thomas effect does not occur.
Such a critical mass was first conjectured in [M2] and is precisely the unique root of an explicit
equation (see (2.16) and (2.18) in Section 2). It turns out that m (N) is increasing with N and
∗
that the condition m > m (N) guarantees the stability also in the limit of infinitely many fermions,
∗
provided that the mass of the extra particle scales as m N.
∝
The second question we address is a sufficient condition for the instability of the model. This can
beseen by plugging suitable trial functions into . An attempt in this direction is in [DFT, Section
α
F
7], but with trial functions that do not satisfy the fermionic symmetry: thus, the result stated there
on the unboundedness from below of the quadratic form for m = 1 and N sufficiently large cannot
be considered valid. Our second main result (Theorem 2.3) fills in this gap and we prove that for
any N 2 the quadratic form is unbounded from below for m < m (2). In analogy with the
α ∗
≥ F
bosonic case, we expect that in such a case H is not self-adjoint and all its self-adjoint extensions
α
are unbounded from below.
Let us make a few remarks on the above-mentioned results. First, we emphasize that in the case
N = 2 we fully characterize stability: the model is stable for m > m (2) and unstable m < m (2).
∗ ∗
Whereas the latter was already found in [S] by means of the theory of self-adjoint extensions, the
former is proved here for the first time.
Second, if N > 2 we expect the condition m > m (N) to befar from optimal for stability. This is
∗
due to the crucial role played by the restriction to antisymmetric wave functions (see the discussion
in Section 4) so that the system might be stable also if our condition is violated.
Third, the fact that the (N+1)-particle system is unstable at least when m is below the same
threshold m (2) for the instability of the (2+1)-particle system has a rather natural interpretation:
∗
the instability of a subsystem made of two fermions plus the different particle is responsible for the
instability of the whole system.
We want to mention that in the final stage of the preparation of this work we became aware of a
recent paper [M4] where the case of two fermions plus a different particle is studied using the theory
of self-adjoint extensions. We believe that a comparison with our methods and results would be of
great interest for the further developments of the subject.
The paper is organized as follows. In Section 2 we introduce the renormalised quadratic form
α
F
and the STM extension H and we formulate our main results. In Section 3 we give the proof of
α
Theorem 2.2. In Section 4 we give the proof of Theorem 2.3. In the Appendix we briefly outline the
formal renormalisation procedure to derive .
α
F
For the convenience of the reader, we collect here some usefulnotation that will beused through-
out the paper. We use the notation L2(Rd) (resp. H1(Rd), H−1/2(Rd), etc.) for the space containing
f f f
totally antisymmetric functions belonging to L2(Rd) (resp. H1(Rd), H 1/2(Rd), etc.). We often use
−
the short-hand notation , , etc. for the associated norms , , etc.
k·kL2 k·kL2f k·kL2(Rd) k·kL2f(Rd)
For a vector x R3 we set x = x. Moreover we define K := (k ,...,k ) and, for i= 1,...,N,
2 N 1
∈ | | −
K˘ := (k ,...,k ,k ,...,k ).
i 1 i 1 i+1 N
−
For any f L2(Rd) the Fourier transform is defined by fˆ(k) = (2π) d/2 dxe ikxf(x).
∈ − Rd − ·
R
4
The functions G , , L , with λ > 0, and D(K) are defined in (2.7), (2.9), (2.8) and (3.5) respec-
λ λ λ
G
tively.
2 Main results
In this section we introduce the quadratic form , the STM extension H and we formulate our
α α
F
main results.
2.1 The quadratic form
α
F
For a three dimensional quantum system composed by N identical fermions, with mass one, plus
a different particles, with mass m, with a two-body zero-range interaction the formal many-body
Hamiltonian is
N N
1 1
H˜ := ∆ ∆ +µ δ(x x ), (2.1)
−2m x0 − 2 xi 0 − i
i=1 i=1
X X
where x R3, i = 0,...,N, and µ R. Introducing the centre of mass and relative coordinates
i
∈ ∈
N
1
X := mx + x ,
m+N 0 i (2.2)
y := x x (cid:16), Xi=1 (cid:17) for i = 1,...N ,
i 0 i
−
one obtains
H˜ = H + m+1H (2.3)
cm 2m
here H := [2(m+N)] 1∆ and
cm − X
−
N N
2
H := ∆ +µ δ(y ), (2.4)
− yi − m+1 ∇yi ·∇yj i
i=1 i<j i=1
X X X
denoting the gradient with respect to y . We also introduce the free Hamiltonian
∇yi i
N
2
D(H ) := H2(R3N), H := ∆ , (2.5)
0 f 0 − yi − m+1 ∇yi ·∇yj
i=1 i<j
X X
and its restriction to functions vanishing in a neighbourhood of the hyperplanes y = 0
i
{ }
D(H˙ ):= ψ H2(R3N) dk ψˆ(k ,...,k )= 0, i = 1,...,N , H˙ := H . (2.6)
0 (cid:26) ∈ f (cid:12)ZR3 i 1 N (cid:27) 0 0 D(H˙0)
(cid:12) (cid:12)
(cid:12) (cid:12)
In order to give a rigor(cid:12)ous meaning to the formal expression (2.4) as a self-adjoint operator in
L2(R3N), one can use Krein’s theory to construct self-adjoint extensions of the operator (2.6) (see
f
[M3]). Here, instead, we follow adifferent approach andwe investigate thequadratic formassociated
with the expectation value ψ H ψ . However, because of the singularity of the “potential” δ(y )
i
h | | i
the quadratic form has to be defined via a renormalisation procedure in the Fourier space. The idea
of the construction is given in the Appendix (see also [DFT],[FT]) and here we only give the final
result.
5
Let us denote for any λ > 0
N 1
G (k ,...,k ):= k2+ 2 k k +λ − , (2.7)
λ 1 N i m+1 i· j
(cid:20)i=1 i<j (cid:21)
X X
N 1 1/2
m(m+2) − 2m
L (k ,...,k ):= 2π2 k2+ k k +λ , (2.8)
λ 1 N−1 (m+1)2 i (m+1)2 i· j
(cid:18) i=1 i<j (cid:19)
X X
and, for any “charge” ξ ∈ Hf−1/2(R3N−3), let us denote the “potential” produced by ξ in the Fourier
space by
N
ξ (k ,...,k ):= ( 1)i+1G (k ,...,k )ξˆ(K˘ ), (2.9)
λ 1 N λ 1 N i
G −
(cid:16) (cid:17) Xi=1
where K˘ := (k ,...,k d,k ,...,k ). Then the quadratic form in the Hilbert space L2(R3N)
i 1 i−1 i+1 N Fα f
is defined as follows
D( ) := ψ L2(R3N) ξ D(Φλ) s.t. φλ := ψ ξ H1(R3N) , (2.10)
Fα ∈ f ∃ ∈ α −Gλ ∈ f
n (cid:12) o
[ψ] := [φλ]+(cid:12)λ φλ 2 λ ψ 2 +NΦλ[ξ], (2.11)
Fα F0 (cid:12) k kL2(R3N)− k kL2(R3N) α
where λ > 0, [φ] := φ H φ and Φλ is the following form on the charge ξ
F0 h | 0| i α
D(Φλ):= H1/2(R3N 3), (2.12)
α f −
Φλ[ξ] := Φdiag[ξ]+Φoff[ξ], (2.13)
α α,λ λ
Φdiag[ξ]:= dk dk ξˆ(k ,...,k ) 2[α+L (k ,...,k )], (2.14)
α,λ R3N−3 1··· N−1 1 N−1 λ 1 N−1
Z
(cid:12) (cid:12)
(cid:12) (cid:12)
Φoff[ξ]:= (N 1) dsdtdk dk ξˆ (s,k ,...,k )ξˆ(t,k ,...,k )
λ − R3N 2··· N−1 ∗ 2 N−1 2 N−1 ·
Z
G (s,t,k ,...,k ). (2.15)
λ 2 N 1
−
We notice that for ξ D(Φλ) we have ξ L2(R3N) and ξ / H1(R3N). Therefore the decompo-
∈ α Gλ ∈ f Gλ ∈ f
sition in (2.10) is meaningful. Moreover, as we shall see in Section 3, the form Φλ is well-defined on
α
D(Φλ).
α
It is worth mentioning that the fermionic constraint implies not only that the wave function is
totally antisymmetric but also that the form on the charges (2.13) differs from the bosonic case by a
sign in front of the off-diagonal part (2.15). This fact results in a weaker effective interaction among
the fermions and the stability problem is qualitatively different.
Inorder toformulate our mainresults on theform wefirstintroduceour definitionof stability
α
F
parameter. Let us consider the following function
1 1
Λ(m,N) := 2π 1(N 1)(m+1)2 arcsin . (2.16)
−
− m(m+2) − m+1
(cid:20) (cid:18) (cid:19)(cid:21)
It is easy to check that, for N fixed, the function Λp(m,N) is decreasing in m and
lim Λ(m,N) = , lim Λ(m,N) = 0. (2.17)
m 0 ∞ m
→ →∞
Then we have
6
Definition 2.1 (Stability parameter m (N)).
∗
For N fixed, we define m (N) as the unique solution to the equation
∗
Λ(m,N) = 1. (2.18)
If we define θ := arctan 1+ 2, where θ (π,π), a direct computation shows that equation (2.18)
m ∈ 4 2
can be equivalently writqten as
π 1
cot2θ+2θ 1 cos22θ = 0. (2.19)
− 2 − N 1
(cid:18) − (cid:19)
We remark that (2.19) reduces for N = 2 to the equation found for the critical mass in [PC, p.
12871, note 10]. We also notice that m (N) is positive and increasing with N. In particular, the
∗
condition Λ(m,N) < 1 is equivalent to m > m (N). This condition is crucial to guarantee closure
∗
and boundedness from below of the form .
α
F
Theorem 2.2 (Stability for m > m (N)).
∗
Let N 2 and m > m (N). Then the quadratic form is closed and bounded from below. In
∗ α
≥ F
particular, it is positive for α 0 and
≥
α2
[ψ] ψ 2 , ψ D( ), (2.20)
Fα ≥ −4π4 1 Λ(m,N) k kL2 ∈ Fα
−
for α < 0. (cid:0) (cid:1)
The proof will be given in Section 3.
Concerning the instability problem, our result is the following.
Theorem 2.3 (Instability for m < m (2)).
∗
Let N 2 and m < m (2). Then the quadratic form is unbounded from below for any α R.
∗ α
≥ F ∈
This theorem generalizes the result obtained for N = 2 in [FT] and it is proved in Section 4. In the
case N = 2, Theorems 2.2 and 2.3 show that the system is stable for m > m (2) and unstable for
∗
m < m (2), therefore our analysis is complete. When N > 2 the problem remains open: no rigorous
∗
result is available for m (2) < m < m (N).
∗ ∗
2.2 The STM extension H
α
HerewerecallthestandarddefinitionoftheSTMextensionH anditsconnectionwiththequadratic
α
form . Moreover, as a direct consequence of Theorem 2.2, we prove our main result on H .
α α
F
As we mentioned in the introduction, H is a distinguished symmetric extension of the operator
α
(2.6). For the details of the construction we refer to [M3] and here we only give the definition
D(H ) := ψ L2(R3N) ξ H3/2(R3N 3) s.t. φ := ψ ξ H2(R3N),
α ∈ f ∃ ∈ f − λ −Gλ ∈ f
(cid:26) (cid:12)
(cid:12)
(cid:12) dk φˆ (k ,...,k )= ( λ ξˆ)(K˘ ), i = 1,...,N , (2.21)
(cid:12) ZR3 i λ 1 N Aα,i i (cid:27)
(H +λ)ψ = (H +λ)φ , (2.22)
α 0 λ
7
where λ > 0 and
N
( λ ξˆ)(K˘ ) := ( 1)i+1 α+L (K˘ ) ξˆ(K˘ ) ( 1)j+1 dk G (k ,...,k )ξˆ(K˘ ). (2.23)
Aα,i i − (cid:20) λ i (cid:21) i −j=1,j=i− ZR3 i λ 1 N j
X6
The last equality in (2.21) should be understood as the boundary condition satisfied by any ψ
∈
D(H ). In fact, by a straightforward computation, one verifies that such equality implies the follow-
α
ing asymptotic condition for R
→ ∞
dk ψˆ(k ,...,k )= 4πR( 1)i+1ξˆ(K˘ )+α( 1)i+1ξˆ(K˘ )+o(1) (2.24)
i 1 N i i
− −
Zki<R
and this is exactly the standard boundary condition, formulated in the Fourier space, characterizing
the STM extension (see [M3]).
The next step is to establish the connection with the form . For any ψ D(H ) one has
α α
F ∈
ψ H +λ ψ = ψ H +λ φ = φ H +λ φ + ξ H +λ φ
α 0 λ λ 0 λ λ 0 λ
h | | i h | | i h | | i hG | | i
N
= φ H +λ φ + ( 1)i+1 dK˘ ξˆ (K˘ ) dk φˆ (k ,...,k )
λ 0 λ i ∗ i i λ 1 N
h | | i − R3N−3 R3
i=1 Z Z
X
N
= φ H +λ φ + ( 1)i+1 dK˘ ξˆ (K˘ )( λ ξˆ)(K˘ ). (2.25)
h λ| 0 | λi − R3N−3 i ∗ i Aα,i i
i=1 Z
X
Exploiting (2.23) and the antisymmetry of ξˆwe have
N
( 1)i+1 dK˘ ξˆ (K˘ )( λ ξˆ)(K˘ )
− R3N−3 i ∗ i Aα,i i
i=1 Z
X
N N
= dK˘ ξˆ(K˘ )2 α+L (K˘ ) ( 1)i+j dk dk ξˆ (K˘ )ξˆ(K˘ )G (k ,...,k )
i i λ i 1 N ∗ i j λ 1 N
R3N−3 | | − − R3N ···
i=1Z i,j=1,i=j Z
X (cid:2) (cid:3) X6
= NΦdiag[ξ]+N(N 1)Φoff[ξ]. (2.26)
α,λ − λ
Therefore we conclude
ψ H ψ = [ψ], ψ D(H ). (2.27)
α α α
h | | i F ∈
From the above relation and Theorems 2.2, 2.3, we obtain the following result.
Theorem 2.4 (Self-adjointness and boundedness from below).
One has the following two possible alternatives:
(i) If N 2 and m > m (N), then defines a unique self-adjoint and bounded from below
∗ α
≥ F
extension Hˆ of the operator H . In particular, Hˆ is positive for α 0 and
α α α
≥
α2
infσ(Hˆ ) , for α < 0. (2.28)
α
≥ −4π4 1 Λ(m,N)
−
(cid:0) (cid:1)
(ii) If N 2 and m < m (2), then, for any α R, no self-adjoint extension Hˆ of H such that
∗ α α
≥ ∈
D(Hˆ ) D( ) can be both self-adjoint and bounded from below.
α α
⊂ F
8
Proof. The assertion (i) immediately follows from Theorem 2.2 and the variational characterization
of the infimum of the spectrum of a self-adjoint operator.
The converse (ii) can be obtained by exploiting the Birman-Krein-Vishik theory of positive self-
adjoint extensions (see, e.g., [ASi], [P1],[P2]). We omit the details and refer to [FT, Proposition 4.1]
where an analogous results was proven.
We remark that, in the case N 2 and m < m (2), an analogy with the bosonic case suggests that
∗
≥
any self-adjoint extension of H is unbounded from below and the Thomas effect occurs, although
α
such a complete characterization is beyond our purposes. The most natural tool to approach the
analysis of this case is indeed the theory of self-adjoint extensions (see, e.g., [S],[M4]).
We conclude this section with a brief comment relative to the special case N = 2, m > m (2). In
∗
particular wewant to give theexplicit characterization of our Hamiltonian Hˆ inorderto make more
α
transparent a possible comparison of our result with the result in [M4].
In Section 3 we shall prove the estimate (3.52), which in particular implies that the form Φλ, defined
α
on H1/2(R3), is closed andpositive for λ sufficiently large. Thereforeit definesapositive, self-adjoint
operator Γλ, D(Γλ) in L2(R3) explicitly given by
α α
D(Γλ) = ξ H1/2(R3) Γλξˆ L2(R3) ,
α ∈ α ∈
n (cid:12) o
Γλξˆ (q)= [α+L (q)]ξˆ(q)+ (cid:12)dpG (p,q)ξˆ(p). (2.29)
α λ (cid:12) λ
R3
(cid:16) (cid:17) Z
Exploiting this fact and following the same line of [DFT, Section5], we obtain
D(Hˆ ) = ψ L2(R6) ξ D(Γλ) s.t. φ := ψ ξ H2(R6),
α ∈ f ∃ ∈ α λ −Gλ ∈ f
(cid:26) (cid:12)
(cid:12)
(cid:12) dpφˆ (p,q) = (Γλξˆ)(q) , (2.30)
(cid:12) λ α
ZR3 (cid:27)
(Hˆ +λ)ψ =(H +λ)φ . (2.31)
α 0 λ
We notice that the above operator differs from the operator (2.21), (2.22) only in a larger class of
admissible charges ξ, i.e., the domain D(Γλ) strictly contains H3/2(R3). We also underline that
α
the boundary condition satisfied on the hyperplanes by an element of (2.30) is the standard STM
boundary condition.
3 Closure and boundedness from below of
α
F
The proof of Theorem 2.2 is based on a careful estimate from below and from above of the form Φλ
α
on the charge ξ. If N > 2 (with an obvious modification in the case N = 2) we rewrite both the
diagonal and the off-diagonal terms of Φλ, defined in (2.14), (2.15), in a more manageable form (see
α
[M3]), by introducing the change of coordinates
N 1 N 1
1 − 1 −
s σ := s+ k , t τ := t+ k .
i i
−→ m+2 −→ m+2
i=2 i=2
X X
Then
Φdiag[ξ] = α ξ 2 +2π2 dσdK ξ˜(σ,K) 2 m(m+2)σ2+D(K)+λ, (3.1)
α,λ k kL2(R3N−3) R3N−3 (m+1)2
Z q
(cid:12) (cid:12)
(cid:12) (cid:12)
9
1
Φoff[ξ] = (N 1) dσdτdKξ˜ (σ,K)ξ˜(τ,K) σ2+τ2+ 2 σ τ +D(K)+λ − , (3.2)
λ − R3N ∗ m+1 ·
Z (cid:16) (cid:17)
where
K := k ,...,k , (3.3)
2 N 1
−
N 1
ξ˜(σ,K) := ξˆ σ 1 − k ,k ,...,k , (3.4)
i 2 N 1
− m+2 −
(cid:18) i=2 (cid:19)
X
N 1
m −
D(K):= (m+3) k2+2 k k . (3.5)
(m+1)(m+2) i i · j
(cid:18) i=2 i<j (cid:19)
X X
Notice that D(K) satisfies the bound (see (3.49) in the following)
N 1 N 1
m − m(m+N +1) −
k2 D(K) k2. (3.6)
m+1 i ≤ ≤ (m+1)(m+2) i
i=2 i=2
X X
Last, setting
σ := D(K)+λp, τ := D(K)+λq, (3.7)
and p p
Q (p) := (D(K)+λ)3/4ξ˜ D(K)+λp,K , (3.8)
K
we obtain (cid:16)p (cid:17)
Φλ[ξ] = Φλ[ξ] α ξ 2 = dK D(K)+λ F [Q ] , (3.9)
0 α − k kL2(R3N−3) R3N−6 1 K
Z
p
where for any ζ 0 we introduced the quadratic form in L2(R3)
≥
D(F ):=D(F )= f L2(R3) dp p2+1 f(p)2 < , F [f]:= Fdiag[f]+Foff[f], (3.10)
ζ 1 (cid:26) ∈ (cid:12)ZR3 | | ∞(cid:27) ζ ζ ζ
(cid:12) p
(cid:12)
and
(cid:12)
Fdiag[f]= 2π2 dp m(m+2)p2+ζ f(p)2 , (3.11)
ζ R3 (m+1)2 | |
Z q
f (p)f(q)
Foff[f]= (N 1) dpdq ∗ . (3.12)
ζ − ZR6 p2+q2+ m2+1p·q+ζ
Using the representation (3.9), (3.10), (3.11), (3.12) we obtain the following estimate for Φλ, which
0
is the crucial ingredient for the proof of Theorem 2.2.
Proposition 3.1 (Upper and lower bounds for Φλ).
0
For any ξ D(Φλ) we have
∈ α
1 Λ(m,N) Φdiag[ξ] Φλ[ξ] 1+Γ(m,N) Φdiag[ξ], (3.13)
− 0,λ ≤ 0 ≤ 0,λ
(cid:16) (cid:17) (cid:16) (cid:17)
where
(N 1)(m+1)2 1
Γ(m,N) := − arcsin . (3.14)
m(m+2) m+1
(cid:18) (cid:19)
p
10
