# On Pointlike Interaction between Three Particles: Two Fermions and Another Particle

Authors: Robert Adol'fovich Minlos
Journal: ISRN Mathematical Physics
 (2012)

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## Abstract

The problem of construction of self-adjoint Hamiltonian for quantum system consisting of three pointlike interacting particles (two fermions with mass 1 plus a particle of another nature with massm>0) was studied in many works. In most of these works, a family of one-parametric symmetrical operators {Hε,ε∈ℝ1} is considered as such Hamiltonians. In addition, the question about the self-adjointness of Hε is equivalent to the one concerning the self-adjointness of some auxiliary operators {𝒯l,l=0,1,…} acting in the space L2(ℝ+1,r2dr). In this work, we establish a simple general criterion of self-adjointness for operators 𝒯l and apply it to the cases l=0 and l=1. It turns out that the operator 𝒯l=0 is self-adjoint for any m, while the operator 𝒯l=1 is self-adjoint for m>m0, where the value of m0 is given explicitly in the paper.


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## Body

## 1. Introduction and Statement of the Problem

This paper is continuation of works [1–4] studying the problem of construction of Hamiltonian for a quantum system which consists of two fermions with mass 1 interacting pointwise with a particle of another nature having mass m.Originally, the construction of such Hamiltonian begins with introduction of the symmetric operator:(1.1)H0=-12(1mΔy+Δx1+Δx2)
acting in a Hilbert space ℋ=L2(ℝ3)⊗L2asym(ℝ3×ℝ3). Here, x1, x2∈ℝ3 are the positions of fermions, y is the position of a separate particle, and Δy, Δx1, and Δx2 are Laplacians with respect to y, x1, and x2, respectively. The domain of definition of H0, D(H0)⊂ℋ consists of smooth rapidly decreasing functions ψ(y,x1,x2)∈ℋ on infinity, antisymmetrical with respect to x1, x2 and satisfying the following conditions:
(1.2)ψ(y,x1,x2)|xi=y=0,i=1,2.
Usually, some family {Hɛ,ɛ∈ℝ1} of symmetric extensions of the operator H0 is proposed as a possible “true” Hamiltonian of the system (the so-called Ter-Martirosian-Skornyakov extensions, see [5]). These extensions were constructed in [1–4]. For some values of mass m, the extensions of Ter-Martirosian-Skornyakov are self-adjoint (for all values of the parameter ɛ); however, for the other values of m they are only symmetric with nonzero deficiency indexes (equal for all ɛ). It turns out (see [3]) that the self-adjointness of all operators {Hɛ} is equivalent to the one for some auxiliary symmetric operator 𝒯 acting in the space L2(ℝ3) (see below). This operator commutes with the operators {Ug,g∈O3} of the representation of the rotation group O3 that acts in L2(ℝ3) by  the usual formula:
(1.3)(Ugf)(k)=f(g-1k),g∈O3,f∈L2(R3).
Let us denote by ℋl⊂L2(ℝ3) the maximal subspace, where the representation (1.3) is multiplied by the irreducible representation of O3 with weight l, l=0,1,2,… (see [6]). Evidently, the space ℋl is invariant with respect to the operator 𝒯, and the restriction 𝒯l=𝒯|ℋl of this operator to the space ℋl is symmetric operator. The operator 𝒯 is self-adjoint if all the operators {𝒯l,l=0,1,…} are self-adjoint. In this paper, we find general simple conditions of self-adjointness of 𝒯l and the form of the defect subspaces (with small exclusions) when these conditions are broken. Then, we apply these conditions to the cases l=0 and l=1 and get that the operator 𝒯l=0 is self-adjoint for all values of m>0, while the operator 𝒯l=1 is self-adjoint for m>m0 and has nonzero deficiency indexes for m≤m0, the constant m0>0 is indicated below (see (5.4)).By the way, we note that the value ofm0 obtained in this paper differs from that one given by mistake in [2].


## 2. A Short Explanation of the Constructions from Papers [1–3]

(1) After Fourier transformation:(2.1)ψ(y,x1,x2)⟶ψ~(q,k1,k2)=12π9/2∫(R3)3ψ(y,x1x2)exp⁡{-i(q,y)-i(k1,x1)-i(k2,x2)}dydx1dx2≡(Fψ)(q,k1,k2),
and change of variables:
(2.2)P=q+k1+k2,pj=Pm+2-kj,j=1,2,
the operator
(2.3)H~0=FH0F-1,
can be represented as a tensor sum:
(2.4)H~0=H~0(1)+mm+1H~0(2),
where H0(1) is a self-adjoint operator in L2(ℝ3):
(2.5)(H~0(1)f)(P)=P2m+2f(P),P∈R3,f∈L2(R3),
and H~0(2) acts in L2asym(ℝ3×ℝ3) by formula
(2.6)(H~0(2)g)(p1,p2)=G(p1,p2)g(p1,p2),g∈L2asym(R3×R3),
with
(2.7)G(p1,p2)=p12+p22+2m+1(p1,p2)>0.
The operator H~0(2) is symmetric, and its domain is
(2.8)D(H~0(2))={g∈L2asym(R3×R3):∫R3g(p1,p2)dpj=0,j=1,2},(2) the deficiency subspaceℛ-1⊂L2asym(ℝ3×ℝ3) of the operator H~0(2) consists of the functions of the form:
(2.9)U℘(p1,p2)=℘(p1)-℘(p2)G(p1,p2)+1,
where the function ℘(p) belongs to Hilbert space
(2.10)L={℘:∫R3|℘(p)|2p2+1dp<∞},
with inner product
(2.11)〈℘1,℘2〉=(U℘1,U℘2)L2(R3×R3)≡(W℘1,℘2)L2(R3).
Here W is some positive operator acting in L2(ℝ3) (see [3]). The domain of the operator (H~0(2))*, that is, a conjugate to H~0(2), is
(2.12)D((H~0(2))*)={g∈L2asym(R3×R3):g(p1,p2)=f(p1,p2)+U℘(p1,p2)+Uψ(p1,p2)G(p1,p2)+1},
where f∈D(H~0(2)), ℘,ψ∈ℒ. In addition, the operator (H~0(2))* acts by the formula:
(2.13)((H~0(2))*g)(p1,p2)=G(p1,p2)g(p1,p2)-(℘(p1)-℘(p2)),
where ℘ is defined by (2.12).The following asymptotics holds for vectorsg∈D((H~0(2))*)N→∞:
(2.14)∫|p1|<Ng(p1,p2)dp1=4πN℘(p2)+b(p2)+o(1).
Here
(2.15)b(p)=-(T℘)(p)+(Wψ)(p),
where the operator W is defined in (2.11), and (T℘)(p) is given by the following expression (μ=2/(m+1))
(2.16)(T℘)(p)=2π2(1-μ24)p2+1℘(p)+∫R3℘(t)G(t,p)+1dt,
defined on the set:
(2.17)D(T)={℘∈L2(R3):|p|℘(p)∈L2(R3)}.
The above-mentioned Ter-Martirosian-Skornyakov's extension H~ɛ(2) of the operator H~0(2) is obtained by requiring
(2.18)b(p)=ɛ℘(p),
where ɛ∈ℝ1 is an arbitrary parameter.Lemma 2.1.
The operatorT defined in the space L2(ℝ3) by (2.16) is symmetric, and the self-adjointness of the operators Hɛ (for all ɛ) is equivalent to the self-adjointness of the operator T (see [2, 3, 5]).The operatorT can be represented as a sum of two operators:
(2.19)T=T+T′,
where the symmetric operator 𝒯 (with the domain D(𝒯)=D(T)) acts as follows:
(2.20)(T℘)(p)=2π21-μ24|p|℘(p)+∫R3℘(t)dtG(t,p)
and T′ is a bounded self-adjoint operator. Since the deficiency indexes of T coincide with the ones of 𝒯 (see [7]), we shall study the conditions of self-adjointness for the operator 𝒯;(3) as we said, the spaceℋl⊂L2(ℝ3) is invariant with respect to 𝒯; it has the form:
(2.21)Hl=L2(R+1,r2dr)⊗L2l(S),
where L2l(S)⊂L2(S) is the space of spherical functions of weight l (see [6]) on the unit sphere S⊂ℝ3. In addition, the operator 𝒯l=𝒯|ℋl has the form
(2.22)Tl=Ml⊗El,
where El is the unit operator in L2l(s), and Ml acts in L2(ℝ+1,r2dr) by the formula:
(2.23)(Mlf)(r)=2π21-μ24rf(r)+2π∫-11dxPl(x)∫0∞(r′)2f(r′)dr′r2+(r′)2+μrr′x,
on the domain
(2.24)D(Ml)≡V={u∈L2(R+1,r2dr):ru(r)∈L2(R+1,r2dr)}.
Here Pl(x), l=0,1,2,…, x∈[-1,1], are orthogonal polynomials (Legendre polynomials) satisfying Pl(1)=1:
(2.25)Pl(x)=12ll!dldxl(x2-1)l,x∈(-1,1).
The operators {Ml,l=0,1,…} are symmetric in L2(ℝ+1,r2dr), and the self-adjointness of Ml is equivalent to the self-adjointness of 𝒯l. Later on, we shall study the operators Ml and derive a condition of self-adjointness.


## 3. Preparatory Constructions

For every functionu∈V⊂L2(ℝ+1,r2dr), we consider the family of functions
(3.1)h(u)={uα=rαu,α∈[0,1],u0=u},
which we call a chain (with initial element u=u0 and the final one u1). All functions uα∈𝔥(u) belong to L2(ℝ+1,r2dr) and have a uniformly bounded norm:
(3.2)‖uα‖2≤‖u0‖2+‖u1‖2,α∈[0,1].
Consider the unitary map (Mellin's transformation [8]):
(3.3)ω:L2(R+1,r2dr)⟶L2(R1,ds):f(r)⟶f~(s)=12π∫0∞r-is+1/2f(r)dr,s∈R1
and its inverse:
(3.4)(ω-1f~)(r)=12π∫-∞∞ris-3/2f~(s)ds.
For every set of functions B⊂L2(ℝ+1,r2dr), we denote by B~⊂L2(ℝ1,ds) the set of their Mellin's transformations:
(3.5)B~=ωB.
For every chain 𝔥(u), we denote by Γu the family of functions:
(3.6)Γu=h(u)~={γα(s),α∈[0,1]},
where γα(s)=(ωuα)(s), uα∈𝔥(u). The family Γu can be represented as a function Γu(z) of a complex variable z=s+iα in the strip:
(3.7)I={z∈C1:Iz∈[0,1]},Γu(z)=12π∫0∞r-is-1/2+αu(r)dr=12π∫0∞r-izu(r)dr.
The function Γu is said to be associated with the chain 𝔥(u), and its values {γα(s)} on the lines ξα={z=s+iα,s∈ℝ1,0≤α≤1}⊂I are called the sections of Γu.Proposition 3.1.
For every chain𝔥(u), u∈V, the associated function {Γu(z),z∈I} is continuous in a closed strip I and analytic inside this strip. Moreover, its sections {γα} satisfy the following inequality:
(3.8)sup⁡0≤α≤1‖γα(⋅)‖L2(R1)<∞.
Inversely, any function {Γ(z),z∈I} which possesses these properties is associated with some (unique) chain  𝔥(v):Γ=Γv, v∈V. Let call this chain generated by Γ. In addition, the functions {vα,α∈[0,1]} of the chain 𝔥(v) are obtained by the inverse Mellin's transformation from the sections of Γ={γα}:
(3.9)vα=ω-1γα.The proof of this proposition can be obtained by using the arguments given in the book by Paley and Wiener (see [9], Chapter I), which are related to the Fourier transformation of functions analytical in a strip in a complex plane. It is not difficult to reformulate these arguments in terms of Mellin's transformation.Note that the estimate (3.8) for {γα} follows from the estimate (3.2) and the unitary Mellin's transformation. Denote by 𝒢 a linear space of functions Γ satisfying conditions of Proposition 3.1. Let us introduce two maps:
(3.10)Ω:h(u)⟶Γu∈G,Ω-1:Γu⟶h(u).
Let N(z), z∈I, be a bounded, continuous function in the strip I, which is analytic inside I. This function generates the family κ~αN,α∈[0,1] of bounded operators in L2(ℝ1) which act as multiplication on the functions nαN(s)=N(z)|z=s+iα, s∈ℝ1, 0≤α≤1:
(3.11)(κ~αNψ)(s)=nαN(s)ψ(s),ψ∈L2(R1).
Evidently, for any Γ∈𝒢, the function N(z)Γ(z) belongs to 𝒢. If the chain 𝔥(u) is generated by Γ=Γu and the chain 𝔥(v) is generated by N(z)Γ(z)=Γv(z), then
(3.12)vα=καNuα,α∈[0,1],uα∈h(u),
where
(3.13)καN=ω-1κ~αNω.
Denote by Π the following self-adjoint operator in L2(ℝ+1,r2dr):
(3.14)(Πf)(r)=rf(r),
with the domain D(Π)=V.It is clear that for anyu∈V, the power Πβ, 0≤β≤1 of the operator Π is applicable to an element uα∈𝔥(u) if β+α≤1 and
(3.15)Πβuα=uα+β.
For the function Γu that is associated with 𝔥(u),the action of the operator Π~β=ωΠβω-1 on the sections  {γα}  of  Γu has the form:
(3.16)Π~βγα=γα+β.
(again if α+β≤1).


## 4. The OperatorMl

The operatorMl (see (2.23)) can be represented as
(4.1)Ml=Π1/2κlΠ1/2,
where κl=κ1/2l is an operator in L2(ℝ+1,r2dr) acting by the formula:
(4.2)(κ1/2lf)(r)=2π21-μ24f(r)+2π∫-11dxPl0(x)∫0∞(r′)2f(r′)dr′(rr′)1/2(r2+(r′)2+μxrr′).Lemma 4.1.
Operatorκ1/2l is bounded and self-adjoint in L2(ℝ+1,r2dr).Proof.
Pass to the operator:(4.3)κ~1/2l=ωκ1/2lω-1,
acting in L2(ℝ1). It follows from calculations in [2, 3] that κ~1/2l is the operator of multiplication on the function:
(4.4)n1/2l(s)=2π2(1-μ24+λ1/2l(s)),
where
(4.5)λ1/2l(s)={∫01Pl(x)ch(sv(x))dxch(sπ/2)cos⁡⁡(v(x))forevenl,-∫01Pl(x)sh(sv(x))dxsh(sπ/2)cos⁡⁡(v(x))foroddl,
and v(x)=arcsinμx/2, 0≤x≤1. As we see the function n1/2l(s), s∈ℝ1, is bounded and real. The lemma is proved.We see from (4.4) and (4.5) that the functions n1/2l(s) and λ1/2l are continued up to bounded, analytical functions Nl(z) and Λl(z) correspondingly, defined in the strip I~={z∈ℂ1:-1/2≤ℑz≤1/2}. Let us define the functions N^l(z)=N¯l(z-i/2) which we shall consider in the strip I={z∈ℂ:0≤ℑz≤1}. The operator κ~1/2l coincides with the operator κ~1/2N^l from the family {κ~αN^l} generated by the function N^l (see (3.11)). Any other operator of this family acts as multiplication on the function:
(4.6)n^αl(s)=N^l(z)|z=s+iα.
Denote by καl the operators
(4.7)καl=ω-1κ~αNlω,
acting in L2(ℝ+1,r2dr).Note that(4.8)(καl)*=κi-αl.
It is convenient to represent the operator Ml in form of three sequential maps
(4.9)Ml:u0∈h(u0)⟶Π1/2u0=u1/2⟶κ1/2lu1/2=v1/2⟶Π1/2v1/2=v1∈h(v),
where v=v0,v1/2,v1 are elements of the chain 𝔥(v) generated by the function Γv=N^lΓu∈𝒢. Note that the chain (4.9) can be rewritten in the following way:
(4.10)u0∈h(u0)→ΩΓu0⟶Γv=N^lΓu0→Ω-1h(v)⟶v1∈h(v).
From (4.1) and self-adjointness of κ1/2l it follows that the operator Ml with the domain D(Ml)=V is symmetric. For any α∈[0,1], a representation of Ml similar to (4.1) is valid:
(4.11)Ml=Π1-ακαlΠα
as well as decomposition like (4.9).Let us now describe the domainD(Ml*)⊇V of the operator Ml* conjugated to Ml. Let g∈D(Ml*) be a function from D(Ml*) and h=Ml*g∈L2(ℝ+1,r2dr). Then for every u∈V=D(Ml), we can write
(4.12)(Mlu,g)=(κ1lΠu,g)=(Πu,(κ1l)*g)=(u,Πκ0lg)=(u,h).
Here we use the representation (4.11) for α=1 and the equality (4.8). Denote f(r)=h(r)-(Πκ0lg)(r) and apply the following evident assertion.Lemma 4.2.
Let a measurable functionf(r) satisfies condition
(4.13)∫0∞f(r)u(r)r2dr=0,
for any u∈V. Then f=0.From this and (4.12), it follows that
(4.14)Πκ0lg=h.
Hence
(4.15)w0≡κ0lg∈V,
and h=w1∈𝔥(w0) is the final element of the chain 𝔥(w0). Thus the domain D(Ml*) of the operator Ml* is
(4.16)D(Ml*)={g∈L2(R+1,r2dr):κ0lg∈V}.
In the case when the operator κ0l has the inverse one, (κ0l)-1, which is equivalent to the condition:
(4.17)n^0l(s)≠0,foranys∈R1,
the following equality is true:
(4.18)D(Ml*)=(κ0l)-1V.
Let M~l*=ωMl*ω-1 be an operator in L2(ℝ1) with domain D(M~l*)=ωD(Ml*). Then for g~∈D(M~l*), the following representation holds true:
(4.19)g~(s)=(n^0l(s))-1w~0(s)=(N^l(z))-1Γw0(z)|z=s,
if condition (4.17) is fulfilled. Here w~0(s)=(ωw0)(s) where w0 is defined in (4.15).Remarks.
(1) Note that the functionN^l(z) is invariant with respect to reflection of the complex plane around the point z=i/2:
(4.20)z⟶z*=-z+i.
Under this reflection, the strip I is mapped onto itself; hence, for every zero z¯∈I(z¯≠i/2) of the function N^l, there exists another zero, z¯*∈I, of N^l with the same multiplicity. The multiplicity of z¯=i/2=z¯* is even;(2) SinceN^l(z)→2π21-μ2/4>0 as z→∞ inside I, the function N^l(z) has finite number of zeros inside I.We can now formulate the main criterion of self-adjointness of the operatorMl.Theorem 4.3.
The operatorMl is self-adjoint if and only if the function N^l(z) has no zeros in the closed strip I.Proof.
(1) AssumeN^l(z)≠0 in the strip I. Then (N^l)-1(z) is bounded and continuous on I and analytical inside I. Let g~∈D~(M~l*). Since n^l(s)≠0 for s∈ℝ1, the representation (4.19) holds true. Since
(4.21)(N^l(z))-1Γw0(z)=Γv∈G,v∈V,
the element g=ω-1g~∈D(Ml*) coincides with v∈V, that is, D(Ml*)=V=D(Ml); it means the self-adjointness of Ml;
(2) assume now the functionNl(z) has zeros z¯1,…,z¯k∈I. Consider first the case when all zeros are lying inside I and their multiplicities are equal to p1,…,pk, respectively. Again, let g~∈D~(M~l*). Since n^l(s)≠0, the representation (4.19) holds true. The function (N^l(z))-1Γw0(z) is meromorphic in I with poles z¯1,…,z¯k having the order p1,…,pk respectively. For this function the usual canonical representation [10] is true:
(4.22)(N^l(z))-1Γw0(z)=Lw0(z)+∑n=1k‍∑m=1pn‍bm(n)(w0)(z-zn)m,
where Lw0(z) is bounded, continuous function on I, and analytical inside I, and the coefficients bm(n)=bm(n)(w0) depend on w0.
Lemma  4.4.The functionLw0(z) in (4.22) belongs to the space𝒢.
The proof of this lemma is given in The appendix.
From (4.19) and (4.22), for g=ω-1g~∈D(Ml*), we have
(4.23)g(r)=v(r)+∑m,n‍bm(n)(ω-1((1⋅-zn)m))(r),
where the function v∈V is defined from relation
(4.24)Lw0(z)=Γv(z)∈G,dm,n(r):=ω-1((1(⋅-zn))m)(r)=Am(n)r-3/2-tn+isn(ln⁡r)m-1χ(r),
where Am(n) is an absolute constant, zn=sn+itn, 0<tn<1 and
(4.25)χ(r)={1,r>1,0,r≤1.
Since linearly independent functions dm,n∈D(Ml*) do not belong to V, due to (4.23), they form the basis in the defect subspace 𝔙 of the operator Ml (see [7]). Since the dimension of the subspace 𝔙 is equal to ∑1k‍pn and the operator Ml is real, its deficiency indexes n± are equal and have the form:
(4.26)n+=n-=12∑1k‍pn.
(It follows from Remarks that the sum ∑1k‍pn is even). Consider now the case when one of the zeros of Nl(z), say, z¯0=s0∈ℝ1, lies on the boundary of I and has multiplicity p (in addition, there is a zero z¯0*=s0+i). In this case, in a neighborhood of z¯0, the function N^l(z) has the form:
(4.27)N^l(z)=(z-z¯0)pQ(z),
where Q(z) is analytic in this neighborhood. Consider the function,
(4.28)G(z)=1(-i(z-z¯0))1/31(z+2i)2,
whereby (-iw)1/3 for ℑw>0, we mean the branch of a many-valued function (-iw)1/3 that takes positive values on the positive part of the imaginary axis. Evidently, the function G(z) is analytic in the strip I and satisfies condition (3.8). However, this function is discontinuous at z¯0 and does not belong to 𝒢. In addition, the function N^(z)lG(z) now belongs to 𝒢 as follows from (4.27) and (4.28). Thus
(4.29)g~(s)=G(z)|z=s∈¯V~=ωV,
but
(4.30)n^l(s)g~(s)=N^l(z)G(z)|z=s∈V~.
Consequently, g=ωg~∈¯V but κ0lg∈V, that is, g∈D(Ml*). Thus D(Ml*)≠V, and the operator Ml has nonzero deficiency indexes. Theorem 4.3 is proved.


## 5. The OperatorsMl in the Cases l=0 and l=1

Here, we apply Theorem4.3 to the cases l=0 and l=1.Theorem 5.1.
(1) Forl=0, the operator Ml=0  is self-adjoint for any m>0;
(2) the operatorMl=1 is self-adjoint for m>m0 and has nonzero deficiency indexes for m≤m0. In addition, for m<m0 these indexes are equal to (1,1). The constant m0 is a unique zero of (5.4).Proof.
We need the following properties of the functionsΛ^l=0(z) and Λ^l=1(z), z∈I.
Lemma  5.2. (1)For anyl=0,1,2,…  the functionΛ^l(z)  is invariant with respect to reflection (4.20);
(2)The pointz=i/2∈I  is a nondegenerate critical point for both functionsΛ^l=0  andΛ^l=1;
(3)These functions take real values on the line:
(5.1)ξ^1/2={z=s+i2,s∈R1},and on the segment:
(5.2)τ^={z=it,0≤t≤1}.Outside the setB=ξ^1/2∪τ^,  both functions take nonreal values;
(4)the real values ofΛ^l, l=0,1,  are between0  andΛ^l(0)=Λ^l(i). Every value ofΛ^l|B—exceptΛ^l(i/2) —is taken exactly at two points;
(5)the extreme values ofΛ^l, l=0,1,   Λ^l(0)=Λ^l(i)  are given by(5.3)Λ^l=0(0)=82π2μ-1sin⁡(12arcsinμ2)>0,Λ^l=1(0)=-3232π2μ-2sin⁡3(12arcsinμ2)≡-q(μ)<0,
(6)the functionq(μ)  increases monotonically on the interval  0<μ<2.
The proof of this lemma is given in The appendix.
Corollary  5.3.(1) The zeros of  N^l(z), l=0,1can only lie in the setB;
(2)N^l=0(z)|B>0   for any value ofμ, and therefore the operatorMl=0  is self-adjoint for allm∈(0,2);
(3)The functionN^l=1(z)|B  is positive if2π21-μ2/4>q(μ)  and vanishes at some pointz∈B (and also atz*∈B) if2π21-μ2/4≤q(μ).
In Figure1, the curves corresponding to the functions 2π21-μ2/4 and q(μ) are depicted. We see that they intersect at a unique point with abscissa μ0∈(0,2) which satisfies the following equation:
(5.4)2π21-μ024=q(μ0).
Thus, form>m0=2/μ0-1 the operator Ml=1 is self-adjoint, and for m<m0 it has deficiency indexes (1,1). For m=m0, the operator Ml=1 is not self-adjoint as well. Theorem 5.1 is proved.Figure 1



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*Source: 230245-2012-07-30.xml*