# Scattering Theory for 2N Parameter Models of Finitely Many Relativistic δ-Sphere and δ-Sphere plus Coulomb Interactions Supported by Concentric Spheres in Quantum Mechanics

Authors: A. Vyabandi; J. Shabani
Journal: Advances in Mathematical Physics
 (2020)

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## Abstract

We study scattering theory for2N parameter models of finitely many relativistic δ-sphere and δ-sphere plus Coulomb interactions. We provide the mathematical definitions of the Hamiltonians, solve the resolvent equations, and compute the nonrelativistic limits for both models. We obtain new results related to spectral properties and scattering data.


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## Body

## 1. Introduction

Over the past three decades,δ-sphere interactions in quantum mechanics have been the subject of several research studies, both from the mathematical point of view and for their applications to the modeling of physical phenomena. The Hamiltonians describing these interactions are defined using the theory of self-adjoint extensions of closed symmetric operators in Hilbert spaces. Initially, the emphasis was placed on studying these interactions in nonrelativistic mechanics [1–10]. The extension of this research work to relativistic quantum mechanics began in 1989 [11–16].In [17], we studied Dirac Hamiltonians with delta interactions and delta plus Coulomb interactions. We obtained a series of new results for this model including the resolvent equation, the spectral properties, the nonrelativistic limit, and the various quantities related to the scattering theory and the generalization of these results to the case of a delta plus Coulomb interaction.In this paper, we extend the results obtained in [17] to the case of Dirac Hamiltonian with finitely many delta and delta plus Coulomb interactions with support on N concentric spheres.The paper is organized as follows: in Section2, we provide a rigorous mathematical definition of the Dirac Hamiltonian with finitely many delta interactions supported by N concentric spheres and generalize all results obtained in [17] to this case. In Section 3, we expand the results obtained in Section 2 to the case of a Dirac Hamiltonian with finitely many spheres plus Coulomb interactions with support on N concentric spheres.


## 2. Basic Properties for the2N Parameter Model of Finitely Many Relativistic δ-Sphere Interactions: Separated Boundary Conditions

### 2.1. Definition of the Model

In this section, we provide in dimensionn=3 the rigorous mathematical definition of finitely many relativistic δ-sphere interactions. The formal expression of the hamiltonian describing the 2N parameter model of finitely many δ-sphere interactions with support on concentric spheres of radii 0<R1<⋯<RN is given by
(1)H=HD+∑n=1NΓnδ∣x∣−Rn,x∈ℝ3,0<R1<⋯<RN,where we define the Dirac Hamiltonian in Hilbert space H=L2ℝ3⊗ℂ4 by
(2)HD=−iα¯∇+β¯c22,DHD=H1,2ℝ3⊗ℂ4.The4×4 matrix Γn is defined by
(3)Γn=A¯n00B¯n,n=1,⋯,N,and A¯n=An1001 and B¯n=Bn1001.An and Bn,n=1,⋯,N, are 2N parameters corresponding to the relativistic δ-sphere interaction supported by the sphere of radius Rn. c,Hm,n,α¯, and β¯ are, respectively, defined by the following: c is the velocity of the light, Hm,n is the Sobolev space of indices m,n, and α¯ and β¯ are 4×4 Dirac matrices given by
(4)α¯=0σσ0,β¯=1l001l.σ are Pauli’s spin matrices defined by
(5)σ1=0110,σ2=0−ii0,σ3=100−1.ConsideringL2ℝ3⊗ℂ4 the symmetric closed operator, H¯D is defined by
(6)H¯D=−iα¯∇+β¯c22,H¯D=ψ∈H1,2ℝ3⊗ℂ4ψSRn=0,1<n≤N,where SRn=x∈ℝ3:∣x∣=Rn is the sphere of radius Rn.Let us look for self-adjoint extensions ofH¯D which are rotationally and space-reflection symmetric.The Hilbert spaceH can be decomposed as follows:
(7)H=⨁j=1/2∞⨁l=j−1/2j+1/2⨁μ=−jjHjlμ,where
(8)Hjlμ=ψ∈Hψr,n=frΩjlμgrΩjl′μ,f,g∈L20,∞,r2dr.The spherical spinorsΩjlμ are defined by [18]
(9)Ωjlμ=j+μ2l+1Yl,μ−1/2θ,φj−μ2l+1Yl,μ+1/2θ,φ,forl=j−12,Ωjl′μ=−j−μ+12l+1Yl,μ−1/2θ,φj+μ+12l+1Yl,μ+1/2θ,φ,forl=j+12.We havel′=j∓1/2 for l=j±1/2. The Hilbert space H then takes the following form:
(10)H=⨁j=1/2∞⨁l=j−1/2j+1/2⨁μ=−jjUjl−1H~⊗Ωjlμθ,φ,where the isomorphism Ujl is defined by
(11)Ulj:L20,∞;r2dr⊗ℂ2⟶H~=L20,∞;dr⊗ℂ2,Ujlψr=rfr−1j−l−1/2rgr,and Ωjlθ,φ represents the vector space formed by the spherical spinors.Following decomposition (10), the operatorH¯D reads
(12)H¯D=⨁j=1/2∞⨁l=j−1/2j+1/2Ujl−1hjl,RUjl⊗1l,where the radial Dirac operator hjl,R reads
(13)hjl,R=c22−cddr+cκjlrcddr+cκjlr−c22≡τ,κjl=−j−l+1/2j+12,Dhjl,R=ψ∈L20,∞⊗ℂ2∣ψ∈ACloc0,∞,ψRn±=0,τψ∈L20,∞⊗ℂ2,R=R1,R2,⋯,RN,where AClocΩ denotes the set of locally absolutely continuous functions on Ω and
(14)ψx±=limε→0+ψx±ε.The adjointhjl,R∗ of hjl,R reads
(15)hjl,R∗=τ,Dhjl,R∗=f∈L20,∞⊗ℂ2∣f∈ACloc0,∞\R,τf∈L20,∞⊗ℂ2.A straightforward computation shows that the equation(16)hjl,R∗−zφ=0,φ∈ℂ−−∞,−c22⋃c22,∞,has 2N linearly independent solutions
(17)φjl,n1r=Gjlz,RnFjlz,rG~jlz,RnF~jlz,r,r<Rn,00,r>Rn,n=1,⋯,N,φjl,n2r=00,r<Rn,Fjlz,RnGjlz,rF~jlz,RnG~jlz,r,r>Rn,n=1,⋯,N,where
(18)Fjlz,r=k2−κjl−1/2Γκjl+32r1/2Jκjl+1/2kr,(19)F~jlz,r=1c12−κjl−1/2k−κjl+1/2Γκjl+32r1/2Jκjl−1/2kr,(20)Gjlz,r=iπ2k2κjl+1/21/2Γκjl+32r1/2Hκjl+1/21kr,(21)G~jlz,r=iπ2c12κjl+1/2kκjl+3/2Γκjl+32−1r1/2Hκjl−1/21kr.Jν∙ is the Bessel function and Hν1∙ the Hankel function of the first type of order ν.Therefore, all self-adjoint extensions ofhjl,R are given by a 4N2 parameter family of self-adjoint operators.In this section, we consider the following special2N parameter family of self-adjoint extensions of hjl,R:
(22)hjl,Γ,R=c22−cddr+cκjlr−cddr+cκjlr−c22,(23)Dhjl,Γ,R=g∈Dhjl,Γ,R∗1−τ0Γjl,n2cgRn+−1+τ0Γjl,n2cgRn−=0,where the 2×2 matrix Γjl,n reads
(24)Γjl,n=Ajl,n00Bjl,n,Ajl,n,Bjl,n∈ℝ.The Hamiltonianhjl,Γ,R gives the mathematical definition of the formal expression
(25)hΓ,R=hD+∑n=1NΓjl,nδr−Rn,Rn>0,n=1,2,⋯,N,where hD is the radial Dirac operator defined by
(26)hD=c22−cddr+cκjlr−cddr+cκjlr−c22,(27)DhD=H1,20,∞⊗ℂ2.Given decomposition (10), a rigorous mathematical definition of formal expression (1) reads(28)HΓ,R=⨁j=1/2∞⨁l=j−1/2j+1/2Ujl−1hjl,Γ,RUjl⊗1l.The caseΓ=0, i.e., Γjl,n=0 for all j and l in equation (28), yields the Dirac Hamiltonian HD defined by equation (2).The caseAjl,n≠0 and Bjl,n=0 for all j and l in equation (28) yields the relativistic N-parameter δ-sphere interactions of the first type [17].The caseAjl,n=0 and Bjl,n≠0 for all j and l in equation (28) yields the relativistic N-parameter δ-sphere interactions of the second type [17].


### 2.2. Resolvent Equation ofhjl,Γ,R

Theorem 1.
The resolvent ofhjl,Γ,R reads
(29)hjl,Γ,R−z−1=hD−z−1+∑n,m=1Nϑnm1zM^jl,z2∙¯,.M^jl,z2∙+ϑnm2zM~jl,z2∙¯,.M~jl,z2∙+ϑnm3zM^jl,z2∙¯,.M~jl,z1∙+ϑnm4zM~jl,z2∙¯,.M^jl,,z1∙,z∈ρhjl,Γ,R,Imk>0,l∈j−12,j+12,j∈12,∞,where ρ is the resolvent set and
(30)ϑ1znm−1=−1Ajl−Bjl4c2+BjlAjlG22jlz,Rn,Rn+G11jlz,Rn,Rn,n=m,−G11jlz,Rm,Rn,n≠m,(31)ϑ2znm−1=−1Bjl−Ajl4c2+AjlBjlG11jlz,Rn,Rn+G22jlz,Rn,Rn,n=m,−G22jlz,Rm,Rn,n≠m,(32)ϑ3znm−1=−2cBjlAjl−12c+2cBjlG11jlz,Rn,Rn+2cAjlG22jlz,Rn,Rn,n=m,12c,n≠m,(33)ϑ4znm−1=−−2cBjlAjl+12c−2cBjlG11jlz,Rn,Rn−2cAjlG22jlz,Rn,Rn,n=m,−12c,n≠m,(34)M^jl,zmr=Gjlz,r,Rn10,r<Rn,−mGjlz,r,Rn10,r>Rn,m=1,2,n=1,2,⋯,N,(35)M~jl,zmr=Gjlz,r,Rn01,r<Rn,−mGjlz,r,Rn01,r>Rn,m=1,2,n=1,2,⋯,N.
hD−z−1,Imk>0, is the radial Dirac resolvent with kernel:
(36)Gjlz,r,Rn=G11jlz,r,RnG12jlz,r,RnG21jlz,r,RnG22jlz,r,Rn,where
(37)G11jlz,r,Rn=Gjlz,RnFjlz,r,r<Rn,Fjlz,RnGjlz,r,r>Rn,G12jlz,r,Rn=G~jlz,RnFjlz,r,r<Rn,F~jlz,RnGjlz,r,r>Rn,G21jlz,r,Rn=Gjlz,RnF~jlz,r,r<Rn,Fjlz,RnG~jlz,r,r>Rn,G22jlz,r,Rn=G~jlz,RnF~jlz,r,r<Rn,F~jlz,RnG~jlz,r,r>Rn.Proof.
Equation (29) follows from Krein’s formula [19].
Letϕ=ϕ1ϕ2∈L0,∞⊗ℂ2 and define the function ξjl by
(38)ξjlr=hjl,Γ,R−z−1ϕr.
Sinceξjl∈Dhjl,Γ,R, it follows that ξjl satisfies the separated boundary conditions in equation (23).The implementation of these boundary conditions provides equation (29).


### 2.3. Spectral Properties ofhjl,Γ,R

Theorem 2.
ForAjl,n,Bjl,n∈ℝ,l∈j−1/2,j+1/2,j∈1/2,∞, the essential spectrum of hjl,Γ,R is purely absolutely continuous and coincides with −∞,−c2/2∪c2/2,∞. Its singularly continuous and residual spectra are empty.Proof.
Follow step by step the proof of theorem 6.2 and proposition 6.1 in ref. [12].According to decomposition (10), equations (12) and (29), the resolvent equation of HΓ,R reads
(39)HΓ,R−z−1=HD−z−1+⨁j=1/2∞⨁l=j+1/2j−1/2⨁μ=−jj×∑n,m=1Nϑnm1z.−1M^jl,z2.¯⊗Ω~jlμ,.×.−1M^jl,z2.⊗Ω~jlμ+ϑnm2z.−1M~jl,z2.¯⊗Ω~jlμ,.×.−1M~jl,z2.⊗Ω~jlμ+ϑnm3z.−1M^jl,z2.¯⊗Ω~jlμ,.×.−1M~jl,z1.⊗Ω~jlμ+ϑnm4z.−1M~jl,z2.¯⊗Ω~jlμ,.×.−1M^jl,z1.⊗Ω~jlμ,z∈ρhjl,Γ,R,Imk>0,l∈j−12,j+12,j∈12,∞,where the notations M^jl,zn.⊗Ω~jlμ and M~jl,zn.⊗Ω~jlμ,n=1,2, mean
(40)M^jl,zn.⊗Ω~jlμ=M^jl,z,1n.ΩjlμM^jl,z,2n.Ωjl′μ,n=1,2,M~jl,zn.⊗Ω~jlμ=M~jl,z,1n.ΩjlμM~jl,z,2n.Ωjl′μ,n=1,2.


### 2.4. The Nonrelativistic Limit

The nonrelativistic limit ofhjl,Γ,R as c⟶∞ is given by the following.Theorem 3.
For the spin1/2 particules, the operator hjl,Γ,R−c2/2 converges in norm resolvent sense to the Schrödinger operator hl,αl,R times the projector onto H~+=L20,∞:
(41)n·limc→∞hjl,Γ,R−c22−z−1=hjl,α,R−z−1⊗1000,z∈ℂ\ℝ,where the operator hjl,α,R is defined by
(42)hjl,α,R=−d2dr2+ll+1r2,Dhjl,α,R=f∈L20,∞f,f′∈ACloc0,∞\R1,⋯,RN,g0+=0,fRn+−fRn−=βl,n2f′Rn++f′Rn−,f′Rn+−f′Rn−=αl,n2fRn++fRn−,−f″+ll+1r−2f∈L20,∞,l≥1,αl,nβl,n−4=0,αl,n,βl,n∈ℝ,1≤n≤N.Proof.
One can use the strategy of Gesztesy and Seba [15] where a similar case was discussed for point interactions.
The Hamiltonianhjl,α,R describes a 2N parameter model of finitely many nonrelativistic δ-sphere interactions in quantum mechanics.


### 2.5. Scattering Theory for the Pairhjl,α,R,hD

Let us define fork>0 the function
(43)Fjl,Γ,Rr=FjlrF~jlr+∑n,m=1ϑnm1zFjlz,RnM^jl,z2r+ϑnm2zF~jlz,RnM~jl,z2r+ϑnm3zFjlz,RnM~jl,z1r+ϑnm4zF~jlz,RnM^jl,z1r,where functions Fjlz,r,F~jlz,r,M~jl,z2r,M^jl,z2r,M~jl,z1r,M^jl,z1r, and ϑnmiz,i=1,2,3,4, are, respectively, defined by equations (18), (35), and (33).The asymptotic behaviour of the functionFjl,Γ,Rr as r⟶∞ yields [20]
(44)Fjl,Γ,Rr→r→∞k>0A^jlzsinkr−κjlπ2B^jlzsinkr−κjl−1π2+∑n,m=1Nϑnm′zC^jlzexp−ikr−κjlπ2O^jlzexp−ikr−κjl−1π2=A^jlz−iC^jlz∑n,m=1Nϑnm′zsinkr−κjlπ2B^jlz−iO^jlz∑n,m=1Nϑnm′zsinkr−κjl−1π2+C^jlz∑n,m=1Nϑnm′zcoskr−κjlπ2O^jlz∑n,m=1Nϑnm′zcoskr−κjl−1π2,where
(45)ϑnm′z=ϑnm1zFjlz,RnFjlz,Rn+ϑnm2zF~jlz,RnF~jlz,Rn,A^jlz=2−κjlk−κjl−1Γ2κjl+2Γκjl+1−1,C^jlz=2κjlkκjlΓ2κjl+2−1Γκjl+1,B^jlz=1c2−κjlk−κjlΓ2κjl+2Γκjl+1−1,O^jlz=1c2κjlkκjl+1Γ2κjl+2−1Γκjl+1.A straight computation shows that equation (44) reads
(46)Fjl,Γ,Rr→r→∞k>0P~jl,12z+P~jl,22z1/2sinkr−κjlπ2+δjl,Γ,R+01P~jl,32z+P~jl,42z1/2sinkr−κjl−1π2+δjl,Γ,R.Then, the phase shiftδjl,Γ,R corresponding to hjl,α,R is defined by
(47)δjl,Γ,R=−arctanP~jl,22zP~jl,12z,=−arctanP~jl,42zP~jl,32z,=−arctanC^jlz∑n,m=1Nϑnm′zA^jlz−iC^jlz∑n,m=1Nϑnm′z.The elements of the on-shell scattering matrix are given by(48)Sjl,Γ,R=exp2iδjl,Γ,Rz.The partial wave scattering amplitude is given by(49)fjl,Γ,R=exp2iδjl,Γ,Rz−12ik.


## 2.1. Definition of the Model

In this section, we provide in dimensionn=3 the rigorous mathematical definition of finitely many relativistic δ-sphere interactions. The formal expression of the hamiltonian describing the 2N parameter model of finitely many δ-sphere interactions with support on concentric spheres of radii 0<R1<⋯<RN is given by
(1)H=HD+∑n=1NΓnδ∣x∣−Rn,x∈ℝ3,0<R1<⋯<RN,where we define the Dirac Hamiltonian in Hilbert space H=L2ℝ3⊗ℂ4 by
(2)HD=−iα¯∇+β¯c22,DHD=H1,2ℝ3⊗ℂ4.The4×4 matrix Γn is defined by
(3)Γn=A¯n00B¯n,n=1,⋯,N,and A¯n=An1001 and B¯n=Bn1001.An and Bn,n=1,⋯,N, are 2N parameters corresponding to the relativistic δ-sphere interaction supported by the sphere of radius Rn. c,Hm,n,α¯, and β¯ are, respectively, defined by the following: c is the velocity of the light, Hm,n is the Sobolev space of indices m,n, and α¯ and β¯ are 4×4 Dirac matrices given by
(4)α¯=0σσ0,β¯=1l001l.σ are Pauli’s spin matrices defined by
(5)σ1=0110,σ2=0−ii0,σ3=100−1.ConsideringL2ℝ3⊗ℂ4 the symmetric closed operator, H¯D is defined by
(6)H¯D=−iα¯∇+β¯c22,H¯D=ψ∈H1,2ℝ3⊗ℂ4ψSRn=0,1<n≤N,where SRn=x∈ℝ3:∣x∣=Rn is the sphere of radius Rn.Let us look for self-adjoint extensions ofH¯D which are rotationally and space-reflection symmetric.The Hilbert spaceH can be decomposed as follows:
(7)H=⨁j=1/2∞⨁l=j−1/2j+1/2⨁μ=−jjHjlμ,where
(8)Hjlμ=ψ∈Hψr,n=frΩjlμgrΩjl′μ,f,g∈L20,∞,r2dr.The spherical spinorsΩjlμ are defined by [18]
(9)Ωjlμ=j+μ2l+1Yl,μ−1/2θ,φj−μ2l+1Yl,μ+1/2θ,φ,forl=j−12,Ωjl′μ=−j−μ+12l+1Yl,μ−1/2θ,φj+μ+12l+1Yl,μ+1/2θ,φ,forl=j+12.We havel′=j∓1/2 for l=j±1/2. The Hilbert space H then takes the following form:
(10)H=⨁j=1/2∞⨁l=j−1/2j+1/2⨁μ=−jjUjl−1H~⊗Ωjlμθ,φ,where the isomorphism Ujl is defined by
(11)Ulj:L20,∞;r2dr⊗ℂ2⟶H~=L20,∞;dr⊗ℂ2,Ujlψr=rfr−1j−l−1/2rgr,and Ωjlθ,φ represents the vector space formed by the spherical spinors.Following decomposition (10), the operatorH¯D reads
(12)H¯D=⨁j=1/2∞⨁l=j−1/2j+1/2Ujl−1hjl,RUjl⊗1l,where the radial Dirac operator hjl,R reads
(13)hjl,R=c22−cddr+cκjlrcddr+cκjlr−c22≡τ,κjl=−j−l+1/2j+12,Dhjl,R=ψ∈L20,∞⊗ℂ2∣ψ∈ACloc0,∞,ψRn±=0,τψ∈L20,∞⊗ℂ2,R=R1,R2,⋯,RN,where AClocΩ denotes the set of locally absolutely continuous functions on Ω and
(14)ψx±=limε→0+ψx±ε.The adjointhjl,R∗ of hjl,R reads
(15)hjl,R∗=τ,Dhjl,R∗=f∈L20,∞⊗ℂ2∣f∈ACloc0,∞\R,τf∈L20,∞⊗ℂ2.A straightforward computation shows that the equation(16)hjl,R∗−zφ=0,φ∈ℂ−−∞,−c22⋃c22,∞,has 2N linearly independent solutions
(17)φjl,n1r=Gjlz,RnFjlz,rG~jlz,RnF~jlz,r,r<Rn,00,r>Rn,n=1,⋯,N,φjl,n2r=00,r<Rn,Fjlz,RnGjlz,rF~jlz,RnG~jlz,r,r>Rn,n=1,⋯,N,where
(18)Fjlz,r=k2−κjl−1/2Γκjl+32r1/2Jκjl+1/2kr,(19)F~jlz,r=1c12−κjl−1/2k−κjl+1/2Γκjl+32r1/2Jκjl−1/2kr,(20)Gjlz,r=iπ2k2κjl+1/21/2Γκjl+32r1/2Hκjl+1/21kr,(21)G~jlz,r=iπ2c12κjl+1/2kκjl+3/2Γκjl+32−1r1/2Hκjl−1/21kr.Jν∙ is the Bessel function and Hν1∙ the Hankel function of the first type of order ν.Therefore, all self-adjoint extensions ofhjl,R are given by a 4N2 parameter family of self-adjoint operators.In this section, we consider the following special2N parameter family of self-adjoint extensions of hjl,R:
(22)hjl,Γ,R=c22−cddr+cκjlr−cddr+cκjlr−c22,(23)Dhjl,Γ,R=g∈Dhjl,Γ,R∗1−τ0Γjl,n2cgRn+−1+τ0Γjl,n2cgRn−=0,where the 2×2 matrix Γjl,n reads
(24)Γjl,n=Ajl,n00Bjl,n,Ajl,n,Bjl,n∈ℝ.The Hamiltonianhjl,Γ,R gives the mathematical definition of the formal expression
(25)hΓ,R=hD+∑n=1NΓjl,nδr−Rn,Rn>0,n=1,2,⋯,N,where hD is the radial Dirac operator defined by
(26)hD=c22−cddr+cκjlr−cddr+cκjlr−c22,(27)DhD=H1,20,∞⊗ℂ2.Given decomposition (10), a rigorous mathematical definition of formal expression (1) reads(28)HΓ,R=⨁j=1/2∞⨁l=j−1/2j+1/2Ujl−1hjl,Γ,RUjl⊗1l.The caseΓ=0, i.e., Γjl,n=0 for all j and l in equation (28), yields the Dirac Hamiltonian HD defined by equation (2).The caseAjl,n≠0 and Bjl,n=0 for all j and l in equation (28) yields the relativistic N-parameter δ-sphere interactions of the first type [17].The caseAjl,n=0 and Bjl,n≠0 for all j and l in equation (28) yields the relativistic N-parameter δ-sphere interactions of the second type [17].


## 2.2. Resolvent Equation ofhjl,Γ,R

Theorem 1.
The resolvent ofhjl,Γ,R reads
(29)hjl,Γ,R−z−1=hD−z−1+∑n,m=1Nϑnm1zM^jl,z2∙¯,.M^jl,z2∙+ϑnm2zM~jl,z2∙¯,.M~jl,z2∙+ϑnm3zM^jl,z2∙¯,.M~jl,z1∙+ϑnm4zM~jl,z2∙¯,.M^jl,,z1∙,z∈ρhjl,Γ,R,Imk>0,l∈j−12,j+12,j∈12,∞,where ρ is the resolvent set and
(30)ϑ1znm−1=−1Ajl−Bjl4c2+BjlAjlG22jlz,Rn,Rn+G11jlz,Rn,Rn,n=m,−G11jlz,Rm,Rn,n≠m,(31)ϑ2znm−1=−1Bjl−Ajl4c2+AjlBjlG11jlz,Rn,Rn+G22jlz,Rn,Rn,n=m,−G22jlz,Rm,Rn,n≠m,(32)ϑ3znm−1=−2cBjlAjl−12c+2cBjlG11jlz,Rn,Rn+2cAjlG22jlz,Rn,Rn,n=m,12c,n≠m,(33)ϑ4znm−1=−−2cBjlAjl+12c−2cBjlG11jlz,Rn,Rn−2cAjlG22jlz,Rn,Rn,n=m,−12c,n≠m,(34)M^jl,zmr=Gjlz,r,Rn10,r<Rn,−mGjlz,r,Rn10,r>Rn,m=1,2,n=1,2,⋯,N,(35)M~jl,zmr=Gjlz,r,Rn01,r<Rn,−mGjlz,r,Rn01,r>Rn,m=1,2,n=1,2,⋯,N.
hD−z−1,Imk>0, is the radial Dirac resolvent with kernel:
(36)Gjlz,r,Rn=G11jlz,r,RnG12jlz,r,RnG21jlz,r,RnG22jlz,r,Rn,where
(37)G11jlz,r,Rn=Gjlz,RnFjlz,r,r<Rn,Fjlz,RnGjlz,r,r>Rn,G12jlz,r,Rn=G~jlz,RnFjlz,r,r<Rn,F~jlz,RnGjlz,r,r>Rn,G21jlz,r,Rn=Gjlz,RnF~jlz,r,r<Rn,Fjlz,RnG~jlz,r,r>Rn,G22jlz,r,Rn=G~jlz,RnF~jlz,r,r<Rn,F~jlz,RnG~jlz,r,r>Rn.Proof.
Equation (29) follows from Krein’s formula [19].
Letϕ=ϕ1ϕ2∈L0,∞⊗ℂ2 and define the function ξjl by
(38)ξjlr=hjl,Γ,R−z−1ϕr.
Sinceξjl∈Dhjl,Γ,R, it follows that ξjl satisfies the separated boundary conditions in equation (23).The implementation of these boundary conditions provides equation (29).


## 2.3. Spectral Properties ofhjl,Γ,R

Theorem 2.
ForAjl,n,Bjl,n∈ℝ,l∈j−1/2,j+1/2,j∈1/2,∞, the essential spectrum of hjl,Γ,R is purely absolutely continuous and coincides with −∞,−c2/2∪c2/2,∞. Its singularly continuous and residual spectra are empty.Proof.
Follow step by step the proof of theorem 6.2 and proposition 6.1 in ref. [12].According to decomposition (10), equations (12) and (29), the resolvent equation of HΓ,R reads
(39)HΓ,R−z−1=HD−z−1+⨁j=1/2∞⨁l=j+1/2j−1/2⨁μ=−jj×∑n,m=1Nϑnm1z.−1M^jl,z2.¯⊗Ω~jlμ,.×.−1M^jl,z2.⊗Ω~jlμ+ϑnm2z.−1M~jl,z2.¯⊗Ω~jlμ,.×.−1M~jl,z2.⊗Ω~jlμ+ϑnm3z.−1M^jl,z2.¯⊗Ω~jlμ,.×.−1M~jl,z1.⊗Ω~jlμ+ϑnm4z.−1M~jl,z2.¯⊗Ω~jlμ,.×.−1M^jl,z1.⊗Ω~jlμ,z∈ρhjl,Γ,R,Imk>0,l∈j−12,j+12,j∈12,∞,where the notations M^jl,zn.⊗Ω~jlμ and M~jl,zn.⊗Ω~jlμ,n=1,2, mean
(40)M^jl,zn.⊗Ω~jlμ=M^jl,z,1n.ΩjlμM^jl,z,2n.Ωjl′μ,n=1,2,M~jl,zn.⊗Ω~jlμ=M~jl,z,1n.ΩjlμM~jl,z,2n.Ωjl′μ,n=1,2.


## 2.4. The Nonrelativistic Limit

The nonrelativistic limit ofhjl,Γ,R as c⟶∞ is given by the following.Theorem 3.
For the spin1/2 particules, the operator hjl,Γ,R−c2/2 converges in norm resolvent sense to the Schrödinger operator hl,αl,R times the projector onto H~+=L20,∞:
(41)n·limc→∞hjl,Γ,R−c22−z−1=hjl,α,R−z−1⊗1000,z∈ℂ\ℝ,where the operator hjl,α,R is defined by
(42)hjl,α,R=−d2dr2+ll+1r2,Dhjl,α,R=f∈L20,∞f,f′∈ACloc0,∞\R1,⋯,RN,g0+=0,fRn+−fRn−=βl,n2f′Rn++f′Rn−,f′Rn+−f′Rn−=αl,n2fRn++fRn−,−f″+ll+1r−2f∈L20,∞,l≥1,αl,nβl,n−4=0,αl,n,βl,n∈ℝ,1≤n≤N.Proof.
One can use the strategy of Gesztesy and Seba [15] where a similar case was discussed for point interactions.
The Hamiltonianhjl,α,R describes a 2N parameter model of finitely many nonrelativistic δ-sphere interactions in quantum mechanics.


## 2.5. Scattering Theory for the Pairhjl,α,R,hD

Let us define fork>0 the function
(43)Fjl,Γ,Rr=FjlrF~jlr+∑n,m=1ϑnm1zFjlz,RnM^jl,z2r+ϑnm2zF~jlz,RnM~jl,z2r+ϑnm3zFjlz,RnM~jl,z1r+ϑnm4zF~jlz,RnM^jl,z1r,where functions Fjlz,r,F~jlz,r,M~jl,z2r,M^jl,z2r,M~jl,z1r,M^jl,z1r, and ϑnmiz,i=1,2,3,4, are, respectively, defined by equations (18), (35), and (33).The asymptotic behaviour of the functionFjl,Γ,Rr as r⟶∞ yields [20]
(44)Fjl,Γ,Rr→r→∞k>0A^jlzsinkr−κjlπ2B^jlzsinkr−κjl−1π2+∑n,m=1Nϑnm′zC^jlzexp−ikr−κjlπ2O^jlzexp−ikr−κjl−1π2=A^jlz−iC^jlz∑n,m=1Nϑnm′zsinkr−κjlπ2B^jlz−iO^jlz∑n,m=1Nϑnm′zsinkr−κjl−1π2+C^jlz∑n,m=1Nϑnm′zcoskr−κjlπ2O^jlz∑n,m=1Nϑnm′zcoskr−κjl−1π2,where
(45)ϑnm′z=ϑnm1zFjlz,RnFjlz,Rn+ϑnm2zF~jlz,RnF~jlz,Rn,A^jlz=2−κjlk−κjl−1Γ2κjl+2Γκjl+1−1,C^jlz=2κjlkκjlΓ2κjl+2−1Γκjl+1,B^jlz=1c2−κjlk−κjlΓ2κjl+2Γκjl+1−1,O^jlz=1c2κjlkκjl+1Γ2κjl+2−1Γκjl+1.A straight computation shows that equation (44) reads
(46)Fjl,Γ,Rr→r→∞k>0P~jl,12z+P~jl,22z1/2sinkr−κjlπ2+δjl,Γ,R+01P~jl,32z+P~jl,42z1/2sinkr−κjl−1π2+δjl,Γ,R.Then, the phase shiftδjl,Γ,R corresponding to hjl,α,R is defined by
(47)δjl,Γ,R=−arctanP~jl,22zP~jl,12z,=−arctanP~jl,42zP~jl,32z,=−arctanC^jlz∑n,m=1Nϑnm′zA^jlz−iC^jlz∑n,m=1Nϑnm′z.The elements of the on-shell scattering matrix are given by(48)Sjl,Γ,R=exp2iδjl,Γ,Rz.The partial wave scattering amplitude is given by(49)fjl,Γ,R=exp2iδjl,Γ,Rz−12ik.


## 3. Basic Properties for the2N Parameter Model of Relativistic Many δ-Sphere plus Coulomb Interactions: Separated Boundary Conditions

### 3.1. Definition of the Hamiltonian

In this section, we give in dimensionn=3 the rigorous mathematical definition of relativistic many δ-sphere plus Coulomb interactions. The formal expression of the Hamiltonian describing N finitely δ-sphere plus Coulomb interactions with support on concentric spheres of radii 0<R1<⋯<RN is given by
(50)Hγ,Γ=HD+γ∣x∣+∑n=1NΓnδx−Rn,x∈ℝ3,0<R1<⋯<RN,where the operator HD is defined by equation (2).Following decomposition (10), we consider the operatorH¯γ,Γ that reads
(51)H¯γ,Γ=⨁j=1/2∞⨁l=j−1/2j+1/2Ujl−1hjl,γ,RUjl⊗1l,where the operator hjl,γ,R is defined in L20,∞⊗ℂ2 by
(52)hjl,γ,R=c22+γr−cddr+cκjlrcddr+cκjlr−c22+γr≡τγ,Dhjl,γ,R=ψ∈L20,∞⊗ℂ2∣ψ∈ACloc0,∞,ψRn±=0,τγψ∈L20,∞⊗ℂ2,R=R1,R2,⋯,RN.The adjointhjl,γ,R∗ of hjl,γ,R reads
(53)hjl,γ,R∗=τγ,Dhjl,γ,R∗=f∈L20,∞⊗ℂ2∣f∈ACloc0,∞\R,τγf∈L20,∞⊗ℂ2.The deficiency index equation(54)hjl,γ,R∗−zφ=0,f∈ℂ−−∞,−c22⋃c22,∞has 2N linearly independent solutions
(55)φjl,γ1=gjl,γ,1z,Rnfjl,γ,1z,rgjl,γ,2z,Rnfjl,γ,2z,r,r<Rn,00,r>Rn,n=1,⋯,N,φjl,γ2=00,r<Rn,fjl,γz,Rngjl,γz,rf~jl,γz,Rng~jl,γz,r,r>Rn,n=1,⋯,N,where
(56)fjl,γ,1z,r=1−γ2κjlc+ζ2−1/2cosarctanγ~2kζ~−1/2Fjl,γz,r−γκjlc+ζF~jl,γz,r,(57)fjl,γ,2z,r=1−γ2κjlc+ζ2−1/2cosarctanγ~2kζ~−1/2F~jl,γz,r−γκjlc+ζFjl,γz,r,(58)gjl,γ,1z,r=1−γ2κjlc+ζ2−1/2cosarctanγ~2kζ~−1/2Gjl,γz,r−γκjlc+ζG~jl,γz,r,(59)gjl,γ,2z,r=1−γ2κjlc+ζ2−1/2cosarctanγ~2kζ~−1/2G~jl,γz,r−γκjlc+ζGjl,γz,r,where
(60)Fjl,γz,r=rζ~+1e−ikr1F1ζ~+1−iγ~2k,2ζ~+2,2ikr,Gjl,γz,r=Γ2ζ~+2−1Γζ~+1−iγ~2k2ik2ζ~+1rζ~+1e−ikrUζ~+1−iγ~2k,2ζ~+2,2ikr,F~jl,γz,r=ζ~c2ζ~+1Γζ~+iγ~2kΓζ~+1+iγ~2k−1rζ~e−ikr×1F1ζ~−iγ~2k,2ζ~,2ikr,G~jl,γz,r=ζ~−1c2ζ~+1−1Γ2ζ~−1Γζ~+iγ~2kΓζ~+iγ~2k−1Γζ~+1+iγ~2k×k22ik2ζ~−1rζ~e−ikrUζ~−iγ~2k,2ζ~,2ikr.We use the following notations:(61)k=1cz2−c44,ζ=κjl2c2−γ2,ζ~=1cζ,γ~=2zγc2.The operatorhjl,γ,R has deficiency indices 2N,2N, and therefore, all self-adjoint extensions of hjl,γ,R are given by 4N2 parameter family of self-adjoint operators.Consider the following special2N-parameter family of self adjoint extensions of hjl,γ,R:
(62)hjl,γ,Γ,R=c22+γr−cddr+cκjlr−cddr+cκjlr−c22+γr,Dhjl,γ,Γ,R=g∈Dhjl,Γ,R∗1−τ0Γjl,n2cgRn+−1+τ0Γjl,n2cgRn−=0,where the 2×2 matrix Γjl,n is defined by equation (24).The Hamiltonianhjl,γ,Γ,R gives the mathematical definition of the formal expression
(63)hγ,Γ,R=hD+γr+∑n=1NΓjl,nδr−Rn,Rn>0,n=1,2,⋯,N,where hD is the radial Dirac operator defined by equation (27).The rigous mathematical definition of formal expression (50) reads
(64)Hγ,Γ,R=⨁j=1/2∞⨁l=j−1/2j+1/2Ujl−1hjl,γ,Γ,RUjl⊗1l.The caseγ=0 and Γ=0, i.e., Γjl,n=0, for all j and l in equation (64), yields the Dirac Hamiltonian HD defined by equation (2).The caseΓ=0, for all j and l in equation (64), yields Hγ,R≡Hγ,D.The caseAjl,n≠0 and Bjl,n=0, for all j and l in equation (64), yields the relativistic many δ-sphere plus Coulomb interactions of the first type.The caseAjl,n=0 and Bjl,n≠0, for all j and l in equation (64), yields the relativistic many δ-sphere plus Coulomb interactions of the second type.


### 3.2. Resolvent Equation ofhjl,γ,Γ,R

Theorem 43.1.
The resolvent ofhjl,γ,Γ,R reads
(65)hjl,γ,Γ,R−z−1=hγ,D−z−1+∑n,m=1Nϑγ,nm1zM^jl,γ2.¯,.M^jl,γ2.+ϑγ,nm2zM~jl,γ2.¯,.M~jl,γ2.+ϑγ,nm3zM^jl,γ2.¯,.M~jl,γ1.+ϑγ,nm4zM~jl,γ2.¯,.M^jl,,γ1.,z∈ρhjl,γ,Γ,R,Imk>0,l∈j−12,j+12,j∈12,∞,where ρ is the resolvent set and
(66)ϑγ1znm−1=−1Ajl−Bjl4c2+BjlAjlG22jl,γz,Rn,Rn+G11jl,γz,Rn,Rn,n=m,−G11jl,γz,Rm,Rn,n≠m,(67)ϑγ2znm−1=−1Bjl−Ajl4c2+AjlBjlG11jl,γz,Rn,Rn+G22jl,γz,Rn,Rn,n=m,−G22jl,γz,Rm,Rn,n≠m,(68)ϑγ3znm−1=−2cBjlAjl−12c+2cBjlG11jl,γz,Rn,Rn+2cAjlG22jl,γz,Rn,Rn,n=m,12c,n≠m,(69)ϑγ4znm−1=−−2cBjlAjl+12c−2cBjlG11jl,γz,Rn,Rn−2cAjlG22jl,γz,Rn,Rn,n=m,−12c,n≠m,(70)M^jl,γmr=Gjl,γz,r,Rn10,r<Rn,−mGjl,γz,r,Rn10,r>Rn,m=1,2,n=1,2,⋯,N,(71)M~jl,γmr=Gjl,γz,r,Rn01,r<Rn,−mGjl,γz,r,Rn01,r>Rn,m=1,2,n=1,2,⋯,N.
hγ,D−z−1,Imk>0, is the radial Dirac resolvent with kernel:
(72)Gjl,γz,r,Rn=G11jl,γz,r,RnG12jl,γz,r,RnG21jl,γz,r,RnG22jl,γz,r,Rn,where
(73)G11jl,γz,r,Rn=gγ,1z,Rnfγ,1z,r,r<Rn,fγ,1z,Rngγ,1z,r,r>Rn,G12jl,γz,r,Rn=gγ,2z,Rnfγ,1z,r,r<Rn,fγ,2z,Rngγ,1z,r,r>Rn,G21jl,γz,r,Rn=gγ,1z,Rnfγ,2z,r,r<Rn,fγ,1z,Rngγ,2z,r,r>Rn,G22jl,γz,r,Rn=gγ,2z,Rnfγ,2z,r,r<Rn,fγ,2z,Rngγ,2z,r,r>Rn.Proof.
Similar to the proof of Theorem2.
Consider to decomposition (10), equations (51) and (65), the resolvent equation of Hγ,Γ,R reads
(74)Hγ,Γ,R−z−1=Hγ,D−z−1+⨁j=1/2∞⨁l=j+1/2j−1/2⨁μ=−jj×∑n,m=1Nϑγ,nm1z.−1M^jl,γ2.¯⊗Ω~jlμ,.×.−1M^jl,γ2.⊗Ω~jlμ+ϑγ,nm2z.−1M~jl,γ2.¯⊗Ω~jlμ,.×.−1M~jl,γ2.⊗Ω~jlμ+ϑγ,nm3z.−1M^jl,γ2.¯⊗Ω~jlμ,.×.−1M~jl,γ1.⊗Ω~jlμ+ϑγ,nm4z.−1M~jl,γ2.¯⊗Ω~jlμ,.×.−1M^jl,γ1.⊗Ω~jlμ,z∈ρhjl,γ,Γ,R,Imk>0,l∈j−12,j+12,j∈12,∞.


### 3.3. The Nonrelativistic Limit

The nonrelativistic limit ofhjl,γ,Γ,R as c⟶∞ is given by the following.Theorem 53.2.
For the spin 1/2 particles, the operatorhjl,γ,Γ,R−c2/2 converges in norm resolvent sense to the Schrödinger operator hl,αl,R times the projector onto H~+=L20,∞:
(75)n·limc→∞hjl,γ,Γ,R−c22−z−1=hjl,γ,α,R−z−1⊗1000,z∈ℂ\ℝ,where the operator hjl,γ,α,R is defined by
(76)hjl,γ,α,R=−d2dr2+ll+1r2+γr,Dhjl,α,R=f∈L20,∞f,f′∈ACloc0,∞\R1,⋯,RN,g0+=0,fRn+−fRn−=βl,n2f′Rn++f′Rn−,f′Rn+−f′Rn−=αl,n2fRn++fRn−,−f″+ll+1r−2f+γr−1f∈L20,∞,l≥1,αl,nβl,n−4=0,αl,n,βl,n∈ℝ,1≤n≤N.Proof.
Similar to the proof of Theorem3.
The Hamiltonianhjl,γ,α,R describes a 2N parameter model of finitely many nonrelativistic δ-sphere plus Coulomb interactions in quantum mechanics.


### 3.4. Scattering Theory for the Pairhjl,γ,α,R,hγ,D

Let us define fork>0 the function
(77)Fjl,γ,Γ,Rr=fγ,1z,rfγ,2z,r+∑n,m=1ϑγ,nm1zfγ,1z,RnM^jl,γ2r+ϑγ,nm2zfγ,2z,RnM~jl,γ2r+ϑγ,nm3zfγ,1z,RnM~jl,γ1r+ϑγ,nm4zfγ,2z,RnM^jl,γ1r,=fγ,1z,rfγ,2z,r+∑n,m=1ϑγ,nm1Gjl,γz,r,Rnfγ,1z,Rn0+ϑγ,nm2Gjl,γz,r,Rn0fγ,1z,Rn−ϑγ,nm3Gjl,γz,r,Rn0fγ,1z,Rn−ϑγ,nm4Gjl,γz,r,Rnfγ,2z,Rn0,=fγ,1z,rfγ,2z,r+∑n,m=1ϑγ,nm1Gjl,γz,r,Rnfγ,1z,Rn0+ϑγ,nm2Gjl,γz,Rn0fγ,1z,Rn−ϑγ,nm3Gjl,γz,r,Rn0fγ,1z,Rn−fγ,2z,Rn0,where the functions fγ,1z,r,fγ,2z,r,M~jl,γ2r,M^jl,γ2r,M~jl,γ1r,M^jl,γ1r, and ϑγ,nmiz,i=1,2,3,4, are defined by equations (59), (71), and (69), respectively.The asymptotic behaviour of the functionFjl,γ,Γ,Rr as r⟶∞ yields [20]:
(78)Fjl,γ,Γ,Rrk>0r⟶∞⟶d1zsinkr−γ~2kln2kr−ζ~π2+δζ0z−γκjlc+ζd1zsinkr−γ~2kln2kr−ζ~π2+δζ0z+d2zsinkr−γ~2kln2kr−ζ~−1π2+δζ−10z−γκjlc+ζ−1d2zsinkr−γ~2kln2kr−ζ~−1π2+δζ−10z+∑n,m=1ϑγ,nm1fγ,1z,Rnfγ,1z,Rn+ϑγ,nm2fγ,2z,Rnfγ,2z,Rn×d3zexp−ikr−γ~2kln2kr−ζ~π2+δζ0z−γκjlc+ζd3zexp−ikr−γ~2kln2kr−ζ~π2+δζ0z+d4zsinkr−γ~2kln2kr−ζ~−1π2+δζ−10z−γκjlc+ζ−1d4zsinkr−γ~2kln2kr−ζ~−1π2+δζ−10z,where the constants di,i=1,2,3,4, read
(79)d1z=1−γ2κjlc+ζ2−1/2cosarctanγ~2kζ~−1/22−ζ~k−ζ~−1Γ2ζ~+2×Γζ~+1+iγ~2k−1eπγ~/2k,d2z=−γcκjlc+ζ1−γ2κjlc+ζ2−1/2cosarctanγ~2kζ~−1/22−ζ~k−ζ~×Γ2ζ~+2Γζ~+1+iγ~2k−1eπγ~/2k,d3z=1−γ2κjlc+ζ2−1/2cosarctanγ~2kζ~−1/22ζ~kζ~Γ2ζ~+2−1×Γζ~+1+iγ~2ke−πγ/2k,d4z=−γcκjlc+ζ1−γ2κjlc+ζ2−1/2cosarctanγ~2kζ~−1/22ζ~kζ~+1×Γ2ζ~+2−1Γζ~+1+iγ~2ke−πγ/2k.Let us introduce the following notations:(80)x1=k−γ~kln2kr−ζ~π2,x2=k−−γ~kln2kr−ζ~−1π2.A straightforward computation shows that(81)Fjl,γ,Γ,Rrk>0r⟶∞⟶d1zsinx1+δζ0z−γκjlc+ζd1zcosx2+δζ0z+d2zcosx1+δζ−10z−γκjlc+ζ−1d2zsinx2+δζ−10z+∑n,m=1ϑγ,nm1fγ,1z,Rnfγ,1z,Rn+ϑγ,nm2fγ,2z,Rnfγ,2z,Rn×d3zcosx1+δζ0−isinx1+δζ0−iγcκjl+ζd3zcosx2+δζ0−isinx2+δζ0+−id4zcosx1+δζ−10−isinx1+δζ−10−γcκjl+ζ−1d4zcosx2+δζ−10−isinx2+δζ−10,where
(82)δζ0z=δζ−10z+arctanγ~2kζ~,δζ0z=argΓζ~+1+iγ~2k.Equation (81) reads
(83)Fjl,γ,Γ,Rrk>0r⟶∞⟶V1zsinx1+δζ~+V2zcosx1+δζ~V3zsinx2+δζ~−1+V4zcosx2+δζ~−1,=V12z+V22z1/2sinx1+δζ~0+δγ,Γ,1CV32z+V42z1/2sinx2+δζ~−10+δγ,Γ,2C,where the Coulomb-modified phase shift δγ,ΓC is given by
(84)δγ,Γ,1Cδγ,Γ,2C=−arctanV2zV1z−arctanV4zV3z,where
(85)V1z=d1z+d2z−i∑n,mϑγ,nm′zd4zsinarctanγ~2kζ~−∑n,mϑγ,nm′zid3z+d4zcosarctanγ~2kζ~,V2z=d2z−i∑n,mϑγ,nm′zd4zcosarctanγ~2kζ~+∑n,mϑγ,nm′zd3z+d4zsinarctanγ~2kζ~,V3z=γκjlc+ζ−d1z+i∑n,mϑγ,nm′zd3zsinarctanγ~2kζ~−γκjlc+ζ−1d2z+∑n,mϑγ,nm′z−γκjlc+ζd3zcosarctanγ~2kζ~+iγκjlc+ζ−1d4z,V4z=γκjlc+ζ−d1z+i∑n,mϑγ,nm′zd3zcosarctanγ~2kζ~+∑n,mϑγ,nm′z−γκjlc+ζd3zsinarctanγ~2kζ~−γκjlc+ζ−1d4z,ϑγ,nm′z=ϑγ,nm1fγ,1z,Rnfγ,1z,Rn+ϑγ,nm2fγ,2z,Rnfγ,2z,Rn.The Coulomb-modified on-shell scattering matrix is given by(86)Sγ,Γ=exp2iδγ,Γ,nCz,n=1,2.The partial wave scattering amplitude is given by(87)fγ,Γ=exp2iδγ,Γ,nCz−12ik,n=1,2.


## 3.1. Definition of the Hamiltonian

In this section, we give in dimensionn=3 the rigorous mathematical definition of relativistic many δ-sphere plus Coulomb interactions. The formal expression of the Hamiltonian describing N finitely δ-sphere plus Coulomb interactions with support on concentric spheres of radii 0<R1<⋯<RN is given by
(50)Hγ,Γ=HD+γ∣x∣+∑n=1NΓnδx−Rn,x∈ℝ3,0<R1<⋯<RN,where the operator HD is defined by equation (2).Following decomposition (10), we consider the operatorH¯γ,Γ that reads
(51)H¯γ,Γ=⨁j=1/2∞⨁l=j−1/2j+1/2Ujl−1hjl,γ,RUjl⊗1l,where the operator hjl,γ,R is defined in L20,∞⊗ℂ2 by
(52)hjl,γ,R=c22+γr−cddr+cκjlrcddr+cκjlr−c22+γr≡τγ,Dhjl,γ,R=ψ∈L20,∞⊗ℂ2∣ψ∈ACloc0,∞,ψRn±=0,τγψ∈L20,∞⊗ℂ2,R=R1,R2,⋯,RN.The adjointhjl,γ,R∗ of hjl,γ,R reads
(53)hjl,γ,R∗=τγ,Dhjl,γ,R∗=f∈L20,∞⊗ℂ2∣f∈ACloc0,∞\R,τγf∈L20,∞⊗ℂ2.The deficiency index equation(54)hjl,γ,R∗−zφ=0,f∈ℂ−−∞,−c22⋃c22,∞has 2N linearly independent solutions
(55)φjl,γ1=gjl,γ,1z,Rnfjl,γ,1z,rgjl,γ,2z,Rnfjl,γ,2z,r,r<Rn,00,r>Rn,n=1,⋯,N,φjl,γ2=00,r<Rn,fjl,γz,Rngjl,γz,rf~jl,γz,Rng~jl,γz,r,r>Rn,n=1,⋯,N,where
(56)fjl,γ,1z,r=1−γ2κjlc+ζ2−1/2cosarctanγ~2kζ~−1/2Fjl,γz,r−γκjlc+ζF~jl,γz,r,(57)fjl,γ,2z,r=1−γ2κjlc+ζ2−1/2cosarctanγ~2kζ~−1/2F~jl,γz,r−γκjlc+ζFjl,γz,r,(58)gjl,γ,1z,r=1−γ2κjlc+ζ2−1/2cosarctanγ~2kζ~−1/2Gjl,γz,r−γκjlc+ζG~jl,γz,r,(59)gjl,γ,2z,r=1−γ2κjlc+ζ2−1/2cosarctanγ~2kζ~−1/2G~jl,γz,r−γκjlc+ζGjl,γz,r,where
(60)Fjl,γz,r=rζ~+1e−ikr1F1ζ~+1−iγ~2k,2ζ~+2,2ikr,Gjl,γz,r=Γ2ζ~+2−1Γζ~+1−iγ~2k2ik2ζ~+1rζ~+1e−ikrUζ~+1−iγ~2k,2ζ~+2,2ikr,F~jl,γz,r=ζ~c2ζ~+1Γζ~+iγ~2kΓζ~+1+iγ~2k−1rζ~e−ikr×1F1ζ~−iγ~2k,2ζ~,2ikr,G~jl,γz,r=ζ~−1c2ζ~+1−1Γ2ζ~−1Γζ~+iγ~2kΓζ~+iγ~2k−1Γζ~+1+iγ~2k×k22ik2ζ~−1rζ~e−ikrUζ~−iγ~2k,2ζ~,2ikr.We use the following notations:(61)k=1cz2−c44,ζ=κjl2c2−γ2,ζ~=1cζ,γ~=2zγc2.The operatorhjl,γ,R has deficiency indices 2N,2N, and therefore, all self-adjoint extensions of hjl,γ,R are given by 4N2 parameter family of self-adjoint operators.Consider the following special2N-parameter family of self adjoint extensions of hjl,γ,R:
(62)hjl,γ,Γ,R=c22+γr−cddr+cκjlr−cddr+cκjlr−c22+γr,Dhjl,γ,Γ,R=g∈Dhjl,Γ,R∗1−τ0Γjl,n2cgRn+−1+τ0Γjl,n2cgRn−=0,where the 2×2 matrix Γjl,n is defined by equation (24).The Hamiltonianhjl,γ,Γ,R gives the mathematical definition of the formal expression
(63)hγ,Γ,R=hD+γr+∑n=1NΓjl,nδr−Rn,Rn>0,n=1,2,⋯,N,where hD is the radial Dirac operator defined by equation (27).The rigous mathematical definition of formal expression (50) reads
(64)Hγ,Γ,R=⨁j=1/2∞⨁l=j−1/2j+1/2Ujl−1hjl,γ,Γ,RUjl⊗1l.The caseγ=0 and Γ=0, i.e., Γjl,n=0, for all j and l in equation (64), yields the Dirac Hamiltonian HD defined by equation (2).The caseΓ=0, for all j and l in equation (64), yields Hγ,R≡Hγ,D.The caseAjl,n≠0 and Bjl,n=0, for all j and l in equation (64), yields the relativistic many δ-sphere plus Coulomb interactions of the first type.The caseAjl,n=0 and Bjl,n≠0, for all j and l in equation (64), yields the relativistic many δ-sphere plus Coulomb interactions of the second type.


## 3.2. Resolvent Equation ofhjl,γ,Γ,R

Theorem 43.1.
The resolvent ofhjl,γ,Γ,R reads
(65)hjl,γ,Γ,R−z−1=hγ,D−z−1+∑n,m=1Nϑγ,nm1zM^jl,γ2.¯,.M^jl,γ2.+ϑγ,nm2zM~jl,γ2.¯,.M~jl,γ2.+ϑγ,nm3zM^jl,γ2.¯,.M~jl,γ1.+ϑγ,nm4zM~jl,γ2.¯,.M^jl,,γ1.,z∈ρhjl,γ,Γ,R,Imk>0,l∈j−12,j+12,j∈12,∞,where ρ is the resolvent set and
(66)ϑγ1znm−1=−1Ajl−Bjl4c2+BjlAjlG22jl,γz,Rn,Rn+G11jl,γz,Rn,Rn,n=m,−G11jl,γz,Rm,Rn,n≠m,(67)ϑγ2znm−1=−1Bjl−Ajl4c2+AjlBjlG11jl,γz,Rn,Rn+G22jl,γz,Rn,Rn,n=m,−G22jl,γz,Rm,Rn,n≠m,(68)ϑγ3znm−1=−2cBjlAjl−12c+2cBjlG11jl,γz,Rn,Rn+2cAjlG22jl,γz,Rn,Rn,n=m,12c,n≠m,(69)ϑγ4znm−1=−−2cBjlAjl+12c−2cBjlG11jl,γz,Rn,Rn−2cAjlG22jl,γz,Rn,Rn,n=m,−12c,n≠m,(70)M^jl,γmr=Gjl,γz,r,Rn10,r<Rn,−mGjl,γz,r,Rn10,r>Rn,m=1,2,n=1,2,⋯,N,(71)M~jl,γmr=Gjl,γz,r,Rn01,r<Rn,−mGjl,γz,r,Rn01,r>Rn,m=1,2,n=1,2,⋯,N.
hγ,D−z−1,Imk>0, is the radial Dirac resolvent with kernel:
(72)Gjl,γz,r,Rn=G11jl,γz,r,RnG12jl,γz,r,RnG21jl,γz,r,RnG22jl,γz,r,Rn,where
(73)G11jl,γz,r,Rn=gγ,1z,Rnfγ,1z,r,r<Rn,fγ,1z,Rngγ,1z,r,r>Rn,G12jl,γz,r,Rn=gγ,2z,Rnfγ,1z,r,r<Rn,fγ,2z,Rngγ,1z,r,r>Rn,G21jl,γz,r,Rn=gγ,1z,Rnfγ,2z,r,r<Rn,fγ,1z,Rngγ,2z,r,r>Rn,G22jl,γz,r,Rn=gγ,2z,Rnfγ,2z,r,r<Rn,fγ,2z,Rngγ,2z,r,r>Rn.Proof.
Similar to the proof of Theorem2.
Consider to decomposition (10), equations (51) and (65), the resolvent equation of Hγ,Γ,R reads
(74)Hγ,Γ,R−z−1=Hγ,D−z−1+⨁j=1/2∞⨁l=j+1/2j−1/2⨁μ=−jj×∑n,m=1Nϑγ,nm1z.−1M^jl,γ2.¯⊗Ω~jlμ,.×.−1M^jl,γ2.⊗Ω~jlμ+ϑγ,nm2z.−1M~jl,γ2.¯⊗Ω~jlμ,.×.−1M~jl,γ2.⊗Ω~jlμ+ϑγ,nm3z.−1M^jl,γ2.¯⊗Ω~jlμ,.×.−1M~jl,γ1.⊗Ω~jlμ+ϑγ,nm4z.−1M~jl,γ2.¯⊗Ω~jlμ,.×.−1M^jl,γ1.⊗Ω~jlμ,z∈ρhjl,γ,Γ,R,Imk>0,l∈j−12,j+12,j∈12,∞.


## 3.3. The Nonrelativistic Limit

The nonrelativistic limit ofhjl,γ,Γ,R as c⟶∞ is given by the following.Theorem 53.2.
For the spin 1/2 particles, the operatorhjl,γ,Γ,R−c2/2 converges in norm resolvent sense to the Schrödinger operator hl,αl,R times the projector onto H~+=L20,∞:
(75)n·limc→∞hjl,γ,Γ,R−c22−z−1=hjl,γ,α,R−z−1⊗1000,z∈ℂ\ℝ,where the operator hjl,γ,α,R is defined by
(76)hjl,γ,α,R=−d2dr2+ll+1r2+γr,Dhjl,α,R=f∈L20,∞f,f′∈ACloc0,∞\R1,⋯,RN,g0+=0,fRn+−fRn−=βl,n2f′Rn++f′Rn−,f′Rn+−f′Rn−=αl,n2fRn++fRn−,−f″+ll+1r−2f+γr−1f∈L20,∞,l≥1,αl,nβl,n−4=0,αl,n,βl,n∈ℝ,1≤n≤N.Proof.
Similar to the proof of Theorem3.
The Hamiltonianhjl,γ,α,R describes a 2N parameter model of finitely many nonrelativistic δ-sphere plus Coulomb interactions in quantum mechanics.


## 3.4. Scattering Theory for the Pairhjl,γ,α,R,hγ,D

Let us define fork>0 the function
(77)Fjl,γ,Γ,Rr=fγ,1z,rfγ,2z,r+∑n,m=1ϑγ,nm1zfγ,1z,RnM^jl,γ2r+ϑγ,nm2zfγ,2z,RnM~jl,γ2r+ϑγ,nm3zfγ,1z,RnM~jl,γ1r+ϑγ,nm4zfγ,2z,RnM^jl,γ1r,=fγ,1z,rfγ,2z,r+∑n,m=1ϑγ,nm1Gjl,γz,r,Rnfγ,1z,Rn0+ϑγ,nm2Gjl,γz,r,Rn0fγ,1z,Rn−ϑγ,nm3Gjl,γz,r,Rn0fγ,1z,Rn−ϑγ,nm4Gjl,γz,r,Rnfγ,2z,Rn0,=fγ,1z,rfγ,2z,r+∑n,m=1ϑγ,nm1Gjl,γz,r,Rnfγ,1z,Rn0+ϑγ,nm2Gjl,γz,Rn0fγ,1z,Rn−ϑγ,nm3Gjl,γz,r,Rn0fγ,1z,Rn−fγ,2z,Rn0,where the functions fγ,1z,r,fγ,2z,r,M~jl,γ2r,M^jl,γ2r,M~jl,γ1r,M^jl,γ1r, and ϑγ,nmiz,i=1,2,3,4, are defined by equations (59), (71), and (69), respectively.The asymptotic behaviour of the functionFjl,γ,Γ,Rr as r⟶∞ yields [20]:
(78)Fjl,γ,Γ,Rrk>0r⟶∞⟶d1zsinkr−γ~2kln2kr−ζ~π2+δζ0z−γκjlc+ζd1zsinkr−γ~2kln2kr−ζ~π2+δζ0z+d2zsinkr−γ~2kln2kr−ζ~−1π2+δζ−10z−γκjlc+ζ−1d2zsinkr−γ~2kln2kr−ζ~−1π2+δζ−10z+∑n,m=1ϑγ,nm1fγ,1z,Rnfγ,1z,Rn+ϑγ,nm2fγ,2z,Rnfγ,2z,Rn×d3zexp−ikr−γ~2kln2kr−ζ~π2+δζ0z−γκjlc+ζd3zexp−ikr−γ~2kln2kr−ζ~π2+δζ0z+d4zsinkr−γ~2kln2kr−ζ~−1π2+δζ−10z−γκjlc+ζ−1d4zsinkr−γ~2kln2kr−ζ~−1π2+δζ−10z,where the constants di,i=1,2,3,4, read
(79)d1z=1−γ2κjlc+ζ2−1/2cosarctanγ~2kζ~−1/22−ζ~k−ζ~−1Γ2ζ~+2×Γζ~+1+iγ~2k−1eπγ~/2k,d2z=−γcκjlc+ζ1−γ2κjlc+ζ2−1/2cosarctanγ~2kζ~−1/22−ζ~k−ζ~×Γ2ζ~+2Γζ~+1+iγ~2k−1eπγ~/2k,d3z=1−γ2κjlc+ζ2−1/2cosarctanγ~2kζ~−1/22ζ~kζ~Γ2ζ~+2−1×Γζ~+1+iγ~2ke−πγ/2k,d4z=−γcκjlc+ζ1−γ2κjlc+ζ2−1/2cosarctanγ~2kζ~−1/22ζ~kζ~+1×Γ2ζ~+2−1Γζ~+1+iγ~2ke−πγ/2k.Let us introduce the following notations:(80)x1=k−γ~kln2kr−ζ~π2,x2=k−−γ~kln2kr−ζ~−1π2.A straightforward computation shows that(81)Fjl,γ,Γ,Rrk>0r⟶∞⟶d1zsinx1+δζ0z−γκjlc+ζd1zcosx2+δζ0z+d2zcosx1+δζ−10z−γκjlc+ζ−1d2zsinx2+δζ−10z+∑n,m=1ϑγ,nm1fγ,1z,Rnfγ,1z,Rn+ϑγ,nm2fγ,2z,Rnfγ,2z,Rn×d3zcosx1+δζ0−isinx1+δζ0−iγcκjl+ζd3zcosx2+δζ0−isinx2+δζ0+−id4zcosx1+δζ−10−isinx1+δζ−10−γcκjl+ζ−1d4zcosx2+δζ−10−isinx2+δζ−10,where
(82)δζ0z=δζ−10z+arctanγ~2kζ~,δζ0z=argΓζ~+1+iγ~2k.Equation (81) reads
(83)Fjl,γ,Γ,Rrk>0r⟶∞⟶V1zsinx1+δζ~+V2zcosx1+δζ~V3zsinx2+δζ~−1+V4zcosx2+δζ~−1,=V12z+V22z1/2sinx1+δζ~0+δγ,Γ,1CV32z+V42z1/2sinx2+δζ~−10+δγ,Γ,2C,where the Coulomb-modified phase shift δγ,ΓC is given by
(84)δγ,Γ,1Cδγ,Γ,2C=−arctanV2zV1z−arctanV4zV3z,where
(85)V1z=d1z+d2z−i∑n,mϑγ,nm′zd4zsinarctanγ~2kζ~−∑n,mϑγ,nm′zid3z+d4zcosarctanγ~2kζ~,V2z=d2z−i∑n,mϑγ,nm′zd4zcosarctanγ~2kζ~+∑n,mϑγ,nm′zd3z+d4zsinarctanγ~2kζ~,V3z=γκjlc+ζ−d1z+i∑n,mϑγ,nm′zd3zsinarctanγ~2kζ~−γκjlc+ζ−1d2z+∑n,mϑγ,nm′z−γκjlc+ζd3zcosarctanγ~2kζ~+iγκjlc+ζ−1d4z,V4z=γκjlc+ζ−d1z+i∑n,mϑγ,nm′zd3zcosarctanγ~2kζ~+∑n,mϑγ,nm′z−γκjlc+ζd3zsinarctanγ~2kζ~−γκjlc+ζ−1d4z,ϑγ,nm′z=ϑγ,nm1fγ,1z,Rnfγ,1z,Rn+ϑγ,nm2fγ,2z,Rnfγ,2z,Rn.The Coulomb-modified on-shell scattering matrix is given by(86)Sγ,Γ=exp2iδγ,Γ,nCz,n=1,2.The partial wave scattering amplitude is given by(87)fγ,Γ=exp2iδγ,Γ,nCz−12ik,n=1,2.



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*Source: 5825397-2020-04-15.xml*