In this section, we discuss the properties of the δ ′ -sphere interaction of second kind called “ δ ′ s -sphere interaction”. Using the theory of self-adjoint of symmetric closed operators in Hilbert space, we provide the mathematical definition of quantum Hamiltonian describing the δ ′ s -sphere interaction formally given by:

H G ^ = H D + G ^ δ ′ s ( | x | − R ) ,   x ∈ IR 3 ,   R ∈ IR (1)

where H D is the Dirac Hamiltonian and G ^ is a real 4 × 4 matrix defined by:

G ^ = ( 1 l A j l 0 0 1 l B j l ) ,

A j l and B j l are real constants and R is the radius of a sphere in IR 3 centered at the origin.

In the Hilbert space H = L 2 ( IR 3 ) ⊗ C 4 I , consider the Dirac Hamiltonian H D defined by:

H D = −   i   α ^ ∇ + β ^ c 2 2 , D ( H D ) = H 1,2 ( IR 3 ) ⊗ C 4 I (2)

where we have used the following definitions and notations:

1) c is the velocity of the light;

2) H m , n is the Sobolev space of indices ( m , n ) ;

3) α ^ and β ^ are 4 × 4 Dirac matrix given by:

α ^ = ( 0 σ σ 0 ) ,     β ^ = ( 1 l 0 0 − 1 l ) (3)

where σ are Pauli’s spin matrix defined by:

σ 1 = ( 0 1 1 0 ) ,   σ 2 = ( 0 − i i 0 ) ,   σ 3 = ( 1 0 0 − 1 ) . (4)

Consider in L 2 ( IR 3 ) ⊗ C 4 I , the symmetric closed operator H ^ D defined by:

H ^ D = −   i   α ^ ∇ + β ^ c 2 2 , D ( H ^ D ) = { χ j l ∈ H 1,2 ( IR 3 ) ⊗ C 4 I / χ j l ( S R ) = 0 } , (5)

where S R = { x ∈ IR 3 : | x | = R n } is the sphere of radius R in IR 3 centered at the origin.

The operator H ^ D admits a large number of self-adjoint extensions [25] . In this case, only those of H ^ D corresponding to H G ^ , which are rotationally and space-reflection symmetric, will be considered.

Under these assumptions, one may decompose the space H in the following way:

H = ⊕ j = 1 2 ∞   ⊕ l = j − 1 2 j + 1 2   ⊕ μ = − j j   H j l μ , (6)

where we have:

1) H j l μ = { χ j l ∈ H | χ j l ( r , n ) = ( f ( r ) Ω j l μ g ( r ) Ω j l ′ μ ) ; f , g ∈ L 2 ( ( 0, ∞ ) , r 2 d r ) } , (7)

2) The spherical spinors Ω j l μ are defined by:

Ω j l μ = ( j + μ 2 l + 1 Y l , μ − 1 2 ( θ , φ ) j − μ 2 l + 1 Y l , μ + 1 2 ( θ , φ ) )     for       l = j − 1 2 , Ω j l ′ μ = ( − j − μ + 1 2 l + 1 Y l , μ − 1 2 ( θ , φ ) j + μ + 1 2 l + 1 Y l , μ + 1 2 ( θ , φ ) )     for       l = j + 1 2 . (8)

3) l ′ = j ∓ 1 2 for l = j ± 1 2 .

The following isomorphism U j l defined by:

U l j : L 2 ( ( 0, ∞ ) ; r 2 d   r ) ⊗ C 2 I → H ˜ = L 2 ( ( 0, ∞ ) ; d   r ) ⊗ C 2 I (9)

( U j l ) χ j l ψ ( r ) = ( r f ( r ) ( − 1 ) j − l − 1 2 r g ( r ) ) (10)

is introduced to allow us to represent H in the form:

H = ⊕ j = 1 2 ∞   ⊕ l = j − 1 2 j + 1 2   ⊕ μ = − j j [ U j l − 1 H ˜ ] ⊗ [ Ω j l μ ( θ , φ ) ] , (11)

where [ Ω j l ( θ , φ ) ] stands for the vector space generated by the spherical spinors.

With respect to the decomposition (11), H ^ D reads:

H ^ D = ⊕ j = 1 2 ∞   ⊕ l = j − 1 2 j + 1 2 [ U j l − 1 h j l U j l ] ⊗ 1 l . (12)

The operator h j l in L 2 ( ( 0, ∞ ) ) ⊗ C 2 I is given by:

h j l = ( c 2 2 − c d d r + c κ j l r c d d r + c κ j l r − c 2 2 ) ≡ τ ,   κ j l = ( − 1 ) j − l + 1 2 ( j + 1 2 ) , D ( h j l ) = { χ j l ∈ L 2 ( ( 0, ∞ ) ) ⊗ C 2 I | χ j l ∈ A C l o c ( ( 0, ∞ ) ) ;   χ j l ( R ± ) = 0 ;                             τ χ j l ∈ L 2 ( ( 0, ∞ ) ) ⊗ C 2 I } , (13)

where A C l o c ( Ω ) denotes the set of locally absolutely continuous functions on Ω and

χ j l ( x ± ) = l i m ε → 0 + χ j l ( x ± ε ) . (14)

The adjoint h j l * of h j l is given by:

h j l * = τ , D ( h j l * ) = { f ∈ L 2 ( ( 0, ∞ ) ) ⊗ C 2 I | f ∈ A C l o c ( ( 0, ∞ ) \ { R } ) ,   τ f ∈ L 2 ( ( 0, ∞ ) ) ⊗ C 2 I } . (15)

Given the following equation:

( h j l * − z ) χ j l = 0 ,   χ j l ∈ IC − { ( − ∞ , − c 2 2 ] ∪ [ c 2 2 , ∞ ) } (16)

One can write Equation (16) in the following form:

χ ″ j l , 1 + [ k 2 ( z ) − κ j l ( κ j l + 1 ) r 2 ] χ j l , 1 = 0 χ ″ j l , 2 + [ k 2 ( z ) − κ j l ( κ j l − 1 ) r 2 ] χ j l , 2 = 0 (17)

where

k ( z ) = 1 c z 2 − c 4 4 . (18)

Equation (17) has two linearly independent solutions:

χ j l ( 1 ) = { ( F j l ( z , r ) F ˜ j l ( z , r ) ) ,   r < R , ( 0 0 ) ;   r > R , (19)

χ j l ( 2 ) = { ( 0 0 ) ;   r < R , ( G j l ( z , r ) G ˜ j l ( z , r ) ) ,   r > R , (20)

where:

F j l ( z , r ) = ( k 2 ) − κ j l − 1 2 Γ ( κ j l + 3 2 ) ( k 2 − κ j l 2 R 2 ) − 1 r 1 2 J κ j l + 1 2 ( k r ) , F ˜ j l ( z , r ) = 1 c ( 1 2 ) − κ j l − 1 2 k − κ j l + 1 2 Γ ( κ j l + 3 2 ) ( k 2 − κ j l 2 R 2 ) − 1 r 1 2 J κ j l − 1 2 ( k r ) , G j l ( z , r ) = i π 2 ( k 2 ) κ j l + 1 2 Γ ( κ j l + 3 2 ) − 1 r 1 2 H κ j l + 1 2 ( 1 ) ( k r ) , G ˜ j l ( z , r ) = i π 2 c ( 1 2 ) κ j l + 1 2 k κ j l + 3 2 Γ ( κ j l + 3 2 ) − 1 r 1 2 H κ j l − 1 2 ( 1 ) ( k r ) . (21)

J p ( . ) is the Bessel function and H p ( 1 ) ( . ) is the Hankel function of the first kind of order p.

The solutions Equation (19) and Equation (20) have been normalized in such a way that:

det [ G ′ j l ( z , r ) F ′ j l ( z , r ) G ˜ ′ j l ( z , r ) F ˜ ′ j l ( z , r ) ] = 1 c . (22)

Therefore, h j l has indices (2, 2) and consequently, all self-adjoint(sa) extensions of h j l are given by a four-parameter family of self-adjoint operators [26] . Since the matrix G ^ in Equation (28) depends on two parameters, it follows that the self-adjoint extension h j l , G ^ of h j l corresponding to the interaction V ( r ) = G ^ δ ( r − R ) is a special two-parameter family.

The relation h j l , G ^ ⊂ h j l * implies that the domain D ( h j l , G ^ ) contains those functions χ j l ∈ D ( h j l * ) which satisfy suitable boundary conditions at r = R .

Theorem 2.1: Any self-adjoint extension h ^ j l of h j l reads [25] :

h ^ j l = ( c 2 2 − c d d   r + c κ j l r c d d   r + c κ j l r − c 2 2 ) , D ( h ^ j l ) = { χ j l ∈ D ( h j l * ) | χ j l     satifies   cond 1   or   cond 2 } , (23)

where cond₁ and cond₂ are given by [25] :

cond 1 : χ j l ( R − ) = e i φ M ¯ χ j l ( R + ) ,   φ ∈ [ 0, π ) , (24)

and M ¯ is 2 × 2 matrix with det M ¯ = 1 .

cond 2 : ( a 1 a 2 0 0 ) χ j l ( R − ) + ( 0 0 b 1 b 2 ) χ j l ( R + ) = 0 (25)

where a 1 , a 2 , b 1 and b 2 are real and both matrices are nonzero. Conversely, any operator of this form is self-adjoint extensions of h j l .

Theorem 2.2: [25] The general form of boundary conditions is given by:

1 − τ 0 G ^ j l 2 c χ j l ( R + ) − 1 + τ 0 G ^ j l 2 c χ j l ( R − ) = 0 (26)

where τ 0 = ( 0 − 1 1 0 ) and G ^ j l = ( A j l 0 0 B j l ) .

Let us now construct the self-adjoint extension corresponding to the radial Dirac operator with the potential:

V ( r ) = G ^ j l δ ′ s ( r − R ) ,   G ^ j l = ( A j l 0 0 B j l ) ,   A j l , B j l ∈ IR . (27)

Suppose that χ j l satisfies the following equation given by:

[ τ + G ^ j l δ ( r − R ) ] χ j l = E χ j l , (28)

τ = ( c 2 2 − c d d r + c κ j l r c d d r + c κ j l r − c 2 2 ) , (29)

where the limits χ ( R ± ε ) exist when ε → 0 . A simple computation shows that integrating Equation (28) over ( R − ε , R + ε ) , and taking the limit ε → 0 we get the following conditions:

{ χ j l ,   1 ( R + ) − χ j l ,   1 ( R − ) = − 1 2 C B j l [ χ j l ,   2 ( R + ) + χ j l   2 ( R − ) ] χ j l ,   2 ( R + ) − χ j l ,   2 ( R − ) = 1 2 C A j l [ χ j l ,   1 ( R + ) + χ j l ,   1 ( R − ) ] (30)

By replacing the function χ j l in Equation (30) by its derivative, we get the following conditions:

{ χ ′ j l ,   1 ( R + ) − χ ′ j l ,   1 ( R − ) + 1 2 C B j l [ χ ′ j l ,   2 ( R + ) + χ ′ j l ,   2 ( R − ) ] = 0 χ ′ j l ,   2 ( R + ) − χ ′ j l ,   2 ( R − ) − 1 2 C A j l [ χ ′ j l ,   1 ( R + ) + χ ′ j l ,   1 ( R − ) ] = 0 (31)

The boundary conditions Equation (31) can be written in the general form:

1 − τ 0 G ^ j l 2 c χ ′ j l ( R + ) − 1 + τ 0 G ^ j l 2 c χ ′ j l ( R − ) = 0. (32)

A straightforward computation shows that these boundary conditions are symmetric and linearly independent and can be written in the form of Equation (26).

One can construct the self-adjoint extension of h j l considering its adjoint h j l * where:

D ( h ^ j l ) = { χ j l ∈ D ( h j l * ) | χ j l   satifies   Eq . ( 32 )   } .

The boundary conditions of Equation (31) characterize the potential of Equation (27) and introduce a new exactly solvable model of relativistic δ ′ s -sphere interactions in quantum mechanics.

Let us consider in L 2 ( ( 0, ∞ ) ) ⊗ C 2 I the operator h j l , G ^ j l defined by:

h j l , G ^ j l = ( c 2 2 − c d d r + c κ j l r c d d r + c κ j l r − c 2 2 ) ≡ τ D ( h j l , G ^ j l ) = { χ j l ∈ D ( h j l * ) | χ ′ j l ,1 ( R + ) − χ ′ j l ,1 ( R − ) + 1 2 C B j l [ χ ′ j l ,2 ( R + ) + χ ′ j l ,2 ( R − ) ] = 0 χ ′ j l ,2 ( R + ) − χ ′ j l ,2 ( R − ) − 1 2 C A j l [ χ ′ j l ,1 ( R + ) + χ ′ j l ,1 ( R − ) ] = 0 } ,                                       l ∈ [ j − 1 2 , j + 1 2 ] ,   j ∈ [ 1 2 , ∞ ) . (33)

The operator h j l , G ^ j l is the self-adjoint extension of the symetric operator h j l . The operator h j l , G ^ j l gives the mathematical definition of the formal expression:

h j l , G ^ j l = h D + G ^ j l δ ′ s ( r − R ) , (34)

where δ ′ s is the δ ′ -sphere interaction of the second kind characterized by the boundary conditions of Equation (31) and h D is the radial Dirac Hamiltonian defined by:

h D = ( c 2 2 − c d d r + c κ j l r c d d r + c κ j l r − c 2 2 ) , D ( h D ) = { χ j l ∈ D ( h j l * ) | χ ′ j l ( R + ) = χ ′ j l ( R − ) χ j l ( R + ) = χ j l ( R − ) } ,   l ∈ [ j − 1 2 , j + 1 2 ] ,   j ∈ [ 1 2 , ∞ ) . (35)

The particular case G ^ j l = 0 in Equation (33) yields the radial Dirac Hamiltonian h j l ,0 ≡ h D . The case A j l ≠ 0 , B j l = 0 in Equation (33) gives the one parameter δ ′ s -sphere interaction defined by:

h j l , A j l = τ D ( h j l , A j l ) = { χ j l = ( χ j l , 1 χ j l , 2 ) ∈ D ( h j l * ) | χ ′ j l , 1 ( R + ) = χ ′ j l , 1 ( R − ) ≡ χ ′ j l , 1 ( R ) , χ ′ j l , 2 ( R + ) − χ ′ j l , 2 ( R − ) = A j l c χ ′ j l , 1 ( R ) } .

Even, the case A j l = 0 , B j l ≠ 0 in Equation (33) provides the one parameter δ ′ s -sphere interaction defined by:

h j l , B j l = τ D ( h j l , B j l ) = { χ j l = ( χ j l , 1 χ j l , 2 ) ∈ D ( h j l * ) | χ ′ j l , 2 ( R + ) = χ ′ j l , 2 ( R − ) ≡ χ ′ j l , 2 ( R ) , χ ′ j l , 1 ( R + ) − χ ′ j l , 1 ( R − ) = − B j l , 2 c χ ′ j l , 2 ( R ) } . (36)

Let G ^ = { G ^ j l } ,   j − 1 2 ≤ l ≤ j + 1 2 ,   1 2 ≤ j < ∞ . The decomposition of Equation (12) implies that the operator H G ^ in L 2 ( IR 3 ) ⊗ C 4 I defined by:

H G ^ = ⊕ j = 1 2 ∞   ⊕ l = j − 1 2 j + 1 2 [ U j l − 1 h j l , G ^ j l U j l ] ⊗ 1 l . (37)

provides the mathematical definition of the formal expression Equation (2). The operator H G ^ in Equation (35) defines the δ ′ s interaction in the tree-dimensional space.

The case G ^ = 0 , i.e. G ^ j l = 0 for all j and l yields the Dirac Hamiltonian H D defined by Equation (2).

Theorem 2.3: The resolvent of h j l , G ^ j l leads:

( h j l , G ^ j l − z ) − 1 ( . ) = ( h D − z ) − 1 ( . ) + Ω j l ( z , R ) { ( Q j l 2 ( . ) ¯   , . ) [ A j l Q j l ( 2 ) ( . ) + A j l B j l 2 c Q ˜ j l ( 1 ) ( . ) ]       + ( Q ˜ j l ( 2 ) ( . ) ¯   , . ) [ B j l Q ˜ j l ( 2 ) ( . ) − A j l B j l 2 c Q j l ( 1 ) ( . ) ] } ,       z ∈ ρ ( h j l , G ^ j l ) ,   Im k > 0 ,   l ∈ [ j − 1 2 , j + 1 2 ] ,   j ∈ [ 1 2 , ∞ ) , (38)

where ρ ( . ) is the resolvent set and where ( h D − z ) − 1 , Im k ( z ) > 0 is the radial Dirac resolvent with kernel:

G j l ( z , r , R ) = ( G 11 j l ( z , r , R ) G 12 j l ( z , r , R ) G 21 j l ( z , r , R ) G 22 j l ( z , r , R ) ) , (39)

where

G 11 j l ( z , r , R ) = { G j l ( z , R ) F j l ( z , r ) ,   r < R F j l ( z , R ) G j l ( z , r ) ,   r > R , G 12 j l ( z , r , R ) = { G ˜ j l ( z , R ) F j l ( z , r ) ,   r < R F ˜ j l ( z , R ) G j l ( z , r ) ,   r > R , G 21 j l ( z , r , R ) = { G j l ( z , R ) F ˜ j l ( z , r ) ,   r < R F j l ( z , R ) G ˜ j l ( z , r ) ,   r > R , G 22 j l ( z , r , R ) = { G ˜ j l ( z , R ) F ˜ j l ( z , r ) ,   r < R F ˜ j l ( z , R ) G ˜ j l ( z , r ) ,   r > R . (40)

and

Ω j l ( z , A j l , B j l , R ) = − [ 1 − B j l A j l 4 c 2 + B j l F ˜ ′ j l ( z , R ) G ˜ ′ j l ( z , R ) + A j l F ′ j l ( z , R ) G ′ j l ( z , R ) ] − 1 , (41)

Q j l ( 2 ) = { ( G ′ j l ( z , R ) F j l ( z , r ) G ′ j l ( z , R ) F ˜ j l ( z , r ) ) ;   r < R , ( F ′ j l ( z , R ) G j l ( z , r ) F ′ j l ( z , R ) G ˜ j l ( z , r ) ) ;   r > R , (42)

Q ˜ j l ( 2 ) = { ( G ′ ˜ j l ( z , R ) F j l ( z , r ) G ′ ˜ j l ( z , R ) F ˜ j l ( z , r ) ) ;   r < R , ( F ′ ˜ j l ( z , R ) G j l ( z , r ) F ′ ˜ j l ( z , R ) G ˜ j l ( z , r ) ) ;   r > R , (43)

Q ˜ j l ( 1 ) = { ( G ′ ˜ j l ( z , R ) F j l ( z , r ) G ′ ˜ j l ( z , R ) F ˜ j l ( z , r ) ) ;   r < R , − ( F ′ ˜ j l ( z , R ) G j l ( z , r ) F ′ ˜ j l ( z , R ) G ˜ j l ( z , r ) ) ;   r > R , (44)

Q j l ( 1 ) = { ( G ′ j l ( z , R ) F j l ( z , r ) G ′ j l ( z , R ) F ˜ j l ( z , r ) ) ;   r < R , − ( F ′ j l ( z , R ) G j l ( z , r ) F ′ j l ( z , R ) G ˜ j l ( z , r ) ) ;   r > R , (45)

with F j l ( z , r ) , F ˜ j l ( z , r ) , G j l ( z , r ) , G ˜ j l ( z , r ) and k ( z ) defined by Equation (21) and Equation (18).

Proof

One can use the Krein resolvent formula which yields the following relation for the resolvent of h j l , G ^ j l :

( h j l , G ^ j l − z ) − 1 = ( h D − z ) − 1 + ∑ p , q = 1 2   γ q p ( χ j l ( p ) ( . ) ¯   , . ) χ j l ( q ) ( . ) ,   z ∈ ρ ( h j l , G ^ j l ) ,     Im k ( z ) > 0 , (46)

where χ j l ( q ) ( r ) ,   q = 1 , 2 are given by Equations (19) and (20) respectively.

Taking the function χ j l = ( χ j l , 1 χ j l , 2 ) ∈ L ( ( 0 , ∞ ) ) ⊗ C 2 I , we can define the function η j l by:

η j l ( z , r ) = ( ( h j l , G ^ j l − z ) − 1 χ j l ) ( r ) . (47)

As the function η j l ∈ D ( h G ^ j l ) , it follows that η j l satisfies the boundary conditions in Equation (33). The implementation of these boundary conditions provides the constants γ q p ( z ) . When we insert γ q p ( z ) into Equation (46), we obtain Equation (38).

In particular case A j l ≠ 0 , B j l = 0 and A j l = 0, B j l ≠ 0 , the resolvent Equation (40) provides respectively:

( h j l , A j l − z ) − 1 ( . ) = ( h D − z ) − 1 ( . ) + A j l Ω j l ( z , A j l , 0 , R ) ( Q j l ( 2 ) ( . ) ¯   , . ) Q j l ( 2 ) ( . ) ,                                               z ∈ ρ ( h j l , A j l ) ,   Im k > 0 ,   l ∈ [ j − 1 2 , j + 1 2 ] ,   j ∈ [ 1 2 , ∞ ) , (48)

and

( h j l , B j l − z ) − 1 ( . ) = ( h D − z ) − 1 ( . ) + B j l Ω j l ( z , 0 , B j l , R ) ( Q ˜ j l ( 2 ) ( . ) ¯   , . ) Q ˜ j l ( 2 ) ( . ) ,                                                 z ∈ ρ ( h j l , B j l ) ,   Im k > 0 ,   l ∈ [ j − 1 2 , j + 1 2 ] ,   j ∈ [ 1 2 , ∞ ) , (49)

Theorem 2.4: For A j l , B j l ∈ ( ∞ , ∞ ) , l ∈ [ j − 1 2 , j + 1 2 ] , j ∈ [ 1 2 , ∞ ) , the essential spectrum of h j l , G ^ j l is purely absolutely continuous and coincides with ( − ∞ , − c 2 2 ] ∪ [ c 2 2 , ∞ ) . Its singularly continuous and residual spectra are empty.

Proof

Proposition 6.1 and Theorem 6.2.in Ref. [25] provide detailed proof.

The eigenvalues E of h j l , G ^ j l located in [ − c 2 2 , c 2 2 ] are given by the pole of the resolvent equation in physical sheet Im k > 0 , i.e. the solution of:

1 − B j l A j l 4 c 2 + B j l F ˜ ′ j l ( k , R ) G ˜ ′ j l ( k , R ) + A j l F ′ j l ( k , R ) G ′ j l ( k , R ) | k ( z ) = i − P = 0 , P < 0 (50)

where k = i − P = 1 c E 2 − c 4 4 .

A straightforward computation shows that Equation (50) reads:

B j l A j l 4 c 2 − 1 = ( P − κ j l 2 R 2 ) − 1 ( B j l c 2 P [ r 1 2 K κ j l − 1 2 ( − P r ) ] ′ r = R [ r 1 2 I κ j l − 1 2 ( − P r ) ] ′ r = R     + A j l [ r 1 2 K κ j l + 1 2 ( − P r ) ] ′ r = R [ r 1 2 I κ j l + 1 2 ( − P r ) ] ′ r = R ) (51)

In Figure 1, we consider the following normalization of the energy ℏ = 2 m = c 2 = 1 . Consider also K κ j l = − 2 with the following condition for the coupling constants: B j l A j l < 0 . Y 2 = B j l A j l 4 c 2 − 1 and Y 1 represents the second member of Equation (51). Using the graphical resolution method, i.e. by analyzing the intersection of the curves Y 2 and Y 1 , one can show easily that Equation (51) has two solutions which correspond to the two eigenvalues { E 1 , E 2 } of h j l , G ^ j l in [ − c 2 2 , c 2 2 ] .

The resonances of h j l , G ^ j l are defined as poles of the resolvent Equation (38) in the unphysical sheet Im k < 0 .

Following the strategy of Gesztesy et al. [27] , in the case of point interactions, one can discuss the nonrelativistic limit of h j l , G ^ j l as c → ∞ .

Theorem 2.5: For spin- 1 2 particules, the operator h j l , G ^ j l − 1 2 converges in norm resolvent sense to the Shrödinger operator h l , α ^ l times the projector onto H + = L 2 ( ( 0, ∞ ) ) :

n ⋅ l i m c → ∞ ( h j l , G ^ j l − c 2 2 − z ) − 1 = ( h l , α ^ l − z ) − 1 ⊗ ( 1 0 0 0 ) ,     z ∈ C \ IRI (52)

where h l , α ^ l is defined by:

h l , α ^ l = − d 2 d   r 2 + l ( l + 1 ) r 2 , D ( h l , α ^ l ) = { f ∈ L 2 ( ( 0 , ∞ ) ) | f , f ′ ∈ A C l o c ( ( 0 , ∞ ) \ { R } ) ,   f ( 0 + ) = 0 ,     if     l = 0 ,                                   ( 1 + α l 2 ) f ′ ( R + ) + ( 1 − α l 2 ) f ′ ( R − ) = 0 ,                                   ( 1 + β l 2 ) f ( R + ) + ( 1 − β l 2 ) f ( R − ) = 0 ,                                   f ″ + l ( l + 1 ) r − 2 f ∈ L 2 ( ( 0 , ∞ ) ) } ,                                   α ^ l = { α l , β l } ,   − ∞ < α l , β l < ∞ ,   l ∈ IN 0 (53)

The boundary conditions in (53) define a self-adjoint extension of the radial schrödinger operator h ˙ l defined by:

h ˙ l = − d 2 d   r 2 + l ( l + 1 ) r 2 , D ( h ˙ l ) = { f ∈ L 2 ( ( 0 , ∞ ) ) | f , f ′ ∈ A C l o c ( ( 0 , ∞ ) ) ,   f ( 0 + ) = 0 ,     i f     l = 0 ,                               f ( R ± ) = f ′ ( R ± ) = 0 , − f ″ + l ( l + 1 ) r − 2 f ∈ L 2 ( ( 0 , ∞ ) ) } , l ∈ IN 0 (54)

Proof

One may follow step by step Ref. [27] , where a similar result was obtained in the case of the point interactions. The Hamiltonian h l , α ^ l defines a two-parameter model of nonrelativistic δ ′ -sphere interaction in quantum mechanics.

Let us define for k > 0 the following function:

( η j l , 1 ( z , r ) η j l , 2 ( z , r ) ) = ( F j l ( z , r ) F ˜ j l ( z , r ) ) + Ω j l ( z , A j l , B j l , R ) { F ′ j l ( z , R ) [ A j l Q j l ( 2 ) ( r ) + A j l B j l 2 c Q ˜ j l ( 1 ) ( r ) ]     + F ˜ ′ j l ( z , R ) [ B j l Q ˜ j l ( 2 ) ( r ) − A j l B j l 2 c Q j l ( 1 ) ( r ) ] } (55)

where the functions F j l ( z , r ) ,   F ˜ j l ( z , r ) ,   Ω j l ( z , A j l , B j l , R ) ,   Q j l ( 2 ) ,   Q ˜ j l ( 2 ) ,   Q ˜ j l ( 1 ) and Q j l ( 1 ) are defined by Equation (21), Equations (41)-(45) respectively. A straightforward computation shows that ( η j l , 1 ( z , r ) η j l , 2 ( z , r ) ) are scattering wave functions of h j l , G ^ j l .

For the cases A j l ≠ 0 and B j l = 0 and A j l = 0 ,   B j l ≠ 0 , (55) yields, respectively to:

( η j l , 1 ( z , r ) η j l , 2 ( z , r ) ) = F j l ( z , r ) F ˜ j l ( z , r ) + A j l Ω j l ( z , A j l ,0, R ) F ′ j l ( z , r ) Q j l ( 2 ) ( . ) (56)

and

( η j l , 1 ( z , r ) η j l , 2 ( z , r ) ) = F j l ( z , r ) F ˜ j l ( z , r ) + B j l Ω j l ( z ,0, B j l , R ) F ˜ ′ j l ( z , r ) Q ˜ j l ( 2 ) ( . ) (57)

Equation (56) and Equation (57) define the scattering wave functions corresponding to the Hamiltonian h j l , A j l and h j l , B j l describing two one parameter relativistic δ ′ s -sphere interactions.

Let us determine the phase shift and the elements of the on-shell scattering matrix corresponding to h j l , G ^ j l using the asymptotic behavior of η j l ( z , r ) .

The asymptotic behavior of ( η j l , 1 ( z , r ) η j l , 2 ( z , r ) ) as r → ∞ yields to:

( η j l , 1 ( z , r ) η j l , 2 ( z , r ) ) → r → ∞ k > 0 ( C j l ( z ) sin [ k r − κ j l π 2 ] C ˜ j l ( z ) sin [ k r − ( κ j l − 1 ) π 2 ] ) + Ω j l ( z , A j l , B j l , R ) [ A j l F ′ j l 2 ( z , R ) + B j l F ˜ ′ j l 2 ( z , R ) ] ( D j l ( z ) exp   − i [ k r − κ j l π 2 ] D ˜ j l ( z ) exp   − i [ k r − ( κ j l − 1 ) π 2 ] ) = ( [ D 1 2 ( z ) + D 2 2 ( z ) ] 1 2 sin [ k r − κ j l π 2 + φ G j l ( z ) ] [ D 3 2 ( z ) + D 4 2 ( z ) ] 1 2 sin [ k r − ( κ j l − 1 ) π 2 + φ ˜ G j l ( z ) ] ) (58)

where

C j l ( z ) = 2 − κ j l k − κ j l − 1 ( k 2 − κ j l 2 R 2 ) − 1 Γ ( 2 κ j l + 2 ) Γ ( κ j l + 1 ) − 1 , C ˜ j l ( z ) = 1 c 2 − κ j l k − κ j l ( k 2 − κ j l 2 R 2 ) − 1 Γ ( 2 κ j l + 2 ) Γ ( κ j l + 1 ) − 1 , D j l ( z ) = 2 κ j l k κ j l Γ ( 2 κ j l + 2 ) − 1 Γ ( κ j l + 1 ) , D ˜ j l ( z ) = 1 c 2 κ j l k κ j l + 1 Γ ( 2 κ j l + 2 ) − 1 Γ ( κ j l + 1 ) . (59)

In this case, the phase shifts corresponding to h j l , G ^ j l are defined as:

φ G ^ j l = − arctan D 2 D 1 = − arctan D 4 D 3 = − arctan D j l ( z ) Ω ¯ j l ( z ) C j l ( z ) − i   D j l ( z ) Ω ¯ j l ( z ) = − arctan D ˜ j l ( z ) Ω ¯ j l ( z ) C ˜ j l ( z ) − i   D ˜ j l ( z ) Ω ¯ j l ( z ) , (60)

where

Ω ¯ j l ( z ) = Ω j l ( z , A j l , B j l , R ) [ A j l F ′ j l ( z , R ) F ′ j l ( z , R ) + B j l F ′ ˜ j l ( z , R ) F ′ ˜ j l ( z , R ) ] (61)

The elements of the on-shell scattering matrix are given by:

S G ^ j l ( z ) = exp [ 2   i   φ G ^ j l ( z ) ] . (62)

The partial wave scattering amplitude is given by:

f G ^ j l ( z ) = exp [ 2   i   φ G ^ j l ( z ) ] − 1 2   i   k . (63)

Let us consider the formal expression given by:

H α , G ^ = H D + α | x | + G ^ δ ′ s ( | x | − R ) ,   α ∈ IR ,   x ∈ IR 3 ,   R > 0, (64)

where G ^ is a real matrix of the form:

G ^ = ( A j l 0 0 B j l ) (65)

Let use the decomposition Equation (12) and introduce in L 2 ( ( 0, ∞ ) ) ⊗ C 2 I the operator:

H ^ α , G ^ = ⊕ j = 1 2 ∞   ⊕ l = j − 1 2 j + 1 2 [ U j l − 1 h j l , α U j l ] ⊗ 1 l , (66)

where h j l , α is given by:

h j l , α = ( c 2 2 + α r − c d d   r + c κ j l r c d d   r + c κ j l r − c 2 2 + α r ) ≡ τ α , D ( h j l , α ) = { χ j l ∈ L 2 ( ( 0 , ∞ ) ) ⊗ C 2 I | χ j l ∈ A C l o c ( ( 0 , ∞ ) ) ;   χ j l ( R ± ) = 0 ;                                 τ α χ j l ∈ L 2 ( ( 0 , ∞ ) ) ⊗ C 2 I } . (67)

κ j l is defined by Equation (13).

The adjoint H ^ α , G ^ * of H ^ α , G ^ reads:

H ^ α , G ^ * = ⊕ j = 1 2 ∞   ⊕ l = j − 1 2 j + 1 2 [ U j l − 1 h j l , α * U j l ] ⊗ 1 l , h j l , α * = ( c 2 2 + α r − c d d   r + c κ j l r c d d   r + c κ j l r − c 2 2 + α r ) ≡ τ α , D ( h j l , α * ) = { χ j l ∈ L 2 ( ( 0 , ∞ ) ) ⊗ C 2 I | χ j l ∈ A C l o c ( ( 0 , ∞ ) \ { R } ) ,                                 τ α χ j l ∈ L 2 ( ( 0 , ∞ ) ) ⊗ C 2 I } . (68)

Let us consider the following equation:

( h j l , α * − z ) χ j l = 0,   χ j l = ( f g ) ∈ D ( h j l , α * ) ,   z ∈ IC − { ( − ∞ , − c 2 2 ] ∪ [ c 2 2 , ∞ ) } , (69)

and introduce the following notations:

k = 1 c z 2 − c 4 4 , ζ = κ j l 2 c 2 − α 2 , ζ ˜ = 1 c ζ , α ˜ = 2 z α c 2 . (70)

Equation (69) has two linearly independent solutions:

χ j l , α , z ( 1 ) = { ( f j l , α , 1 ( z , r ) f j l , α , 2 ( z , r ) ) ,   r < R , ( 0 0 ) ;   r > R , (71)

χ j l , α , z ( 2 ) = { ( 0 0 ) ;   r < R , ( g j l , α , 1 ( z , r ) g j l , α , 2 ( z , r ) ) ,   r > R , (72)

where

f j l , α , 1 ( z , r ) = ( 1 − α 2 ( κ j l c + ζ ) 2 ) − 1 2 [ cos ( arctan α ˜ 2 k ζ ˜ ) ] − 1 2 [ F j l , α ( z , r ) − α κ j l c + ζ F ˜ j l , α ( z , r ) ] , f j l , α , 2 ( z , r ) = ( 1 − α 2 ( κ j l c + ζ ) 2 ) − 1 2 [ cos ( arctan α ˜ 2 k ζ ˜ ) ] − 1 2 [ F ˜ j l , α ( z , r ) − α κ j l c + ζ F j l , α ( z , r ) ] ,

g j l , α , 1 ( z , r ) = γ ( κ j l , R ) ( 1 − α 2 ( κ j l c + ζ ) 2 ) − 1 2 [ cos ( arctan α ˜ 2 k ζ ˜ ) ] − 1 2 [ G j l , α ( z , r ) − α κ j l c + ζ G ˜ j l , α ( z , r ) ] , g j l , α , 2 ( z , r ) = γ ( κ j l , R ) ( 1 − α 2 ( κ j l c + ζ ) 2 ) − 1 2 [ cos ( arctan α ˜ 2 k ζ ˜ ) ] − 1 2 [ G ˜ j l , α ( z , r ) − α κ j l c + ζ G j l , α ( z , r ) ] . (73)

and

F j l , α ( z , r ) = r ζ ˜ + 1   e − i k r F 1 1 ( ζ ˜ + 1 − i α ˜ 2 k , 2 ζ ˜ + 2 , 2   i   k r ) , G j l , α ( z , r ) = Γ ( 2 ζ ˜ + 2 ) − 1 Γ ( ζ ˜ + 1 − i   α ˜ 2 k ) ( 2   i   k ) 2 ζ ˜ + 1 r ζ ˜ + 1   e − i k r U ( ζ ˜ + 1 − i α ˜ 2 k , 2 ζ ˜ + 2 , 2   i   k r ) , F ˜ j l , α ( z , r ) = ζ ˜ c ( 2 ζ ˜ + 1 ) | Γ ( ζ ˜ + i   α ˜ 2 k ) | | Γ ( ζ ˜ + 1 + i   α ˜ 2 k ) | − 1 r ζ ˜   e − i k r × F 1 1 ( ζ ˜ −   i   α ˜ 2 k , 2 ζ ˜ , 2   i   k r ) , G ˜ j l , α ( z , r ) = ζ ˜ − 1 c ( 2 ζ ˜ + 1 ) − 1 Γ ( 2 ζ ˜ ) − 1 Γ ( ζ ˜ + i   α ˜ 2 k ) | Γ ( ζ ˜ + i   α ˜ 2 k ) | − 1 | Γ ( ζ ˜ + 1 + i   α ˜ 2 k ) |                                     × k 2 ( 2   i   k ) 2 ζ ˜ − 1 r ζ ˜   e − i k r U ( ζ ˜ − i α ˜ 2 k , 2 ζ ˜ , 2   i   k r ) , (74)

F 1 1 ( a , b , r ) ( U ( a , b , r ) ) denotes the regular (respectively, irregular) confluent hypergeometric functions.

γ ( κ j l , R ) is the normalization constants chosen in such way:

det [ g ′ j l , α , 1 ( z , r ) f ′ j l , α , 1 ( z , r ) g ′ j l , α , 2 ( z , r ) f ′ j l , α , 2 ( z , r ) ] = 1 c (75)

For the particular case α → 0 + , we obtain:

( f ′ j l , α , 1 ( z , r ) f ′ j l , α , 2 ( z , r ) ) → α → 0 + ( F ′ j l ( z , r ) F ˜ ′ j l ( z , r ) ) (76)

( g ′ j l , α , 1 ( z , r ) g ′ j l , α , 2 ( z , r ) ) → α → 0 + ( G ′ j l ( z , r ) G ˜ ′ j l ( z , r ) ) (77)

where F ′ j l ( z , r ) , F ˜ ′ j l ( z , r ) , G ′ j l ( z , r ) and G ˜ ′ j l ( z , r ) are respectively the derivatives of the functions F j l ( z , r ) , F ˜ j l ( z , r ) , G j l ( z , r ) and G ˜ j l ( z , r ) defined by Equation (21).

The operator h j l , α has deficiency indices (2, 2), and consequently, all its self-adjoint extensions may be parametrized by a four-parameter family of self-adjoint operators.

Consider the following two-parameters family of self-adjoint extensions of h j l , α :

h j l , α , G j l = ( c 2 2 + α r − c d d   r + c κ j l r − c d d   r + c κ j l r − c 2 2 + α r ) ,

D ( h j l , α , G j l ) = { χ j l ∈ D ( h j l * ) | { χ ′ j l , 1 ( R + ) − χ ′ j l , 1 ( R − ) + 1 2 C B j l [ χ ′ j l , 2 ( R + ) + χ ′ j l , 2 ( R − ) ] = 0 χ ′ j l , 2 ( R + ) − χ ′ j l , 2 ( R − ) − 1 2 C A j l [ χ ′ j l , 1 ( R + ) + χ ′ j l , 1 ( R − ) ] = 0 } }                                       A j l , B j l ∈ IR ,   l ∈ [ j − 1 2 , j + 1 2 ] ;   j ∈ [ 1 2 , ∞ ) . (78)

The operator h j l , α , G j l gives the mathematical definition of the formal expression:

h j l , α , G j l = h D + α r + G j l δ ′ s ( r − R ) ,   R > 0. (79)

The case G j l = 0 in Equation (78) gives the radial Dirac-Coulomb Hamiltonian h j l , α ,0 ≡ h α , D ,

h α , D = h D + α r ≡ τ α , D ( h α , D ) = { χ j l ∈ D ( h j l , α * ) | χ j l ( R + ) = χ j l ( R − ) χ ′ j l ( R + ) = χ ′ j l ( R − ) } ;   [ j − 1 2 , j + 1 2 ] , [ 1 2 , ∞ ) . (80)

In particular case when A j l ≠ 0 , B j l = 0 and A j l = 0 , B j l ≠ 0 in Equation (78) simplifies respectively to one parameter δ ′ s -sphere plus Coulomb interaction. Therefore, the model in Equation (64) is defined in L 2 ( IR 3 ) ⊗ C 4 I by:

H ^ α , G ^ = ⊕ j = 1 2 ∞   ⊕ l = j − 1 2 j + 1 2 [ U j l − 1 h j l , α , G ^ j l U j l ] ⊗ 1 l . (81)

The particular case G ^ j l = 0 for all j and l provides the Dirac-Coulomb Hamiltonian H α , D ,

H α , D = H D + α | x | ,   D ( H α , D ) = H 1,2 ( IR 3 ) ⊗ C 4 I . (82)

Theorem 3.1: The resolvent of h j l , α , G ^ j l reads:

( h j l , α , G j l − z ) − 1 ( . ) = ( h α , D − z ) − 1 ( . ) + Ω j l , α ( z , R ) { ( Q j l , α ( 2 ) ( . ) ¯   , . )     × [ A j l Q j l , α ( 2 ) ( . ) + A j l B j l 2 c Q ˜ j l , α ( 1 ) ( . ) ] + ( Q ˜ j l , α ( 2 ) ( . ) ¯   , . ) [ B j l Q ˜ j l , α ( 2 ) ( . ) − A j l B j l 2 c Q j l , α ( 1 ) ( . ) ] } , z ∈ ρ ( h j l , α , G ^ j l ) ,   Im k > 0 ,   l ∈ [ j − 1 2 , j + 1 2 ] ,   j ∈ [ 1 2 , ∞ ) , (83)

where ρ ( . ) is the resolvent set and where ( h α , D − z ) − 1 , Im k ( z ) > 0 is the radial Dirac resolvent with kernel:

G j l , α ( z , r , R ) = ( G 11 j l , α ( z , r , R ) G 12 j l , α ( z , r , R ) G 21 j l , α ( z , r , R ) G 22 j l , α ( z , r , R ) ) , (84)

and

G 11 j l , α ( z , r , R ) = { g ′ j l , α , 1 ( z , R ) f j l , α , 1 ( z , r ) ,     r < R f ′ j l , α , 1 ( z , R ) g j l , 1 ( z , r ) ,     r > R , G 12 j l , α ( z , r , R ) = { g ′ j l , α , 2 ( z , R ) f j l , α , 1 ( z , r ) ,     r < R f ′ j l , α , 2 ( z , R ) g j l , α , 1 ( z , r ) ,     r > R , G 21 j l , α ( z , r , R ) = { g ′ j l , α , 1 ( z , R ) f j l , α , 2 ( z , r ) ,     r < R f ′ j l , α , 1 ( z , R ) g j l , α , 2 ( z , r ) ,     r > R , G 22 j l , α ( z , r , R ) = { g ′ j l , α , 2 ( z , R ) f j l , α , 2 ( z , r ) ,     r < R f ′ j l , α , 2 ( z , R ) g j l , α , 2 ( z , r ) ,     r > R (85)

Ω j l , α ( z , R ) = − [ 1 − B j l A j l 4 c 2 + B j l f ′ j l , α , 2 ( z , R ) g ′ j l , α , 2 ( z , R ) + A j l f ′ j l , α , 1 ( z , R ) g ′ j l , α , 1 ( z , R ) ] − 1 , (86)

Q j l , α ( 2 ) = { ( g ′ j l , α , 1 ( z , R ) f j l , α , 1 ( z , r ) g ′ j l , α , 1 ( z , R ) f j l , α , 2 ( z , r ) ) ,     r < R , ( f ′ j l , α , 1 ( z , R ) g j l , α , 1 ( z , r ) f ′ j l , α , 1 ( z , R ) g j l , α , 2 ( z , r ) ) ,     r > R , (87)

Q ˜ j l , α ( 2 ) = { ( g ′ j l , α , 2 ( z , R ) f j l , α , 1 ( z , r ) g ′ j l , α , 2 ( z , R ) f j l , α , 2 ( z , r ) ) ,     r < R , ( f ′ j l , α , 2 ( z , R ) g j l , α , 1 ( z , r ) f ′ j l , α , 2 ( z , R ) g j l , α , 2 ( z , r ) ) ,     r > R , (88)

Q ˜ j l , α ( 1 ) = { ( g ′ j l , α , 2 ( z , R ) f j l , α , 1 ( z , r ) g ′ j l , α , 2 ( z , R ) f j l , α , 2 ( z , r ) ) ,     r < R , ( − f ′ j l , α , 2 ( z , R ) g j l , α , 1 ( z , r ) f ′ j l , α , 2 ( z , R ) g j l , α , 2 ( z , r ) ) ,     r > R , (89)

Q j l , α ( 1 ) = { ( g ′ j l , α , 1 ( z , R ) f j l , α , 1 ( z , r ) g ′ j l , α , 1 ( z , R ) f j l , α , 2 ( z , r ) ) ,     r < R , ( − f ′ j l , α , 1 ( z , R ) g j l , α , 1 ( z , r ) f ′ j l , α , 1 ( z , R ) g j l , α , 2 ( z , r ) ) ,     r > R , (90)

with the functions f j l , α ,1 ( z , r ) , f j l , α ,2 ( z , r ) , g j l , α ,1 ( z , r ) , g j l , α ,2 ( z , r ) defined by Equation (73).

Theorem 3.2: For A j l , B j l ∈ ( − ∞ , ∞ ) ,   l ∈ [ j − 1 2 , j + 1 2 ] ,   j ∈ [ 1 2 , ∞ ) and α ∈ IR , the essential spectrum of h j l , α , G j l is purely absolutely continuous and coincide with ( ∞ , − c 2 2 ] ∪ [ c 2 2 , ∞ ) . Its singular continuous and residual spectra are empty. The bound states of h j l , α , G ^ j l in the gap ( − c 2 2 , c 2 2 ) coincide with the poles of the resolvent Equation (90) in Im k > 0 .

Proof

Follow step by step the proof of the Theorem 2.4.

Let us define, for k > 0 , the following function:

( η j l , α ,1 ( z , r ) η j l , α ,2 ( z , r ) ) = ( f j l , α ,1 ( z , r ) f j l , α ,2 ( z , r ) ) + Ω j l , α ( z , R ) { f ′ j l , α ,1 ( z , R ) [ A j l Q j l , α ( 2 ) ( z , r )     + A j l B j l 2 c Q ˜ j l , α ( 1 ) ( z , r ) ] + f ′ j l , α ,2 ( z , R ) [ B j l Q ˜ j l , α ( 2 ) ( z , r ) − A j l B j l 2 c Q j l , α ( 1 ) ( z , r ) ] } , (91)

with the functions f j l , α ,1 ( z , r ) , f j l , α ,2 ( z , r ) , Ω j l , α ( z , R ) , Q j l , α ( 2 ) ( . ) , Q ˜ j l , α ( 2 ) , Q ˜ j l , α ( 1 ) and Q j l , α ( 1 ) ( . ) defined by Equation (73), Equations (86)-(90) respectively.

A straightforward computation shows that ( η j l , α ,1 η j l , α ,2 ) are scattering wave functions of h j l , α , G ^ j l . The cases A j l ≠ 0 , B j l = 0 and A j l = 0 , B j l ≠ 0 in Equation (91) simplifies, respectively to:

( η j l , α , 1 ( z , r ) η j l , α , 2 ( z , r ) ) = ( f j l , α , 1 ( z , r ) F j l , α , 2 ( z , r ) ) + A j l Ω j l , α ( z , A j l , 0 , R ) f ′ j l , α , 1 ( z , R ) Q j l ( 2 ) ( z , r ) (92)

and

( η j l , α , 1 ( z , r ) η j l , α , 2 ( z , r ) ) = ( f j l , α , 1 ( z , r ) f j l , α , 2 ( z , r ) ) + B j l Ω j l , α ( z , 0 , B j l , R ) f ′ j l , α , 2 ( z , R ) Q ˜ j l , α ( 2 ) ( z , r ) (93)

Equation (92) and Equation (93) define the scattering wave functions corresponding to the Hamiltonian h j l , α , A j l and h j l , α , B j l describing two one parameter relativistic δ ′ s -sphere interactions.

Let us determine the phase shift and the elements of the on-shell scattering matrix corresponding to h j l , α , G ^ j l using the asymptotic behavior of η j l , α ( z , r ) . The asymptotic behavior of ( η j l , α ,1 η j l , α ,2 ) as r → ∞ yields [28] :

( η j l , α , 1 ( z , r ) η j l , α , 2 ( z , r ) ) → r → ∞ k > 0 ( V 1 ( z ) sin [ y 1 + δ ζ ˜ 0 ] + V 2 ( z ) cos [ y 1 + δ ζ ˜ 0 ] V 3 ( z ) sin [ y 2 + δ ζ ˜ − 1 0 ] + V 4 ( z ) cos [ y 2 + δ ζ ˜ − 1 0 ] )                                             = ( [ V 1 2 ( z ) + V 2 2 ( z ) ] 1 2 sin [ y 1 + δ ζ ˜ 0 + δ α , G j l , 1 C ] [ V 3 2 ( z ) + V 4 2 ( z ) ] 1 2 sin [ y 2 + δ ζ ˜ − 1 0 + δ α , G j l , 2 C ] ) , (94)

where

y 1 = k − α ˜ k ln ( 2 k r ) − ζ ˜ π 2 , y 2 = k − α ˜ k ln ( 2 k r ) − ( ζ ˜ − 1 ) π 2 . (95)

and

δ ζ 0 ( z ) = δ ζ − 1 0 ( z ) + arctan ( α ˜ 2 k ζ ˜ ) , δ ζ 0 ( z ) = arg Γ ( ζ ˜ + 1 + i α ˜ 2 k ) . (96)

The Coulomb modified phase shift φ α , G ^ j l C corresponding to h j l , α , G ^ j l is given by:

( φ α , G ^ j l , 1 C φ α , G ^ j l , 2 C ) = ( − arctan V 2 ( z ) V 1 ( z ) − arctan V 4 ( z ) V 3 ( z ) ) . (97)

The constants V i ( i = 1 , 2 , 3,4 ) are defined by:

V 1 ( z ) = d 1 ( z ) + [ d 2 ( z ) − i   Ω ¯ j l , α ( z ) d 4 ( z ) ] sin [ arctan ( α ˜ 2 k ζ ˜ ) ]                       − Ω ¯ j l , α ( z ) (   i   d 3 ( z ) + d 4 ( z ) ) cos [ arctan ( α ˜ 2 k ζ ˜ ) ] , V 2 ( z ) = [ d 2 ( z ) − i   Ω ¯ j l , α ( z ) d 4 ( z ) ] cos [ arctan ( α ˜ 2 k ζ ˜ ) ]                       + Ω ¯ j l , α ( z ) ( d 3 ( z ) + d 4 ( z ) ) sin [ arctan ( α ˜ 2 k ζ ˜ ) ] ,

V 3 ( z ) = α κ j l c + ζ [ − d 1 ( z ) + i   Ω ¯ j l , α ( z ) d 3 ( z ) ] sin [ arctan ( α ˜ 2 k ζ ˜ ) ]                       − ( α κ j l c + ζ ) − 1 d 2 ( z ) + Ω ¯ j l , α ( z ) [ − α κ j l c + ζ d 3 ( z ) cos [ arctan ( α ˜ 2 k ζ ˜ ) ]                       + i ( α κ j l c + ζ ) − 1 d 4 ( z ) ] , V 4 ( z ) = α κ j l c + ζ [ − d 1 ( z ) + i   Ω ¯ j l , α ( z ) d 3 ( z ) ] cos [ arctan ( α ˜ 2 k ζ ˜ ) ]                       + Ω ¯ j l , α ( z ) [ − α κ j l c + ζ d 3 ( z ) sin [ arctan ( α ˜ 2 k ζ ˜ ) ] − ( α κ j l c + ζ ) − 1 d 4 ( z ) ] , (98)

with Ω ¯ j l , α ( z ) defined by:

Ω ¯ j l , α ( z ) = Ω j l , α ( z , R ) [ A j l f ′ j l , α ,1 ( z , R ) f ′ j l , α ,1 ( z , R ) + B j l f ′ j l , α ,2 ( z , R ) f ′ j l , α ,2 ( z , R ) ] . (99)

The constants d i ,   i = 1,2 , 3,4 are given by:

d 1 ( z ) = ( 1 − α 2 ( κ j l c + ζ ) 2 ) − 1 2 [ cos ( arctan α ˜ 2 k ζ ˜ ) ] − 1 2 2 − ζ ˜ k − ζ ˜ − 1 Γ ( 2 ζ ˜ + 2 )                       × | Γ ( ζ ˜ + 1 + i α ˜ 2 k ) | − 1 e π α ˜ 2 k , d 2 ( z ) = − α c ( κ j l c + ζ ) ( 1 − α 2 ( κ j l c + ζ ) 2 ) − 1 2 [ cos ( arctan α ˜ 2 k ζ ˜ ) ] − 1 2 2 − ζ ˜ k − ζ ˜                       × Γ ( 2 ζ ˜ + 2 ) | Γ ( ζ ˜ + 1 + i α ˜ 2 k ) | − 1 e π α ˜ 2 k ,

d 3 ( z ) = ( 1 − α 2 ( κ j l c + ζ ) 2 ) − 1 2 [ cos ( arctan α ˜ 2 k ζ ˜ ) ] − 1 2 2 ζ ˜ k ζ ˜ Γ ( 2 ζ ˜ + 2 ) − 1                       × | Γ ( ζ ˜ + 1 + i α ˜ 2 k ) | e − π α ˜ 2 k , d 4 ( z ) = − α c ( κ j l c + ζ ) ( 1 − α 2 ( κ j l c + ζ ) 2 ) − 1 2 [ cos ( arctan α ˜ 2 k ζ ˜ ) ] − 1 2 2 ζ ˜ k ζ ˜ + 1                       × Γ ( 2 ζ ˜ + 2 ) − 1 | Γ ( ζ ˜ + 1 + i α ˜ 2 k ) | e − π α ˜ 2 k . (100)

The limit α → 0 + , in (97) yields:

l i m α → 0 + δ α , G ^ j l , 1 C = l i m α → 0 + φ α , G ^ j l , 2 C = φ G ^ j l ( z ) (101)

where φ G ^ j l ( z ) is defined by (60).

The Coulomb-modified on-shell scattering matrix is given by:

S α , G ^ j l , n = exp [ 2   i   ϕ α , G ^ j l , n C ( z ) ] ,   n = 1 , 2. (102)