1. Introduction
The study of the
-sphere interactions in quantum mechanics has interested many authors since the twentieth century both from the mathematical point of view and for their applications in modeling of physical phenomena [1] - [6] . In nuclear physics, the delta shell interaction has been applied in a calculation of the energy levels of isotopes of Pb, Sn and Ni, Po 210, and nuclei belonging to the 82-neutron shell [7] . In Ref. [8] , the authors show that the effective two-nucleon interaction in 2 s 12-1 d 32-shell nuclei can be well approximated by a delta function, which acts only at the nuclear surface. The application in solid state physics and molecular physics may be found respectively in Ref. [9] and Ref. [10] .
These studies have used the Von Neumann formalism of self-adjoint extensions of symmetric linear operators in Hilbert space [11] [12] . In Ref. [13] , the authors have built a rigorous mathematical model based on the self-adjoint extensions of symmetric operators in the Hilbert space defining the
-sphere interactions in non-relativistic quantum mechanics.
In relativistic quantum mechanics, Dittrich et al. study Dirac operator with a contact interaction supported by a sphere by restricting their attention to the operators that are rotationally and space-reflection symmetric. They define the self-adjoint extensions of the Dirac radial operator and discuss spectral properties.
In the continuation of this work, other researches were made in this field, thus enriching the knowledge of the basic properties of Dirac operator with a
shell potential [14] [15] [16] [17] [18] .
To the best of our knowledge, the study of
-sphere interactions using the theory of self-adjoint extensions of symmetric operators in the Hilbert space began in the twentieth century [19] . And so far, little work has been done in this area both in relativistic and non-relativistic quantum mechanics [20] [21] . Yet these interactions are exactly solvable models and their systematic study allows us to better understand their properties.
In Ref. [21] and Ref. [22] , authors study the one and N parameters models of relativistic
-sphere interactions called the first and the second kind. For these models, their work provides basic properties and discusses the stationary scattering theory. Nevertheless, in Ref. [23] , using the theory mentioned above, the authors introduced in a different way than in the previous case a rigorous mathematical definition of
-sphere interactions. But, in Ref. [23] , the study of the basic properties of relativistic
-sphere interactions was missing. This is the aim of this paper. In addition, as indicated in Ref. [24] , the study of the 2 N parameters models that unify the N parameters models of
-sphere interactions of the first and the second kind allows us to better understand the dynamic of the perturbed physical system in term of scattering data. Therefore, we discuss the basic properties of two-parameter models of relativistic
-sphere and
-sphere plus Coulomb interactions in three space dimensions using the theory of the self-adjoint extensions of symmetric closed operators in Hilbert space, where
interaction denotes
interaction of the second kind.
The paper is organised as follows. In Section 2, we provide a mathematical definition of the Hamiltonian describing the two-parameter models of relativistic
-sphere interactions and obtain new results on the resolvent equation, the spectral properties and the scattering data (scattering matrix, amplitude, length and the differential scattering cross section). In Section 3, we generalize the results of Section 2 to the case of a two-parameter relativistic
-sphere interaction plus a Coulomb interaction.
2. The
-Sphere Interaction
2.1. The Definition of the Hamiltonian
In this section, we discuss the properties of the
-sphere interaction of second kind called “
-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
-sphere interaction formally given by:
(1)
where
is the Dirac Hamiltonian and
is a real 4 × 4 matrix defined by:
,
and
are real constants and R is the radius of a sphere in
centered at the origin.
2.2. The Radial Operators
In the Hilbert space
, consider the Dirac Hamiltonian
defined by:
(2)
where we have used the following definitions and notations:
1) c is the velocity of the light;
2)
is the Sobolev space of indices
;
3)
and
are 4 × 4 Dirac matrix given by:
(3)
where
are Pauli’s spin matrix defined by:
(4)
Consider in
, the symmetric closed operator
defined by:
(5)
where
is the sphere of radius R in
centered at the origin.
The operator
admits a large number of self-adjoint extensions [25] . In this case, only those of
corresponding to
, which are rotationally and space-reflection symmetric, will be considered.
Under these assumptions, one may decompose the space
in the following way:
(6)
where we have:
1)
(7)
2) The spherical spinors
are defined by:
(8)
3)
for
.
The following isomorphism
defined by:
(9)
(10)
is introduced to allow us to represent
in the form:
(11)
where
stands for the vector space generated by the spherical spinors.
With respect to the decomposition (11),
reads:
(12)
The operator
in
is given by:
(13)
where
denotes the set of locally absolutely continuous functions on
and
(14)
The adjoint
of
is given by:
(15)
2.3. The Self-Adjoint Extensions of
Given the following equation:
(16)
One can write Equation (16) in the following form:
(17)
where
(18)
Equation (17) has two linearly independent solutions:
(19)
(20)
where:
(21)
is the Bessel function and
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:
(22)
Therefore,
has indices (2, 2) and consequently, all self-adjoint(sa) extensions of
are given by a four-parameter family of self-adjoint operators [26] . Since the matrix
in Equation (28) depends on two parameters, it follows that the self-adjoint extension
of
corresponding to the interaction
is a special two-parameter family.
The relation
implies that the domain
contains those functions
which satisfy suitable boundary conditions at
.
Theorem 2.1: Any self-adjoint extension
of
reads [25] :
(23)
where cond1 and cond2 are given by [25] :
(24)
and
is 2 × 2 matrix with det
.
(25)
where
and
are real and both matrices are nonzero. Conversely, any operator of this form is self-adjoint extensions of
.
Theorem 2.2: [25] The general form of boundary conditions is given by:
(26)
where
and
.
Let us now construct the self-adjoint extension corresponding to the radial Dirac operator with the potential:
(27)
Suppose that
satisfies the following equation given by:
(28)
(29)
where the limits
exist when
. A simple computation shows that integrating Equation (28) over
, and taking the limit
we get the following conditions:
(30)
By replacing the function
in Equation (30) by its derivative, we get the following conditions:
(31)
The boundary conditions Equation (31) can be written in the general form:
(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
considering its adjoint
where:
The boundary conditions of Equation (31) characterize the potential of Equation (27) and introduce a new exactly solvable model of relativistic
-sphere interactions in quantum mechanics.
Let us consider in
the operator
defined by:
(33)
The operator
is the self-adjoint extension of the symetric operator
. The operator
gives the mathematical definition of the formal expression:
(34)
where
is the
-sphere interaction of the second kind characterized by the boundary conditions of Equation (31) and
is the radial Dirac Hamiltonian defined by:
(35)
The particular case
in Equation (33) yields the radial Dirac Hamiltonian
. The case
in Equation (33) gives the one parameter
-sphere interaction defined by:
Even, the case
in Equation (33) provides the one parameter
-sphere interaction defined by:
(36)
Let
. The decomposition of Equation (12) implies that the operator
in
defined by:
(37)
provides the mathematical definition of the formal expression Equation (2). The operator
in Equation (35) defines the
interaction in the tree-dimensional space.
The case
, i.e.
for all j and l yields the Dirac Hamiltonian
defined by Equation (2).
2.4. The Resolvent Equation of
Theorem 2.3: The resolvent of
leads:
(38)
where
is the resolvent set and where
,
is the radial Dirac resolvent with kernel:
(39)
where
(40)
and
(41)
(42)
(43)
(44)
(45)
with
and
defined by Equation (21) and Equation (18).
Proof
One can use the Krein resolvent formula which yields the following relation for the resolvent of
:
(46)
where
are given by Equations (19) and (20) respectively.
Taking the function
, we can define the function
by:
(47)
As the function
, it follows that
satisfies the boundary conditions in Equation (33). The implementation of these boundary conditions provides the constants
. When we insert
into Equation (46), we obtain Equation (38).
In particular case
and
, the resolvent Equation (40) provides respectively:
(48)
and
(49)
2.5. The Spectral Properties of
Theorem 2.4: For
,
,
, the essential spectrum of
is purely absolutely continuous and coincides with
. 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
located in
are given by the pole of the resolvent equation in physical sheet
, i.e. the solution of:
(50)
where
.
A straightforward computation shows that Equation (50) reads:
(51)
In Figure 1, we consider the following normalization of the energy
. Consider also
with the following condition for the coupling constants:
.
and
represents the second member of Equation (51). Using the graphical resolution method, i.e. by analyzing the intersection of the curves
and
, one can show easily that Equation (51) has two solutions which correspond to the two eigenvalues
of
in
.
Figure 1. Existence of two eigenvalues in
.
The resonances of
are defined as poles of the resolvent Equation (38) in the unphysical sheet Im
.
2.6. The Nonrelativistic Limit
Following the strategy of Gesztesy et al. [27] , in the case of point interactions, one can discuss the nonrelativistic limit of
as
.
Theorem 2.5: For spin-
particules, the operator
converges in norm resolvent sense to the Shrödinger operator
times the projector onto
:
(52)
where
is defined by:
(53)
The boundary conditions in (53) define a self-adjoint extension of the radial schrödinger operator
defined by:
(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
defines a two-parameter model of nonrelativistic
-sphere interaction in quantum mechanics.
2.7. The Scattering Theory for the Pair (
)
Let us define for
the following function:
(55)
where the functions
and
are defined by Equation (21), Equations (41)-(45) respectively. A straightforward computation shows that
are scattering wave functions of
.
For the cases
and
and
, (55) yields, respectively to:
(56)
and
(57)
Equation (56) and Equation (57) define the scattering wave functions corresponding to the Hamiltonian
and
describing two one parameter relativistic
-sphere interactions.
Let us determine the phase shift and the elements of the on-shell scattering matrix corresponding to
using the asymptotic behavior of
.
The asymptotic behavior of
as
yields to:
(58)
where
(59)
In this case, the phase shifts corresponding to
are defined as:
(60)
where
(61)
The elements of the on-shell scattering matrix are given by:
(62)
The partial wave scattering amplitude is given by:
(63)
3. The
-Sphere plus Coulomb Interaction
3.1. The Definition of the Hamiltonian
Let us consider the formal expression given by:
(64)
where
is a real matrix of the form:
(65)
Let use the decomposition Equation (12) and introduce in
the operator:
(66)
where
is given by:
(67)
is defined by Equation (13).
The adjoint
of
reads:
(68)
3.2. The Self-Adjoint Extension of
Let us consider the following equation:
(69)
and introduce the following notations:
(70)
Equation (69) has two linearly independent solutions:
(71)
(72)
where
(73)
and
(74)
denotes the regular (respectively, irregular) confluent hypergeometric functions.
is the normalization constants chosen in such way:
(75)
For the particular case
, we obtain:
(76)
(77)
where
and
are respectively the derivatives of the functions
and
defined by Equation (21).
The operator
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
:
(78)
The operator
gives the mathematical definition of the formal expression:
(79)
The case
in Equation (78) gives the radial Dirac-Coulomb Hamiltonian
,
(80)
In particular case when
,
and
,
in Equation (78) simplifies respectively to one parameter
-sphere plus Coulomb interaction. Therefore, the model in Equation (64) is defined in
by:
(81)
The particular case
for all j and l provides the Dirac-Coulomb Hamiltonian
,
. (82)
3.3. The Resolvent Equation of
Theorem 3.1: The resolvent of
reads:
(83)
where
is the resolvent set and where
,
is the radial Dirac resolvent with kernel:
(84)
and
(85)
(86)
(87)
(88)
(89)
(90)
with the functions
defined by Equation (73).
3.4. The Spectral Properties of
Theorem 3.2: For
and
, the essential spectrum of
is purely absolutely continuous and coincide with
. Its singular continuous and residual spectra are empty. The bound states of
in the gap
coincide with the poles of the resolvent Equation (90) in Im
.
Proof
Follow step by step the proof of the Theorem 2.4.
3.5. The Scattering Theory for the Pair (
)
Let us define, for
, the following function:
(91)
with the functions
and
defined by Equation (73), Equations (86)-(90) respectively.
A straightforward computation shows that
are scattering wave functions of
. The cases
,
and
in Equation (91) simplifies, respectively to:
(92)
and
(93)
Equation (92) and Equation (93) define the scattering wave functions corresponding to the Hamiltonian
and
describing two one parameter relativistic
-sphere interactions.
Let us determine the phase shift and the elements of the on-shell scattering matrix corresponding to
using the asymptotic behavior of
. The asymptotic behavior of
as
yields [28] :
(94)
where
(95)
and
(96)
The Coulomb modified phase shift
corresponding to
is given by:
(97)
The constants
are defined by:
(98)
with
defined by:
(99)
The constants
are given by:
(100)
The limit
, in (97) yields:
(101)
where
is defined by (60).
The Coulomb-modified on-shell scattering matrix is given by:
(102)
4. Conclusion
In this paper, using the self-adjoint theory of symmetric operator in Hilbert space, we studied the basic properties of two-parameter models of relativistic
-sphere and
-sphere plus Coulomb interaction (where a charged particle is perturbed by a
-sphere interaction). For both interactions, we obtain interesting results on resolvent equations, spectral properties and scattering data (the phase shift, scattering matrix, scattering amplitude, and scattering cross section). As a perspective, one can use simulations or real-world data to validate the theoretical models proposed in this paper. Also, in our future paper in preparation, we intend to study the case where the relativistic
-sphere interaction is centered on finitely many concentric spheres.
Acknowledgments
We would like to thank Ph.D. supervisors Juma Shabani and Alfred Vyabandi for their encouragement, for their precious advice, and fruitful discussions on topics during the preparation of this paper. We are particularly grateful to them.