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

Abstract

We study scattering theory for parameter models of finitely many relativistic -sphere and -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.

1. Introduction

Over the past three decades, -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 concentric spheres.

The paper is organized as follows: in Section 2, we provide a rigorous mathematical definition of the Dirac Hamiltonian with finitely many delta interactions supported by 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 concentric spheres.

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

2.1. Definition of the Model

In this section, we provide in dimension the rigorous mathematical definition of finitely many relativistic -sphere interactions. The formal expression of the hamiltonian describing the parameter model of finitely many -sphere interactions with support on concentric spheres of radii is given by where we define the Dirac Hamiltonian in Hilbert space by

The matrix is defined by and and

and , are parameters corresponding to the relativistic -sphere interaction supported by the sphere of radius . , and are, respectively, defined by the following: is the velocity of the light, is the Sobolev space of indices , and and are Dirac matrices given by

are Pauli’s spin matrices defined by

Considering the symmetric closed operator, is defined by where is the sphere of radius .

Let us look for self-adjoint extensions of which are rotationally and space-reflection symmetric.

The Hilbert space can be decomposed as follows: where

The spherical spinors are defined by [18]

We have for . The Hilbert space then takes the following form: where the isomorphism is defined by and represents the vector space formed by the spherical spinors.

Following decomposition (10), the operator reads where the radial Dirac operator reads where denotes the set of locally absolutely continuous functions on and

The adjoint of reads

A straightforward computation shows that the equation has linearly independent solutions where

is the Bessel function and the Hankel function of the first type of order .

Therefore, all self-adjoint extensions of are given by a parameter family of self-adjoint operators.

In this section, we consider the following special parameter family of self-adjoint extensions of : where the matrix reads

The Hamiltonian gives the mathematical definition of the formal expression where is the radial Dirac operator defined by

Given decomposition (10), a rigorous mathematical definition of formal expression (1) reads

The case , i.e., for all and in equation (28), yields the Dirac Hamiltonian defined by equation (2).

The case and for all and in equation (28) yields the relativistic -parameter -sphere interactions of the first type [17].

The case and for all and in equation (28) yields the relativistic -parameter -sphere interactions of the second type [17].

2.2. Resolvent Equation of

Theorem 1. The resolvent of reads where is the resolvent set and , is the radial Dirac resolvent with kernel: where

Proof. Equation (29) follows from Krein’s formula [19].
Let and define the function by Since , it follows that satisfies the separated boundary conditions in equation (23).

The implementation of these boundary conditions provides equation (29).

2.3. Spectral Properties of

Theorem 2. For , the essential spectrum of is purely absolutely continuous and coincides with . 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 reads where the notations and , mean

2.4. The Nonrelativistic Limit

The nonrelativistic limit of as is given by the following.

Theorem 3. For the spin particules, the operator converges in norm resolvent sense to the Schrödinger operator times the projector onto : where the operator is defined by

Proof. One can use the strategy of Gesztesy and Seba [15] where a similar case was discussed for point interactions.
The Hamiltonian describes a parameter model of finitely many nonrelativistic -sphere interactions in quantum mechanics.

2.5. Scattering Theory for the Pair

Let us define for the function where functions , and , are, respectively, defined by equations (18), (35), and (33).

The asymptotic behaviour of the function as yields [20] where

A straight computation shows that equation (44) reads

Then, the phase shift corresponding to is defined by

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

The partial wave scattering amplitude is given by

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

3.1. Definition of the Hamiltonian

In this section, we give in dimension the rigorous mathematical definition of relativistic many -sphere plus Coulomb interactions. The formal expression of the Hamiltonian describing finitely -sphere plus Coulomb interactions with support on concentric spheres of radii is given by where the operator is defined by equation (2).

Following decomposition (10), we consider the operator that reads where the operator is defined in by

The adjoint of reads

The deficiency index equation has linearly independent solutions where where

We use the following notations:

The operator has deficiency indices , and therefore, all self-adjoint extensions of are given by parameter family of self-adjoint operators.

Consider the following special -parameter family of self adjoint extensions of : where the matrix is defined by equation (24).

The Hamiltonian gives the mathematical definition of the formal expression where is the radial Dirac operator defined by equation (27).

The rigous mathematical definition of formal expression (50) reads

The case and , i.e., , for all and in equation (64), yields the Dirac Hamiltonian defined by equation (2).

The case , for all and in equation (64), yields

The case and , for all and in equation (64), yields the relativistic many -sphere plus Coulomb interactions of the first type.

The case and , for all and in equation (64), yields the relativistic many -sphere plus Coulomb interactions of the second type.

3.2. Resolvent Equation of

Theorem 43.1. The resolvent of reads where is the resolvent set and , is the radial Dirac resolvent with kernel: where

Proof. Similar to the proof of Theorem 2.
Consider to decomposition (10), equations (51) and (65), the resolvent equation of reads

3.3. The Nonrelativistic Limit

The nonrelativistic limit of as is given by the following.

Theorem 53.2. For the spin 1/2 particles, the operator converges in norm resolvent sense to the Schrödinger operator times the projector onto : where the operator is defined by

Proof. Similar to the proof of Theorem 3.
The Hamiltonian describes a parameter model of finitely many nonrelativistic -sphere plus Coulomb interactions in quantum mechanics.

3.4. Scattering Theory for the Pair

Let us define for the function where the functions , and , are defined by equations (59), (71), and (69), respectively.

The asymptotic behaviour of the function as yields [20]: where the constants , read

Let us introduce the following notations:

A straightforward computation shows that where

Equation (81) reads where the Coulomb-modified phase shift is given by where

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

The partial wave scattering amplitude is given by

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.