Asymptotic solution of Sturm–Liouville problem with periodic boundary conditions for relativistic finite-difference Schrödinger equation

The quasi-potential approach is very famous in modern relativistic particles physics. This approach is based on the so-called covariant single-time formulation of quantum field theory in which the dynamics of fields and particles is described on a space-like three-dimensional hypersurface in the Minkowski space. Special attention in this approach is paid to methods for constructing various quasi-potentials. The quasipotentials allow to describe the characteristics of relativistic particles interactions in quark models such as amplitudes of hadron elastic scatterings, mass spectra, widths of meson decays and cross sections of deep inelastic scatterings of leptons on hadrons. In this paper Sturm–Liouville problems with periodic boundary conditions on a segment and a positive half-line for the 2m-order truncated relativistic finite-difference Schrödinger equation (Logunov–Tavkhelidze–Kadyshevsky equation, LTKT-equation) with a small parameter are considered. A method for constructing of asymptotic eigenfunctions and eigenvalues in the form of asymptotic series for singularly perturbed Sturm–Liouville problems with periodic boundary conditions is proposed. It is assumed that eigenfunctions have regular and boundary-layer components. This method is a generalization of asymptotic methods that were proposed in the works of A.N. Tikhonov, A.B. Vasilyeva, and V. F Butuzov. We present proof of theorems that can be used to evaluate the asymptotic convergence for singularly perturbed problems solutions to solutions of degenerate problems when ε → 0 and the asymptotic convergence of truncation equation solutions in the case m → ∞. In addition, the Sturm–Liouville problem on the positive half-line with a periodic boundary conditions for the quantum harmonic oscillator is considered. Eigenfunctions and eigenvalues are constructed for this problem as asymptotic solutions for 4-order LTKT-equation.


Introduction
The relativistic finite-difference analog of the Schrödinger equation (Logunov-Tavkhelidze-Kadyshevsky equation, LTK-equation) with the quasipotential in the relativistic configurational space for the radial wave functions of bound states for two identical elementary particles without spin has the form [1]- [13]: rad 0 = 2 2 ch ( ℏ ) + ℏ 2 ( + 1) where is a mass, is a momentum, is an angular momentum of each elementary particle and ( ) is a quasi-potential (a piecewise continuous function).
Asymptotic solutions in the form of regular and boundary layer parts of boundary value problems for LTK-equation with the quasi-potential on a segment and on a positive half-line were constructed in the works [14]- [16], and the question of the asymptotic behavior of the solutions was investigated when a small parameter → 0. Also in these works the truncation method was applied to LTK-equation. Thus, LTK-equation of infinite order was reduced to the equation of finite 2 -order. Boundary value problems on a segment and on a positive half-line were formulated for this "truncated" equation (Logunov-Tavkhelidze-Kadyshevsky truncated equation, LTKTequation). Eigenfunctions and eigenvalues in the form of asymptotic series were constructed for these problems and the solution behavior was studied when the order of LTKT-equation tends to infinity 2 → ∞.
In the paper [17] mass spectra and probabilities of radiative decays of heavy quarkonia were obtained in the framework of the constituent quark model of hadrons based on the relativistic Logunov-Tavkhelidze-Kadyshevsky equation.
Researchers pay a lot of attention to the description of quantum systems that consist of one-dimensional linear chains of identical harmonic oscillators with a nearest neighbor interaction. Periodic boundary conditions, where the -th oscillator is coupled back to the first oscillator, and fixed wall boundary conditions, where the first oscillator and the -th oscillator are coupled to a fixed wall, was considered in the paper [18], [19].
In this paper Sturm-Liouville problems with periodic boundary conditions on a segment and a positive half-line are formulated for the truncated to order 2 relativistic finite-difference Schrödinger equation (Logunov-Tavkhelidze-Kadyshevsky equation, LTKT-equation) with a small parameter.
For these singularly perturbed problems a method is proposed for constructing eigenfunctions and eigenvalues in the form of asymptotic series. This method allows to obtain asymptotic solutions in the form of regular and boundary-layer parts. It is also possible to investigate the question of asymptotic solutions behavior when → 0 and 2 → ∞. The Sturm-Liouville problem for 4-order LTKT-equation on a positive half-line with periodic boundary conditions is formulated for the quantum harmonic oscillator quasipotential and eigenfunctions and eigenvalues in the form of asymptotic series are constructed.

The Sturm-Liouville problems for the LTKT-equation
We consider the quasi-potential equation [3]- [5] in a relativistic configuration space for the radial wave functions of bounded states for two identical elementary particles where is a mass, is a momentum, is a moment of elementary particles and ( ) is a quasi-potential.
Let physical parameter be ℏ = 1, = 1, = 1 and = 0 (case of -wave) in (1) where We can rewrite the equation (1) in the form as under The equation (1) is an infinite order differential equation with a small parameter ( ≪ 1) at higher derivatives and we can classify it as singularly perturbed equations.
We can truncate the equation (4) to a finite equation of 2 -order with > 1 and it can be rewritten as follows where 2 is the self-adjoint 2-order elliptic operator,̃2 is the self-adjoint 2 -order elliptic operator, ,2 ( ) is the solution of the 2 -order equation.
We can formulate the boundary value problem 2 on a segment [0, 0 ] and the boundary value problem 2 on a positive half-line [0, +∞) for defining the eigenfunctions [ ,2 , ] ∞ =1 and the eigenvalues [ ,2 , ] ∞ =1 for this differential equation as follows where ,2 (0) = ,2 ( 0 ), = 0, 1, … , 2 − 1, are the periodic boundary conditions of the problem 2 , and ,2 (0) = ,2 (+∞), = 0, 1, … , 2 − 1, are the periodic boundary conditions of the problem 2 . If we assume = 0, we can get the degenerate problems 0 and 0 for defining the eigenfunctions [ 0, ] ∞ =1 and the eigenvalues [ 0, ] ∞ =1 of following type as under is the periodic boundary conditions of the problem 0 , and is the periodic boundary conditions of the problem 0 . We can consider the question of the behavior of the eigenfunctions [ ,2 , ] ∞ =1 and the eigenvalues [ ,2 , ] ∞ =1 of the problems 2 and 2 in the case when a small parameter tends to zero ( → 0) but fixed order 2 of the operator̃2 , and in the case when the order is increased but a small parameter is fixed.
Let ‖ (Ω Γ )‖ denotes the norm of operators (Ω Γ ) and we can write We can give the sufficient conditions for the solvability of the problems 0 , 0 and 2 , 2 . Condition 1. The operator 2 for the periodic boundary conditions of the problems 0 or 0 must be positively defined, i.e.

Condition 2.
The operator 2 under boundary conditions of problems 2 or 2 must be positive, i.e.
for any functions ,2 ∈ (Ω Γ ) from domain Ω Γ , and it must satisfy the boundary conditions of the corresponding singularly perturbed problem ( 2 or 2 ).
It is known that the degeneration of the problems 2 , 2 into the problems 0 , 0 are regular if the number of roots with negative real parts and positive real parts of an additional characteristic equation, which in our case has the form coincide with the number of boundary conditions that drop down on the left and, respectively, on the right when we replace the consideration problems 2 , 2 to problems 0 , 0 . Let's now consider the generalized characteristic form of the operator The regular degeneration of the problems 2 , 2 to 0 , 0 is fulfilled if the following condition is true. Condition 3. If the following inequality take place for the real part of the sum ( ) where is not depended on , then problems 2 and 2 degenerate into problems 0 and 0 regularly.
Let's assume that a set of eigenvalues ,2 ,1 ⩽ ,2 ,2 ⩽ … ⩽ ,2 , ⩽ … and 0, , and this set of eigenvalues corresponds to a complete orthonormal set of eigenfunctions Since existence domains Ω of operators̃2 and 2 coincide for the problems 2 and 0 and also for any function ,2 ∈ Ω , that satisfies the boundary conditions of the problem 2 , the following inequality from Condition 2 holds true, then the following estimate inequality occurs ,2 , ⩾ 0, , = 1, 2, …. A similar estimate takes place for the problems 2 and 0 .

General scheme for constructing of the asymptotics. Regular and boundary series
We can use methods of the singular perturbations theory of differential equations and find solutions to problems 2 and 2 . Let's search for a formal solution ,2 ( ) of the problems 2 and 2 in the form of asymptotic series (11) where a partial sum and similar inequalities for the boundary conditions of these problems, where , and ≪ 1, ≪ 1 are positive constants that are independent of and . The asymptotic solution for ,2 have the form as under wherē2 ( ) = +1 2 ( ) is error of the asymptotic approximation of the solution ,2 by a partial sum Θ ,2 .
We can write the regular part of the asymptotic expansion in the form 2 ( , ) ≡̄2 ,0 ( ) +̄2 ,1 ( ) + 2̄2 ,2 ( ) + … , and the singular parts of the asymptotic expansion have the forms as under for describing the behavior of the solution on the left edge of a segment [0, 0 ] or a positive half-line [0, +∞), for describing the behavior of the solution of the problem 2 on the right edge of a segment [0, 0 ]. It is known that the function 2 ( 2 , ) = 0 for the problem 2 , since the solution of the problem 0 is chosen so that it tends to zero when → +∞ together with all its derivatives. Here we use new independent (stretched) variables 1 = / and 2 = ( 0 − )/ for the boundary functions Π 2 , , 2 , . Similarly, we can present the simple eigenvalue of ,2 in the form of the asymptotic series in powers of the small parameter in the form as under where the partial sum satisfies the condition | ,2 − Θ ,2 | <̃+ 1 , wherẽ> 0 is a positive constant that is independent of and .
So an asymptotic approximation of the eigenvalue ,2 has the form as under is an error of the asymptotic approximation of the eigenvalue ,2 for this partial sum.
In addition, we assume that the function ( ) can be decomposed as a convergent series in the neighborhood of the points = 0 and = 0 where 1 = / and 2 = ( 0 − )/ are the stretched variables.

The main terms of the asymptotic series
We can determine the terms of the asymptotic series of the decomposition 2 , , Π 2 , , 2 , and 2 , of the problems 2 and 2 if we substitute the decomposition (11), (12) and (13) in the equation (5) and the boundary conditions (6) of the problem 2 and the equation (5) and the boundary conditions (7) of the problem 2 , and then we equate all members of the series that stand at equal powers of a small parameter .
We should use additional requirements for the boundary functions where → 0 and a fixed . These requirements allows to select the solutions Π 2 , and 2 , that tend to zero outside the boundary layer only.

Building a zero approximation of the asymptotic expansion
We can get the systems of equations and determine the solutions̄2 ,0 , Π 2 ,0 , 2 ,0 and 2 ,0 of the problems 2 and 2 in a zero approximation in the form The eigenfunctions [̄2 ,0, ] ∞ =1 and the eigenvalues [ 2 ,0, ] ∞ =1 coincide with the solutions of the corresponding degenerate problems 0 or 0 .
Thus, we can determine the boundary functions Π 2 ,0 ( 1 ), 2 ,0 ( 2 ) if we find the solutions of the boundary value problems as under We can write the functions Π 2 ,0 ( 1 ) and 2 ,0 ( 2 ) in the forms Hence, the number of arbitrary constants 2 ,1 ,0 and 2 ,2 ,0 equals the number of disappearing boundary conditions of problems 2 or 2 when we try formulate the degenerate problems 0 or 0 .
Let the values 2 ( = 1, … , 2 − 2) be the roots of the additional characteristic equation Since an algebraic equation is biquadrate; thus, it has the same number of roots with positive and negative real parts. We can get the following relations from the boundary conditions where = 0, 1, 2, … , 2 −1, and we can derive a system of 2 linear equations like that for finding coefficients 2 ,1 ,0 , 2 ,2 ,0 ( = 1, 2, … , − 1), where a system has form as under are block matrices.
Since the values of 2 ( = 1, 2 − 2) are pairwise different and the matrices D 2 11 , D 2 22 , D 2 are non-degenerate and there is an inverse of D 2 matrix (D 2 ) −1 , so then the only solution of the algebraic system (14) exists and it has the form: Thus, a zero approximation of̄2 ,0 , Π 2 ,0 , 2 ,0 , 2 ,0 of the problems 2 and 2 could be constructed completely.
Using the method of constant variation, we can find the partial solutions of the inhomogeneous equations (15), (17), i.e.
( 2 ) from the systems as under ⃗ Ω 1 = (D 2 11 ) −1 F 1 , ⃗ Ω 2 = (D 2 22 ) −1 F 2 . After integrating and substituting the solutions in (18), (19), we can find as many arbitrary constants as the boundary conditions of the problems 2 or 2 fall out when we proceed to analysis of the degenerate problems 0 or 0 .
Thus, this algorithm allows us to find the asymptotic solutions of the problems 2 and 2 with any desired degree of accuracy of a small parameter .
Using series for the constructions of a solution, we can get wherē2 is the restricted function (‖̄2 ‖ = (1)).
This implies the estimate for̄2 that is in the conditions of the theorem.

Solutions behavior analysis of the problems 2 and 2 in the case → ∞
Here we investigate the question about the behavior of the eigenfunctions and the eigenvalues of 2 and 2 problems in the case of unlimited increasing of 2 -order LTKT-equation.
Let's consider the problems of 2 , 2 and 2 +2 , 2 +2 for finding Here we assume that the eigenvalues are arranged in order of monotonic increase.

Construction of an asymptotic solution in the case of the oscillator potential
We can consider the boundary value problem 2 on the [0, ∞+) axis with the quasi-potential of a linear harmonic oscillator in the form ( ) = 2 . Analysis of this problem allows to describe the behavior chains of harmonic oscillators with periodic boundary conditions when they are very far apart from each other.

Conclusions
Recently, there is a great interest in studying properties of bound states of a quarkonium such as charmonium̄and bottomonium. These states are similar to the properties of positronium (the bound state of an electron and a positron). Special attention of researchers who deal with bound states of quarks is paid to quasi-potential methods. The quasi-potential approach allows to describe the characteristics of relativistic elementary particles such as amplitudes of hadron elastic scattering, mass spectra and widths of meson decays, and the cross sections of deep inelastic scattering of leptons on hadrons. Since experimental measurements of relativistic elementary particles are carried out with high accuracy, the quark systems models allow to use the precision calculation of various parameters. Experiment has amassed a wealth of high precision data on quarkonium production in relativistic heavy ion collisions at RHIC and LHC in different kinematical regimes that provides a challenging testing ground for theory and phenomenology.
We use a quasi-potential approach in our work. The quasi-potential method in the field theory is based on a two-time Green function for particle systems. The bounded states of such systems are described by a wave function that satisfies a quasi-potential Schrödinger-type equation that depends on energy and non-local potential. The main advantage of this quasi-potential equation is its three-dimensional character. We have shown the absence of a non-physical parameter of relative time for this equation. This quasi-potential wave equation can be obtained for any system numbers of particles with arbitrary spins. This approach was successfully applied to calculate corrections to the energy levels of hydrogen-like systems within the framework of quantum electrodynamics. The great number of properties of the elementary particles amplitude scattering at high energies is explained using a quasi-potential Lippman-Schwinger equation with a Gaussian potential. The quasi-potential method has a number of advantages among the methods of studying the relativistic two-body problem. The advantage of this approach is that quasipotential equations are written out in three-dimensional space, which makes it possible to use the methods of non-relativistic quantum mechanics.
In this paper Sturm-Liouville problems with periodic boundary conditions on a segment and a positive half-line are formulated for the truncated to order 2 relativistic finite-difference Schrödinger equation (Logunov-Tavkhelidze-Kadyshevsky equation, LTKT-equation) with a small parameter. For these singularly perturbed problems a method is proposed for constructing asymptotic solutions with accuracy up to any given order . With the help of this method asymptotic solutions in the form of regular and boundary-layer parts are obtained and the question of asymptotic solutions behavior when → 0 is investigated. The behavior of solutions is investigated in the case → ∞ and estimation of this behavior is given. It makes possible to determine the convergence of solutions of the Sturm-Liouville problems for LTKT-equation with periodic boundary conditions in the case → ∞.
In non-relativistic quantum mechanics, the particle in a one-dimensional lattice is a problem that occurs in the model of a periodic crystal lattice. The potential is caused by ions in the periodic structure of the crystal creating an electromagnetic field, so electrons are subject to a regular potential inside the lattice. This is a generalization of the free electron model, which assumes zero potential inside the lattice.