CC BY
8
Research article
UDC 517.958, 517.963
PACS 02.30.Hq, 02.30.Mv, 03.65.Ge, 11.10.Jj, 03.65.Pm, 02.30.Em DOI: 10.22363/2658-4670-2020-28-3-230-251
Asymptotic solution of Sturm—Liouville problem with periodic boundary conditions for relativistic finite-difference Schrodinger equation
Ilkizar V. Amirkhanov1, Irina S. Kolosova2, Sergey A. Vasilyev2
1 Joint Institute for Nuclear Research 6, Joliot-Curie St., Dubna, Moscow Region, 141980, Russian Federation 2 Peoples' Friendship University of Russia (RUDN University) 6, Miklukho-Maklaya St., Moscow, 117198, Russian Federation
(received: July 2, 2020; accepted: September 14, 2020)
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 Schrodinger 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 e —> 0 and the asymptotic convergence of truncation equation solutions in the case m ^ to. 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.
Key words and phrases: asymptotic analysis, singularly perturbed differential equation, Sturm-Liouville problem, relativistic finite-difference Schrodinger equation, periodic boundary conditions, quasi-potential approach
© Amirkhanov I.V., Kolosova I.S., Vasilyev S.A., 2020
This work is licensed under a Creative Commons Attribution 4.0 International License http://creativecommons.org/licenses/by/4.0/
1. Introduction
The relativistic finite-difference analog of the Schrodinger 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
where m is a mass, q is a momentum, I is an angular momentum of each elementary particle and V(r) 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 e ^ 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 2m-order. Boundary value problems on a segment and on a positive half-line were formulated for this "truncated" equation (Logunov-Tavkhelidze-Kadyshevsky truncated equation, LTKT-equation). 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 2m ^ to.
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 n identical harmonic oscillators with a nearest neighbor interaction. Periodic boundary conditions, where the n-th oscillator is coupled back to the first oscillator, and fixed wall boundary conditions, where the first oscillator and the n-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 2m relativistic finite-difference Schrodinger 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 e ^ 0 and 2m ^ to. The Sturm-Liouville problem for 4-order LTKT-equation on a positive half-line with periodic
form [1]—[13]:
[HT0ad + V(r) - 2c^q2 + m2c2I) _ 0
(1)
^rad _
0
boundary conditions is formulated for the quantum harmonic oscillator quasipotential and eigenfunctions and eigenvalues in the form of asymptotic series are constructed.
2. 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
[#0ad + V(r) - 2c^q2 + m2с2]ф(г, I) = 0, (2)
2 , ( ih \ h21(1+1) ( ih N Trad = 2™r2 ch ( -D) +---——— exp ( -Г>) =
mc J mr(r + — ) \mc
v mc J
f (-!)*> 2шс2 (±\2p D2P + h21(1+1) f (ih_y
(2p)W \mc] mr(r + ^) p\\mc] '
dp
DP =
drp'
where m is a mass, q is a momentum, I is a moment of elementary particles and V(r) is a quasi-potential.
We can limit the speed of light to the infinity (c ^ to) formally. In this case, the equation (1) becomes the non-relativistic Schrodinger equation [20]
[-h2D2 + h21(1 +l)/r2 + mV(r) - q2] i>(r) = 0. (3)
Let physical parameter be h = 1, m=l, £ = 1 and I = 0 (case of ¿"-wave) in (1) where
Ae>TO = 2q2/Vl + e2q2 + 1, v = V(r), q2 = (1 + 0.2he2 We can rewrite the equation (1) in the form as under
[Llo (r)=0, (4)
Ll = ^2 + £2
Ll =Y,e2P~2 L2p + <r), L2p = 22p^d2P , £^(0,1],
L2 = L2 + v(r) = -D2 + v(r), n = £s2*-2 l2p+2 = £ D2p+2.
P=1 P=1 ( r )
The equation (1) is an infinite order differential equation with a small parameter (e < 1) at higher derivatives and we can classify it as singularly perturbed equations.
We can truncate the equation (4) to a finite equation of 2m-order with m > 1 and it can be rewritten as follows
[L2m — \,2m\^e,2m (r) = 0, m
L2m = L2 + e2L'2m = Y,£2p-2L2P + v(r),
p=i
m—1 m—1 2( — 1)P+l
Li™ = E c2*—2l2,,+2 = E 2k-+wr2p—2 D2r+2'
P=1 P=1 ( r ' )••
where L2 is the self-adjoint 2-order elliptic operator, L£2m is the self-adjoint 2m-order elliptic operator, tjj£ 2m (r) is the solution of the 2m-order equation.
We can formulate the boundary value problem A2"' on a segment [0, r0] and the boundary value problem on a positive half-line [0, +oo) for defining the eigenfunctions [^£>2m>1]''=1 and the eigenvalues [A£>2m>1]''=1 for this differential equation as follows
[L2m — ^e,2m] ilJe,2m (r) = 0, (5)
where
D (0) = D^e>2m (r0), i = 0,1,...,2m-1, (6)
are the periodic boundary conditions of the problem , and
D^m (0) = D^£i2rn(+w), 1 = 0,1,... ,2m-1, (7)
are the periodic boundary conditions of the problem .
If we assume £ = 0, we can get the degenerate problems A0 and B0 for defining the eigenfunctions and the eigenvalues [ A0^]'=1 of following
type as under
[L2 - \] A(r) = 0, (8)
where
D^0 (0)= Dirip0 (r0), i = 0,1, (9)
is the periodic boundary conditions of the problem A0, and
^^(0) = D^0 (+<x>), i = 0,1, (10)
is the periodic boundary conditions of the problem 0.
We can consider the question of the behavior of the eigenfunctions Vl)e,2m.ri]'=1 and the eigenvalues [A£>2m>7]'=1 of the problems A2"' and B2m in the case when a small parameter tends to zero (e ^ 0) but fixed order
2m of the operator L2m, and in the case when the order m is increased but a small parameter e is fixed.
The eigenfunctions [^ei2mn\^=1 and are the solutions of the corre-
sponding problems A2™', A0 and B2"', B0. These solutions are elements of a Hilbert space ) with an inner product (^, p)H(nr) = f ^(r) '-P(r) dr
p G )), in which there is a set of a linear continuous self-adjoint operators A(ttT) : H(ttr) ^ H(ttr) of problems B2m, , Bo (¿2m,L2 G A, m > 2), where (r = A,B) is a domain of the operator (a subscript A corresponds to a segment [0, r0] and a subscript B is a positive half-line [0,+rc)).
Let \\A(Qr)\\ H denotes the norm of operators ) and we can write
WL mi
= sup , mH = (M№. \m\H H
We can give the sufficient conditions for the solvability of the problems A0, o and B2m.
Condition 1. The operator L2 for the periodic boundary conditions of the problems A0 or B0 must be positively defined, i.e.
(L2 (^0),^0)H(Qr, = f L2(^0)^0 dr = J \D^012 dr + J dr^0,
for any functions v(r) G Cœ(r) and ^0 G H(QT) from domain , and it must satisfy the boundary conditions of the corresponding degenerate problems (A0 or B0).
Condition 2. The operator L2m under boundary conditions of problems
A1™' or Bmust be positive, i.e.
(.^J2m^e,1m,^e,1m)H(nr) = (2p + 2)\\£2P 1 f №2P+2^s,1m)^e,2m dr =
p=1 1 / \\ JQj-
m—1
for any functions 2m G H(Q,T) from domain Q-p, and it must satisfy the boundary conditions of the corresponding singularly perturbed problem (A1™' or Blm ).
It is known that the degeneration of the problems A2m ,B2m into the problems A0, B0 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
m ( — 1\P
p(a2 m ) = m )1p-1 = 0,
coincide with the number of boundary conditions that drop down on the left and, respectively, on the right when we replace the consideration problems A2rn, B2m to problems A0, B0.
Let's now consider the generalized characteristic form of the operator
m
^ e2p—2L2p, which is obtained by replacing D2p with (i£)2'p
p=1
m 2(—1)p
(i) = Ei22-1T£2p—2 (t^2" ■
The regular degeneration of the problems A2m, to A0, B0 is fulfilled if the following condition is true.
Condition 3. If the following inequality take place for the real part of the
sum tt£ (£)
m 2 m
Re (n£ m = Ew™£ 2p—2?p > CE£ 2p—2 i^i2p > 0, P=1(2p)•• P=1
where C is not depended on then problems A2™' and Bl'm degenerate into problems A0 and B0 regularly.
Let's assume that a set of eigenvalues A £2rn1 < A £ 2rn 2 < ... < A £ 2mn < ... and A0 , 1 < A0 , 2 < ... < A0 , n < ... is ordered in ascending order [X£, 2m, [^0 , 7]'1, and this set of eigenvalues corresponds to a complete orthonormal set of eigenfunctions [ijj£ 2m, 7^, [^0, 7^.
Since existence domains of operators L2m and L2 coincide for the problems A2Jn and A0 and also for any function tjj£ 2m £ , that satisfies the boundary conditions of the problem A2™, the following inequality from
Condition 2
(L 2rnA ,2m, r£,2m >(L 21P £,2m, r£,2m
holds true, then the following estimate inequality occurs A £ 2mi > A 0 7, 7=1,2,.... , , ,
A similar estimate takes place for the problems B^m and B0.
3. Constructing of asymptotic solutions for boundary
value problems
3.1. 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 A2"' and B2"'.
Let's search for a formal solution tjj£ 2m( r) of the problems A2im and B2m in the form of asymptotic series
©(r) = i>2m (r, e) + n2m p! , £) + Q2m P2, £) =
TO
= ^ £ k (^2m,k ( r) + U2rn,k$( P! ) + Q2m,k$(P2^ (11)
k=0
where a partial sum
©j^e^rn^) = ^ £ k ( ^2 m,k ( 0 + ^2rn,k$ ( P ! ) + Q 2rn,k$ ( P2 )),
k=0
satisfies inequalities for solutions of the problem A"2
m
e
maX 1 ije,2rn - ©j^e,2rn 1 < MA ^ ,
re[SA,ro-SA]
2m
and the problem B
max , 1 e,2rn - ®ji>e,2m 1 < MB£^+1,
and similar inequalities for the boundary conditions of these problems, where MA, MB and SA «1, 5B « 1 are positive constants that are independent of r and e.
The asymptotic solution for tjj£ 2m have the form as under
3
i>e,2m ( r) = ^ £ k(. $2 m,k ( r) + n2rn,ki) ( Pi) + Q 2rn,k( P2)) + Z2 ™ (
k=0
Zjm (r) = 1p£,2m — ©jiJ£,r2m,
where z2m (r) = e^+iz'2m(r) is error of the asymptotic approximation of the solution ^£>2m by a partial sum ©j^£2m.
We can write the regular part of the asymptotic expansion in the form
$2 rn ( ^ = $2 rn,0(r) + £ $2 rn,i(r) + £ 2 $2 rn,2 (r) + - ,
and the singular parts of the asymptotic expansion have the forms as under
n2rn Pi = n2rn,0 A. Pi ) + £ n2rn,i A. Pi ) + £ 2 n2rn,2^ ( Pi ) + ■■■,
for describing the behavior of the solution on the left edge of a segment [0, r0] or a positive half-line [0,+to),
Q 2m A. P2,£) = Q 2m,0 P2 ) + £ Q 2rn,i P2 ) + £ 2Q 2m,2^ ( P2 ) +
for describing the behavior of the solution of the problem on the right edge of a segment [0, r0].
It is known that the function Q2mtjj(p2 ,e) = 0 for the problem B£m, since the solution of the problem B0 is chosen so that it tends to zero when r ^ together with all its derivatives. Here we use new independent (stretched)
variables p1 = r/e and p2 = (r0 — r)/e for the boundary functions n2m k-0,
Q2m,k^ .
Similarly, we can present the simple eigenvalue of A£ 2m in the form of the asymptotic series in powers of the small parameter e in the form as under
\ , 2m = ^2m , 0 + £^2m, 1 + £2 ^2m, 2 + ■■• , (12)
where the partial sum
'J
Qj ^e,2m = ""j^' ^2m,k, k=0
satisfies the condition IX£i2rn — QjXs 2m| < M e^+1, where M > 0 is a positive constant that is independent of r and e.
So an asymptotic approximation of the eigenvalue A£ 2m has the form as under
j
^£,2m = '"J£k^2m,,k + ,
k=0
where = e^+1 A2'm, barA?™1 = X£t2m — QjX£i2m is an error of the asymptotic approximation of the eigenvalue A£ 2rn for this partial sum.
In addition, we assume that the function v(r) can be decomposed as a convergent series in the neighborhood of the points r = 0 and r = r0
oo
v(r) =
s=-1 s=-1
}(T) = ^ v\rs, v(r) = ^v2s (r- roy,
and
'(Pi) = Ë V1£Sp*, v(p2) = Ë (-!)|s|v2£sP2, (13)
oo
V(Pi) =
s=-1 s=-1
where p1 = r/e and p2 = (r0 — r)/e are the stretched variables.
3.2. The main terms of the asymptotic series
We can determine the terms of the asymptotic series of the decomposition ^2mk, n2m k0, Q2m k0 and \2mk of the problems A2"' and B2"' if we substitute the decomposition (11), (12) and (13) in the equation (5) and the boundary conditions (6) of the problem A2£m and the equation (5) and the boundary conditions (7) of the problem B:2m, and then we equate all members of the series that stand at equal powers of a small parameter e. We should use additional requirements for the boundary functions
n2rn, k (.Pi Qr2m , k ^(P2 ) ^ 0 k = °),l,2,...,
where £ ^ 0 and a fixed r. These requirements allows to select the solutions n2m k^ and Q2m that tend to zero outside the boundary layer only.
3.2.1. Building a zero approximation of the asymptotic expansion
We can get the systems of equations and determine the solutions 0, n2m 0i>, Q2m 0^ and X2m 0 of the problems A2"' and B2m in a zero approximation in the form
[L2 — X2rn,0] $2m,0 = 0 L2 = —D2 + V(r), Li n é = 0 Li =t> 2{-1)P d2P
2m 2m,0V 0, lj2m ~ (2p)U dp2? '
L2 O é = 0 L2 = fi 2(-1)P d2P
2m^2m,0 r 0, lj2m ~ fo^tl , 2p ,
p=l (0P>!! dP2
D% $2m,0(0) + n2m>0m) = & $2m>0(r) + Q2m,0№)) ,
n2rn,0$(PiQ2m,0№2)^0, e^0, i = 0,1,2,... ,2m - 1,
where r = r0 for A2m and r ^ for B2m.
The eigenfunctions [^2m>0>1 ]'=1 and the eigenvalues [X2rrh0}.1]'L1 coincide with the solutions of the corresponding degenerate problems A0 or B0.
Thus, we can determine the boundary functions n2m 0^>(p1), Q2m 0^(p2) if we find the solutions of the boundary value problems as under
L2m n2m, 0 ^ = 0, L"2m Qr2m, 0 ^ = 0, D% n2m,0 m + & Am,0 (0) = WQ2rn,0W) + D* ^2rn,0 (*),
n2rn,0$(Pi)^0, Q2m,0№2)^0, e^0, i = 0,1,2,... ,2m — 1. We can write the functions n2m 0^>(p1 ) and Q2rn 04>(p2) in the forms
m—1
n2m,0^(P1) = ^ C2™"1 exp(-a2mPl),
<=1
m—1
Q2rn,0^(P2) = ^ Cf0'2 exp(-«2mP2). <=1
Hence, the number of arbitrary constants C^1'1 and C^'2 equals the number
of disappearing boundary conditions of problems A22m or B2m when we try formulate the degenerate problems A0 or B0.
Let the values a2™' (( = 1,..., 2m — 2) be the roots of the additional characteristic equation
m ( — 1)P
p(a2m ) = Y.[(—1v(a2m )2p—2 = 0.
Since an algebraic equation
Re (a2^) >0, ( = I,m-1, Re (a2^) <0, ( = m, 2m-2,
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
Dzn2m,0rn - DiQ2m>0i>(r) = —Di02(0) + D*02(r),
where i = 0,1,2,..., 2m —1, and we can derive a system of 2m linear equations like that
D2mC2m = b2m, (14)
for finding coefficients C^1'1, C^'2 (( = 1,2,... ,m — 1), where a system has form as under
D2m = ( 0 )
V 0 D%) ,
and where
r\2m D11 l = 1, di,r^ _ (- -a2rn y-1,
T-\2m d22 l = 1, d-2,r( _ — (a2m y-1,
C2mT _ pi2m,1 f^2m,2 Um-1,0, °1,0 pi2m,2 \ , •••, ^m-1,0)
rl2mT _ (ir,2m rn2m „2m „2m \
b — (P1 ,---,Pm-1,yi , ••• , "m—1),
p2m = —D%i>2rn,0(0), q2m = Dii>2m,,o(r), i = 0,1,... ,2m — 1,
are block matrices.
Since the values of a2'm(( = 1,2m —2) are pairwise different and the matrices D^, D2^, D2m are non-degenerate and there is an inverse of D2m matrix (D2m)-1, so then the only solution of the algebraic system (14) exists and it has the form: C2m = (D2m)-1 b2m.
Thus, a zero approximation of ^>2m 0, n2m 00, Q2m 00, X2m 0 of the problems A2™ and B2m could be constructed completely.
3.2.2. Further construction of the asymptotic series
We can get the systems of the equations for the problems A2m and B"lm and use the additional conditions for finding the solutions m k, n2m k 0, Q2m k 0 and X2m k in the case k > 0 in this form
[L2 — \2m,0m, k _ ^2m,k^2m, 0 — (r),
^mn2m,k^ _ d^k (fil), Q2m,k^ _ 92k (Pr2),
D% $2m,k(0) + n2rn,k^(0)) _ D% ($2m,k + Q2rn,k ^(r)),
n2m,k Q2m,ki)(p2 )^0, £^0, k = 1,2,..., i = 0,l,...,2m-l,
hkm (r) = Y (2ni2\UD2P+2$2rn,k-2p-
X2m,p ^2 m,k—p,
p=1 \2P+2)-- p=i
y1 k—2
9lkn (Pl) = 1 n2m,k—1 ^ + ^^ {X2m,p — vpPi ) n2m,k—p—2^,
Pi p=0
V2 k—2
9^k(P2) = 1 Qr2m,k—1 Y {X2m,p — (—1)PVpP2) Qlm^k—p—l
Pl p=0
If the parameter A is a simple proper value of the self-adjoint operator A that acting in the Hilbert space H(Qr) and if the function ^ £ H(ilT) is the corresponding normalized eigenfunction ||Vimr> ^ = 1 then in the space
±1 (Qp)
H1 (Qr) (H1(^T) is an orthogonal complement to the function ^ in the space H(QT)) and then there is the operator A — XI that has a bounded inverse operator (A — XI)—}, , (pseudo-resolvent).
Hence, the equation Ap — Xp = — h, h £ H(Qr) can be solved and the solution of this equation could be presented as under
U = (h,^)H(Qr) (A — XI)H11(Qr) (u^ — h),
where (u^ — h) £ H1 (QT).
Thus, we can get the solutions ^2m k n and X2m k n for any k > 0
X2m,k,n = {h2km,^0,n )H{Qr) = j hkm (r)^0,n (r)dr, n=1,2,..., ^2m,k,n = (^2 — X2m,0,n)H q ) ^k™,
where H1 (QT) is the orthogonal complement to eigenfunctions ^0n £ H(QT), (r = A, B) of the degenerate boundary value problem A0 or B0, where
1^0,n ^(fip) = 1.
We can find the boundary functions n2m k^(p1), Q2m ki>(p2) for k > 0 from the boundary value problems in the form
L2m n2m , k ^ = 91k', Q^m , k^ = 92k?', (15)
D% ^^ H0) — & Q2m,k W) = —D% $2 m,k(0) + Di $2 m>k (r), (16) n2m,k4>(p1 )^0, Q2mtk^(p2) ^ 0, £ ^ 0, i = 0,1,... ,2m — 1. (17) We can write the functions n2m k^ and Q2m k^ as under
n2m,ki>(P1) = n2m,k$(P1) + n2m,k A (P1 ), (18)
Q2m,ki)(P2) = Q2m,ki{p2) + Q2m,k(P2), (19)
where
m—1
n2m,ki>(Pi ) = E cf^1 exp (-a2mPi) ,
C=1
m—1
Q"2m,kÎ(P2) = E C2y"k'2 eXP (-a2mp2]
C=1
are the general solutions of the homogeneous equations (15), (17), and
m—1
n~2m,kr(Pl) = E (Pi) eXP —2mPi) ,
<=1
m—1
Q2m,,kP (P2) = E C2T (P2) exP (-a2mP2) ,
<=1
are the partial solutions of these inhomogeneous equations.
Since, the roots a2n are pairwise distinct, then the Vronsky determinants
W[e—a1mPi ,...,e—a2™-iPi ] , W [e—a2mp2, ... , ] ,
that are composed of the function systems [exp (-a2™p1 and
[exp (-a2™P2)]^ , are non-zero.
Using the method of constant variation, we can find the partial solutions of the inhomogeneous equations (15), (17), i.e.
D1? Ù1 = F1, D2^fiÎ2 = F2,
= ( dCf^1 (P1) (P1 )
1 I dp1 , , dp1
oT =
del™'2 (p2) dCt7%—-1 (P2)
dp2 dp2 FT = (0,...,0,g2r), FT = (0,...,0,g2^ ),
where det |D2^| ± 0, det |D2^| + 0.
We can find the functions C2y,1(p1 ) and C2™'2 (p2) from the systems as
under O1 = (D^)—1 F1, O2 = (D2T)—1F2. '
After integrating and substituting the solutions in (18), (19), we can find as many arbitrary constants as the boundary conditions of the problems or B2m fall out when we proceed to analysis of the degenerate problems A0 or B0.
Thus, this algorithm allows us to find the asymptotic solutions of the problems A22m and B^ with any desired degree of accuracy of a small parameter e'J.
4. Asymptotic analysis of the solutions
We can formulate the following theorem for the justification of the asymptotic solutions of the problems Aand B;;m.
Theorem 1. If the self-adjoint elliptic operators L2, L2m satisfy Conditions 1-3 for the problems A^m, B"2m, A0, B0 and the function v(r) G CC is represented as the uniformly converging series in the neighborhood of the point r = 0 and the neighborhood of the point r = r0
CO CO
V(r) = E V±1 r'S, V(r) = E v2 (r-r0)S ,
s=-1 s=-1
the asymptotic solutions of boundary value problems A"2™' and B2rn exist. The corresponding n eigenvalue X£ 2m n and the corresponding n eigenfunction
^e 2m n(r) of the operator L2m have the following asymptotic representations
^e,2m,n = ^2m,0,n + £^2m,1,n + £2 X2m,2,n + ••• + £^+1 A2m,
C
^e,2rn,n (r) = E^ (^2rn,k,n (r) + n2rn,k,n ^(.Pl) + Q2rn,k,n^(.P2 )) + £3+1Z2jm (r), k=0
where X2m 0 n = X0 n is n-th simple eigenvalue and m 0 n(r) = n is the n-th function of the operator L2 for boundary value problems A0 and B0; the
functions fa rn , k , n (r)> n2rn, k , Q-2m , k , n ^ and the values of X2rn , k , n for k>0
are determined from the systems of the equations and the boundary conditions given in Paragraph 2.
The estimations for the residual members z2m(r) and A2m have form as under \Dz2m\H + \\z2m\H = 0(ej+1), = 0(1), for p-order derivative of the partial sum Qj^e>2m>n is \Dq+2z2m= 0(ej-q+1), 1 ^q < s, s > 2m —2, in the inner subdomain [5,r0 — 5] is \Dq+2z2m\H = 0(e^+1), |g| < s, in border regions (0,5] and [r0 —S,r0) is \Dq+2z2m\H = 0(e^-q+1), 1 ^ Iql < s.
Proof. It is assumed that the function Qj^£ 2m n (r) satisfies the boundary conditions of the problems and B2m and
U0\H = hhrn,0\H = 1 Ue,2rn\h = 1 + 0(€). Using series for the constructions of a solution, we can get
[L2m, — Qj\,2m] Qj ^e,2m (r) = fj™,
where ]2m is the restricted function ( \]2m\H = 0(1)).
According to the estimate, we have the evaluation in the form as under
inf IX — K,2m,J < — \H\H,
r>. > >
where ^ £ is an arbitrary function from the scope of the operator U2m and A > 0 is an arbitrary real number.
Using the evaluation \^£ 2m||ff = 1 + 0(e), we can get that
A — Q \ = r^+1 \2m
/x£,2m,n yyj/^£,2m c L-*j ,
where IA2m| < Ifjln/\Qj^£2mlh. Hence, we can get the estimate A2m = 0(1). ,
Let be a closed linear shell consisting of eigenfunctions Qj, 2rn, n(r), corresponding to the corresponding eigenvalues QjX£2mn, that are lying on
a segment [X0n — d, X0n + d], where d is a number d > £ (¡¡2mQj^s 2m —
Qj\,2m Qj^e,2mIh ^ a), then there is such a function £T^, H^llff = 1,
for which the following inequality \Qj^s 2m — ^t\h ^ 2£/d is satisfied.
If e is sufficiently small, then the following inequalities occur
X£,2m,n—1 — ^ d, Xe,2m,n — X0,n ^ d, Xe,2m,n+1 — \),ra+1 ^ d,
where 3d = min[A0,n — \0,n—1; \),n+1 — \),n].
Thus, a segment [A0 n — d,X0n + d] contains the single eigenvalue A£ 2rn n of the operator L2rn, which is relevant to the single normalized eigenfunction 2m n(r), which coinciding with the normalized function , and there is the estimation
\^£, 2m, n —Qj^£, 2m, n/\Qj^£,2m, n \h\h ^ 0(6j+1).
Thus, we can get the estimation ||z?m||ff = o(1), where z2m = ej+1 z2m =
^£,2m,n ^£,2m,n , and ^£,2m,n = \Qj^£,2m,n ¡H ^£,2m,n.
Since the inequality X£ 2m j > X0>7, j = 1,2,... is true and there is the ratio
[L2m — Qj K,2m ]Qj A,2m, (r) = 6j+1 f2™, \f2m\H = 0(1),
we can get the following estimations
\[L2m — K,2m, m\H = 0(ej+1)
and \L^2m^2 ^ ¡H ^ \ [L2m — X£,2m ^¡H + IX£,2m 1 ¡¿j ™\h = 0(6J+1).
Using Conditions 1-3 and assuming that the function z2m satisfies the
boundary conditions of the problems A2m and B£m, we can get the following estimations
m—1
\%m\2H < Y ^IDPz2mIff + \Dz2mIff + Ih < Ce2(j+1)\W2^\h, p=1
where the constant C > 0 which is independent of r and £ and the function
w2m is the restricted function for which the estimation \w2m\\H — 0(1) takes place.
This implies the estimate for zthat is in the conditions of the theorem.n
5. Solutions behavior analysis of the problems and B2m in the case m ^ œ
Here we investigate the question about the behavior of the eigenfunctions and the eigenvalues of A2m and problems in the case of unlimited increasing of 2m-order LTKT-equation.
Let's consider the problems of Â2£m,B"2m and A2£m+2, B2m+2 for finding
]^=1, [^s^m^^l and [^£,2m+2,-/[Xe,2m+2,^]^=1. Here we assume
that the eigenvalues are arranged in order of monotonic increase. Let the relations
\2m+2j — J, \2m+2 \ =\
2m Ye,n Ye,2m+2,n Ye,2m,n, L-^2m /Ks,2m+2,n ^e,2m,n,
take place, where \^£t2m+2\H = 1 Ue,2rn\h = 1 We can formulate the following
Theorem 2. If the positive self-adjoint elliptic operators act in the space H(Q,T), L2, L2m and satisfy Conditions 1-3 for the problems Â2^', B2m, A0, B0, then we have the following estimates for m ^ œ
2P2m
\\2m+2 \ I ^ \\Je — Je \ ^
\^2m Ae,n ^ P2m+2 ±J2mAH ^ (2m + 2)\\,
2ç-2m,
\\^2m+2 j \ / 2fc 2m Ye,n\H ^
(2m+ 2)!!' Proof. We can get the ratios
\2m+2 t — Je — 2(1) £ r)2m+2
2m> ~ 2m+2 ^2m ~ m... , r.\n ^ '
2(-i)^+1 £2™ (2m+ 2)!!
K,2m+2,n ^ sup[((^2m+ L + L<2m )lf, f) H] ^ Xe,2m,n +
Mh = 1,(<P, ^e,2rnrf)H = 0 7=l,n-l,
where A is the largest positive eigenvalue of the operator A2™+2L, and there is the following inequality A < |A^+2L\H.
We get the inequalities IA2™+2\£>n 1 < \L2m+2 - L2m\\h, where
A2T2K,nI ^ 2e2m/(2m + 2)V..
Thus, there is the equality
^ 2,2m,
[f2m+2 — K,2m+2 ]A2m ^e = (2m + 2)!m'
where v2m is the restricted function, \\v2m\\# = 0(1), (v2m,A2™+2^e)H = 0.
We can assume that the operator (L£2m+2 — Xe 2m+2) has a limited inverse operator (L2m+2 — \i2m+2)-1 (a pseudo-resolvent) and there are the ratios
2P2m,
\2m+2 j = 2t (fe — \ )-1 r.
LA2m Ye — (2m + 2)U 2'm+2 Ae,2m+2)H1 u2m,
2,2m,
and \\A2^+2^e n \\H < —-——. Thus, the theorem is proved. □
(2m+ 2)!!'
6. Construction of an asymptotic solution in the case of the oscillator potential
We can consider the boundary value problem B2m on the [0, axis with the quasi-potential of a linear harmonic oscillator in the form v(r) = r2. 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.
The solution of the degenerate boundary value problem B0 is an orthonormal system of Hermite functions
Vv = [n!2n VW1/2 exp(—r2/2)Hn(r), Xn = 2n+1, n= 1,3,5,..., where
[n/21 2rn-2m,
Hn(r) = n! £ ^mKn — 2rn)!-
We can show that the zero approximation has equality ^2m 0n = n.
We can find the functions n2m 0 n^(p1) and Q2m 0 n4>(p2) in the form
m-1
n2rn , 0 , n ^(Pl) = Y C0kn exP (—ak Pi), Q2rn , 0 , n^(P2 ) = 0
dS^2 m, 0 , n
(0)
k=1
m—1
Cokn = ^^ £S 1 ,s
drs
S = 1
Ais = — - -, l,s = 1,^,m-1,
, ll¥s (ai -as)
2q
E (-1)r n2r-2
¿-^ (2r)!as
A2q,s = A2q+I,s = -^=1 ,-n , Q=l,2,..., (m-l)/2, l,s = 1,..., m-l,
Dz (0) = [n\2ny/^\~1/2Di [exp(-r2/2)Hn(r)]|(r=o}, n = l,3,5,... The first approximation of the solution has the forms
^2m,1,n = , X2m,1,n = 0,
m—1
U2m,1,n j>(P1) = E C1kn exP(-afcP), Q2m,1,n^(P2) = 0
k=1
(0)
C1kn = E £S 1
drs
S = 1
The next approximation has the following ratios
^2m,2,n = ^0,n, X2m,2,n = £^(n2 + + l) ), n = l, 3, 5, ,
m—1
n2rn,1,ni)(P1) = E Fkn (r,£) exp(-£—1 akr), Q2rn,2,n^(fi2) = 0
k=1
m—1
Fkn (T,£) Rkn + P1 r^kn, ^kn ^1kn X2m,0,n ^ ^ ^1pnBp,k,
p=0
Tkn = ^1knX2m,0,n Bk,k, Bkk =
n^k (aJ -ak V
m—1
E (ak- Vj)
R = j=1>j^k_ R =
k,k ^ , ,2 , JDp,k
nm(ctj -ak) ' p' K -ak)nm K' - ak)'
Thus, we can continue the procedure for constructing the asymptotic series and building an asymptotic solution of the problem under consideration with accuracy up to any given order e.
7. Conclusions
Recently, there is a great interest in studying properties of bound states of a quarkonium such as charmonium cc and bottomonium bb. 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 Schrodinger-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 2m relativistic finite-difference Schrodinger 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 e. 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 £ —y 0 is investigated.
The behavior of solutions is investigated in the case m — to 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 m — to.
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.
In this work the Sturm-Liouville problem on the positive half-line with a periodic boundary conditions for the quantum harmonic oscillator is considered and eigenfunctions and eigenvalues are constructed as asymptotic solutions for 2m-order LTKT-equation. Their solutions allow to describe the behavior chains of harmonic oscillators with periodic boundary conditions when they are very far apart from each other. We can use more complex quasi-potentials and describe the bounded states of the elementary particles in the quark-gluon plasma.
Acknowledgments
The publication has been prepared with the support of the "RUDN University Program 5-100" and funded by RFBR according to the research projects No. 18-07-00567.
References
[1] A. A. Atanasov and A. T. Marinov, "^-Expansion for bound states described by the relativistic three-dimensional two-particle quasi-potential equation," Theoretical and Mathematical Physics, vol. 129, no. 1, pp. 14001407, 2001. DOI: 10.1023/A:1012423629038.
[2] Y. D. Chernichenko, "On a solution of the relativistic inverse problem for the sum of nonlocal separable quasipotentials," Russian Physics Journal, vol. 55, no. 6, pp. 699-711, 2012. DOI: 10.1007/s11182-012-9869-3.
[3] V. G. Kadyshevsky, R. M. Mir-Kasimov, and N. B. Skachkov, "Quasipotential Approach and the Expansion in Relativistic Spherical Functions," Nuovo Cimento A, vol. 55, no. 2, pp. 233-257, 1968. DOI: 10.1007/ BF02759225.
[4] V. G. Kadyshevsky and M. Mateev, "On a Relativistic Quasipotential Equation in the Case of Particles with Spin," Nuovo Cimento A, vol. 55, no. 2, pp. 275-300, 1968. DOI: 10.1007/BF02759227.
[5] V. G. Kadyshevsky, "Quasipotential type equation for the relativistic scattering amplitude," Nuclear Physics, no. 1, pp. 125-148, 1968.
[6] A. A. Logunov and A. N. Tavkhelidze, "Quasi-optical approach in quantum field theory," Nuovo Cimento, vol. 29, pp. 380-399, 1963. DOI: 10.1007/BF02750359.
[7] A. A. Logunov and A. N. Tavkhelidze, "Quasi-potential character of the scattering amplitude," Nuovo Cimento, vol. 30, pp. 134-142, 1963. DOI: 10.1007/BF02750754.
[8] A. A. Logunov, A. N. Tavkhelidze, and O. A. Khrustalev, "Quasipotential character of the Mandelstam representation," Physics Letters, vol. 4, no. 6, pp. 325-326, 1963.
[9] V. A. Matveev, V. I. Savrin, A. N. Sissakian, and A. N. Tavkhelidze, "Relativistic Quark Models in the Quasipotential Approach," Theoretical and Mathematical Physics, vol. 132, no. 2, pp. 1119-1136, 2002. DOI: 10.1023/A:1019704709192.
[10] A. Pokraka and R. Dick, "Dimensional effects on the density of states in systems with quasi-relativistic dispersion relations and potential wells," Canadian Journal of Physics, vol. 94, no. 8, pp. 773-779, 2016. DOI: 10.1139/cjp-2015-0758.
[11] K. A. Sveshnikov and P. K. Silaev, "Quasi-exact solution of a relativistic finite-difference analogue of the Schrödinger equation for a rectangular potential well," Theoretical and Mathematical Physics, vol. 132, no. 3, pp. 408-433, 2002. DOI: 10.1023/A:1020220104534.
[12] K. A. Sveshnikov and P. K. Silaev, "Quasi-exact solution of the problem of relativistic bound states in the (1+1)-dimensional case," Theoretical and Mathematical Physics, vol. 149, no. 3, pp. 1665-1689, 2006. DOI: 10.1007/s11232-006-0150-1.
[13] V. S. Minh, E. P. Zhidkov, and V. G. Kadyshevsky, "Solutions of relativistic radial quasipotential equations," Theoretical and Mathematical Physics, vol. 63, no. 2, pp. 493-503, 1985. DOI: 10.1007/BF01017906.
[14] I. V. Amirkhanov, E. P. Zhidkov, I. E. Zhidkova, and S. A. Vasilyev, "Construction of an asymptotic approximation of eigenfunctions and eigenvalues of a boundary value problem for the singular perturbed relativistic analog of the Schrödinger equation with an arbitrary potential [Asimptotika sobstvennyh funkcij i sobstvennyh znachenij kraevoj zadachi dlya singulyarno vozmushchennogo relyativistskogo analoga uravneniya Schrödingera pri proizvol'nom potenciale]," Mathematical Models and Computer Simulations [Matematicheskoe modelirovanie], vol. 15, no. 9, pp. 3-16, 2003, in Russian.
[15] I. V. Amirkhanov, E. P. Zhidkov, D. Z. Muzafarov, N. R. Sarker, I. Sarhadov, and Z. A. Sharipov, "Investigation of boundary-value problems for the singular perturbed differential equation of high order [Issledovanie kraevyh zadach dlya singulyarno-vozmushchennogo differ-encial'nogo uravneniya vysokogo poryadka]," Mathematical Models and Computer Simulations [Matematicheskoe modelirovanie], vol. 19, no. 11, pp. 65-79, 2007, in Russian.
[16] I. V. Amirkhanov, N. R. Sarker, I. Sarhadov, Z. K. Tukhliev, and Z. A. Sharipov, "Analytical and Computational Investigations of Solutions of Boundary-Value Problems for the Quasipotential Equation [Analiticheskoe i chislennoe issledovaniya reshenij kraevyh zadach dlya kvazipotencial'nogo uravneniya]," Bulletin of Peoples' Friendship University of Russia. Series Mathematics. Information Sciences. Physics [Vestnik Rossijskogo universiteta druzhby narodov. Seriya: Matematika, informatika, fizika], no. 4, pp. 44-52, 2012, in Russian.
[17] V. O. Galkin, R. N. Faustov, and D. Ebert, "Logunov-Tavkhelidze equation in the relativistic quark model," Theoretical and Mathematical Physics, vol. 191, no. 2, pp. 641-648, 2017. DOI: 10 . 1134/ S0040577917050038.
[18] N. Alam and S. Mandal, "On the quantum phase fluctuations of coherent light in a chain of two anharmonic oscillators coupled through a linear one," Optics Communications, vol. 366, pp. 340-348, 2016. DOI: 10 . 1016/j.optcom.2016.01.019.
[19] Z. Chen, "Mapping quantum many-body system to decoupled harmonic oscillators: general discussions and examples," Physics Letters A, vol. 382, no. 37, pp. 2613-2617, 2018. DOI: 10.1016/j.physleta.2018.07.043.
[20] E. P. Zhidkov, V. G. Kadyshevsky, and Y. V. Katyshev, "Problem of the c ^ ro limit in the relativistic Schrödinger equation," Theoretical and Mathematical Physics, vol. 3, no. 2, pp. 443-446, 1970. DOI: 10. 1007/BF01046508.
For citation:
I. V. Amirkhanov, I. S. Kolosova, S. A. Vasilyev, Asymptotic solution of Sturm-Liouville problem with periodic boundary conditions for relativistic finite-difference Schrödinger equation, Discrete and Continuous Models and Applied Computational Science 28 (3) (2020) 230-251. DOI: 10.22363/2658-46702020-28-3-230-251.
Information about the authors:
Amirkhanov, Ilkizar V. — Candidate of Physical and Mathematical Sciences, head of the group of Methods for Solving Mathematical Physics Problems of Laboratory of Information Technologies (LIT) of Joint Institute for Nuclear Research (e-mail: [email protected], phone: +7(496)2162547, ORCID: https://orcid.org/0000-0003-2621-144X, Scopus Author ID: 6507929197)
Kolosova, Irina S. — PhD's degree student of Department of Applied Probability and Informatics of Peoples' Friendship University of Russia (RUDN University) (e-mail: [email protected], phone: +7(495)9522823, ORCID: https://orcid.org/0000-0002-7594-3375)
Vasilyev, Sergey A. — Candidate of Physical and Mathematical Sciences, assistant professor of Department of Applied Probability and Informatics of Peoples' Friendship University of Russia (RUDN University) (e-mail: [email protected], phone: +7(495)9522823, ORCID: https://orcid.org/0000-0003-1562-0256, ResearcherID: 5806-2016, Scopus Author ID: 56694334800)
УДК 517.958, 517.963
PACS 02.30.Hq, 02.30.Mv, 03.65.Ge, 11.10.Jj, 03.65.Pm, 02.30.Em DOI: 10.22363/2658-4670-2020-28-3-230-251
Асимптотическое решение задачи Штурма—Лиувилля
с периодическими граничными условиями для релятивистского конечно-разностного уравнения
Шрёдингера
И. В. Амирханов1, И. С. Колосова2, С. А. Васильев2
1 Объединённый институт ядерных исследований ул. Жолио-Кюри, д. 6, Дубна, Московская область, Россия, 1^1980 2 Российский университет дружбы народов ул. Миклухо-Маклая, д. 6, Москва, Россия, 117198
Описание взаимодействия релятивистских частиц в рамках квазипотенциального подхода широко применяется в современной физике. Этот подход основан на так называемой ковариантной формулировке квантовой теории поля, в которой эта теория рассматривается на пространственно-подобной трёхмерной гиперповерхности в пространстве Минковского. Особое внимание в этом подходе уделяется методам построения различных квазипотенциалов, а также использованию квазипотенциального подхода для описания характеристик связанных состояний в кварковых моделях, таких как амплитуды адронного упругого рассеяния, масс-спектры и ширины распадов мезонов, сечения глубокого неупругого рассеяния лептонов на адронах.
В настоящей работе сформулированы задачи Штурма—Лиувилля с периодическими граничными условиями на отрезке и на положительной полупрямой для усечённого релятивистского конечно-разностного уравнения Шрёдингера (уравнение Логунова—Тавхелидзе—Кадышевского, ЦГКТ-уравнение) с малым параметром при старшей производной.
Целью работы является построение асимптотических решений (собственных функций и собственных значений) в виде регулярных и погранслойных частей решений для этой сингулярно возмущённой задачи Штурма—Лиувилля. Основная задача исследования состоит в асимптотическом анализе поведенческих решений рассматриваемой задачи в случае е ^ 0 и т ^ <х>. Нами был предложен метод построения асимптотических решений (собственных функций и собственных значений), который является обобщением асимптотических методов решения сингулярно возмущённых задач, представленных в работах А. Н. Тихонова, А. Б. Васильевой и В. Ф. Бутузова. Основной результат данной работы — доказанные теоремы об асимптотической сходимости решений сингулярно возмущённой задачи к решениям вырожденной задач при е ^ 0 и сходимости решений усечённого ЦГКТ-уравнения в случае т ^ ж. Кроме того, в статье нами рассматривается задача Штурма-Лиувилля на положительной полуоси для ЦГКТ-уравнения 4-го порядка с периодическими граничными условиями для квантового гармонического осциллятора. Для этой задачи построены асимптотические приближения собственных функций и собственных значений и показана их сходимость к решению вырожденной задачи.
Ключевые слова: асимптотический анализ, сингулярно возмущённое дифференциальное уравнение, задача Штурма—Лиувилля, релятивистское конечно-разностное уравнение Шрёдингера, периодические краевые условия, квазипотенциальный подход
