Asymptotics of the eigenvalues of a fourth-order functional-differential Текст научной статьи по специальности «Математика»

DOI: https://doi.org/10.23670/IRJ.2019.89.11.002

АСИМПТОТИКА СОБСТВЕННЫХ ЗНАЧЕНИЙ ОДНОГО ФУНКЦИОНАЛЬНО-ДИФФЕРЕНЦИАЛЬНОГО ОПЕРАТОРА ЧЕТВЁРТОГО ПОРЯДКА С СУММИРУЕМЫМ ПОТЕНЦИАЛОМ

Научная статья

Митрохин С.И. *

Московский государственный университет им. М. В. Ломоносова, Москва, Россия * Корреспондирующий автор (mitrokhin-sergeyxtatlyandex.ru)

Аннотация

В статье изучаются спектральные свойства функционально-дифференциального оператора четвёртого порядка с суммируемым потенциалом. Граничные условия являются разделёнными. Решение функционально -дифференциального уравнения сведено к решению интегрального уравнения Вольтерра. При больших значениях спектрального параметра выведена асимптотика решений функционально-дифференциального уравнения, задающего исследуемый оператор. При помощи найденной асимптотики решений изучены граничные условия оператора. Получено уравнение на собственные значения изучаемого оператора. Найдена асимптотика собственных значений исследуемого функционально-дифференциального оператора.

Ключевые слова: спектральная теория, функционально-дифференциальный оператор, спектральный параметр, суммируемый потенциал, асимптотика решений, асимптотика собственных значений, собственные функции.

ASYMPTOTICS OF THE EIGENVALUES OF A FOURTH-ORDER FUNCTIONAL-DIFFERENTIAL

OPERATOR WITH SUMMABLE POTENTIAL

Research article

Mitrokhin S.I. *

Lomonosov Moscow State University, Moscow, Russia

* Corresponding author (mitrokhin-sergeyx[at]yandex.ru)

Abstract

In this paper the spectral properties of a fourth-order functional-differential operator with a summable potential are studied. The boundary conditions are separated. The solution of the functional-differential equation is reduced to the solution of the Volterra integral equation. For large values of the spectral parameter, the asymptotics of the solutions of the functional-differential equation that defines the investigated operator is derived. Using the found asymptotics of solutions, the boundary conditions of the operator are studied. The equation for the eigenvalues of the operator is obtained. The asymptotics of the eigenvalues of the functional-differential operator under investigation is found.

Keywords: spectral theory, functional-differential operator, spectral parameter, summable potential, asymptotics of solutions, asymptotics of eigenvalues, eigenfunctions.

Introduction

Consider a functional-differential operator given by a fourth-order differential equation

y(4>(x) + q(x)• y(p(x)) = A • p(x) • y(x),0 < x < n, p(x) > 0,0 < p(x) < x, (1)

with boundary conditions of the form

/(0) = y'"(0) = y'(n) = y'"(n) = 0, (2)

under the assumption of the summability of the potential q(x ) :

q(x) e Li [0; n](=)jj q(t )dt J = q(x) (3)

almost everywhere on the segment [0; n].

We assume that in equation (1) the weight function p(x) is constant: p(x) = a4 > 0 for all x e[0; n], a > 0 and the delay value is a smooth and increasing function: there is a derivative /3'(x), and /3'(x) > 0 for all x e [0; n].

Historical review

Differential operators with nonsmooth coefficients began to be studied not so long ago. In the classical paper [1], for the second-order differential operator, the convergence of expansions in eigenfunctions at the points of discontinuity of the coefficients was studied.

In the article [2] the differentiation operators of the first and second order with a discontinuous alternating weight function were studied. Regularized traces for the functional-differential operator of the second order were calculated by the

11

author in [3]. The spectral properties of second-order operators with a discontinuous weight function were studied in [4]. In article [5] the second-order operators with summable potential were first studied. The method of article [5] does not carry over to operators of order higher than the second.

In the article [6] the author demonstrated a method for studying the spectral properties of a fourth-order operator with a summable potential. The Sturm-Liouville operators with singular coefficients were studied in [7]. A sixth-order differential operator with summable coefficients with a retarded argument was studied by the author in [8].

In article [9] the regularized trace of a second-order operator whose potential is a Delta function was calculated. Differential operators of odd order with multipoint boundary conditions with summable potential were studied by the author in [10]. Operators of the form (1) - (2) - (3) have not been studied by anyone before.

Asymptotics of solutions of differential equation (1)

Let X = s4, s = VX , moreover, that branch of the root for which Vl = +1 is fixed. Let {wk j4^ be the various roots of the fourth degree of unity:

—(k-l) 4

w\ = 1, wk = e 4 , k = 1,2,3,4; w1 = 1, w2 = i, w3 =-1, w4 = i; X wp = 0(p = 1,2,3). (4)

k=1

The following theorem is proved by the method of variation of arbitrary constants.

Theorem 1. The solution y(x, s) of the functional differential equation (1) is the solution of the following Volterra

integral equation:

4 14 x

y(x, s) = XCk • eflwksx X wk • eflwksxJq(t) e^ • y(e(t), s) dtA. (5)

k=1 4a s k=1 n

To verify the validity of the statement (5) we obtain using the properties (3) and (4):

(6)

4 1 4 I I 1

y'^ s) = S Ck -Ks). - — X Wk -KS). .1J... I - —.

k=1 k=1 V0 Jak

■XWk . eaWksx . q(x). e-aWksx . y{p(x),s).

k=1

The last sum in equality (6) is zero by virtue of property (4). Similarly we have:

4 14 I x I

y (" )(x, s) = X Ck. {awks) . eaWkSX - ^ X wk • (awks )m . eaWksx. |J... | , m = 2; m = 3. (7)

k=1 J k=1

.0 Jak

Let's differentiate the formula (7) by m = 3 for the variable X, substitute the resulting expression and (5) in equation (1), we get the correct equality. So, indeed, the formula (5) of theorem 1 is correct.

Next, we apply the method of successive Picard approximations: find a function y(fi(x), s) from the formula (5), substitute the resulting expression in (5), and get:

4 14 x

y(x,s) = XCk • ecm'kSX - 4a1-3• Xwk • eaWkSX • Jq(t)• e"awksx •

k=1 J k=1

4

X C • eawkse(t).

^ /l„3„3

4 P(t)

■ Xwn • eawksp{t] • Jq{C\ e aw"sx • y(e(Z),s)• dCai

n=1

4a s3

• dtak , x G[0;n].

(8)

By opening the brackets in formula (8) and rearranging the terms, we obtain the following statement. Theorem 2. The general solution of the functionally differentiable equation (1) has the following form:

y(x, s) = X Ck • yk (x, s); y(")(x, s) = X Ck • y[m)(x, s), m = 1,2,3, (9)

k=1 k=1

where Ck (k = 1,2,3,4) are arbitrary constants, and for the fundamental system of solutions {yk (x, s )}4=1 of equation (1) for s ^ +œ the following asymptotic estimates are valid:

Ук (x s ) = e

awks.

- 4k s )

3_3

4a s

+ O

f llm slax Л

e 1

V s У

к = 1,2,3,4,

(10)

yM(x, s ) = (as )Jw,

m awk sx к •e k

- Am (x, s) 4a3s3

+ O

f Imslax Л

e

, m = 1,2,3; к = 1,2,3,4,

(11)

4k (x, s) = £ Wn eawksx • jq(x) • ea(wkв^ЫКкп,

(12)

Am (x, s) = £ Wn-(awns)m • eaWksx • j q(x) • ea(wke^)s

n'rdtakn, m = 1,2,3; к = 1,2,3,4.

(13)

Ук (0, s ) = 1; ykm )(0, s ) = wm • am • sm ; y (0, s ) = £ Ck -1; y(m >(0, s ) = amsm •X C

ЛЛ>т -к • wk,

к=1

к=1

m = 1,2,3; к = 1,2,3,4.

(14)

Estimates of the form (10)-(11) are obtained similarly to the estimates of the monograph [11, chapter 2]. Study of boundary conditions (2). The presence of formulas (9)-(14) allows us to study the boundary conditions (2). From formula (9) we have:

(15)

(16)

System (15) - (16) is a homogeneous system of four linear equations with four unknowns C1, C2, C3, C4. It follows from

Kramer's method that such a system has a nonzero solution

f 4 Л

X Ck4 * 0

V к=1

only when its determinant is zero. Therefore, the

following statement is true.

Theorem 3. The equation for eigenvalue of the functional differential operator (1) - (2) - (3) has the following form:

f (s ) =

y (0, s) y2 (0, s) y3 (0, s) y4 (0, s)

y(0, s ) y2(0, s ) y3(0, s ) y4(0, s )

y (n s) y2 (n s) y3 (П s) y4 (П s)

yfe s ) y2"(n, s ) y3(n, s ) y4(n, s )

= 0,

(17)

and from the formulas (14) it follows that

yk (0, s) = Wk • as, yk"(0, s) = w3k • (as)3, к = 1,2,3,4.

(18)

6

s

n=1

Expanding the determinant for f (s) from (17) - (18), first in the fourth row, and then the resulting determinants in the third row, we find:

f (s) = (as)4. f (s) = (as)4. [F12 - D34 - F . D24 + F14. D23 + F23. D14 - F24. D„ + F34. D12 ] = 0, (19)

where the following notation is introduced:

F =

km

Уk(п, S ) У m S ).

yk(n, s) y'm(n, S)

; Dkm =

.3 ,..3

k, m e {l,2,3,4}.

(20)

Using the formulas (4), all coefficients for D^ from (20) can be calculated explicitly:

D12 = -2/ = D13 = 0; D14 = 2/; В2Ъ = -2/; D2A = 0; ВЪА = -2/.

(2l)

Therefore, the equation (19) - (21) can be rewritten as follows:

f1 (S)= F12 + F14 - F23 + F34 = 0

(22)

Using the formulas (11) - (13), we have:

F12 =

w1e

+a{ X

4a s3 A331 (n S)

s6

33S ' + 4a s 3 4 s 6

D

122 _ eaw1sn

D

w2e

w23eaw1sn

4a s

33

4a s

123 aw^n

+о 7 :

(П S) + J J_' 4a3s3 O '

A332 (П S )

3

г32 '

4a3 s 3

s6, +

= Dn • e'

?(w1+w2 )s

(23)

where the notation is introduced:

D122

w-

w,

^2 A332

, D123

A (П S)

A331(ПS )

SI w

s w:

(24)

Similarly, it is possible to write out functions F14, F23, F34 from (22).

The asymptotics of the eigenvalues of the operator (1) - (2) - (3)

The equation (22) - (24) will be studied by the methods of work [3], [4]. The indicator diagram of this equation (see [12, chapter 12]) is a square ABCD, the coordinates of the vertices of this square are as follows: A(1;1), B(-1;1), C (-1;-1), D(1;-1). From the general theory of finding the roots of equation (22) - (24) (see [12, chapter 12]), it follows that the roots are in four sectors of an infinitesimal angles, the bisectors of which are middle-perpendiculars to the sides of the square ABCD . Studying the roots of equation (22) - (24) using the methods of works [8], [10], we conclude that the following statement is true.

Theorem 3. Asymptotics of eigenvalues of the functional differential operator

(1) - (2) - (3) in the sector 2) (which corresponds to the segment [AB] of the indicator diagram ABCD)) has the following form:

/ • k d3

, 3k,2 , ^ Sk ,2 =-+ —+ O

a

ak,

1

3

— I, k = k + -, k e N.

k6: 2

wk m

2

Substituting the formulas (25) into equation (22) - (24), we apply the formulas

(12) - (13) and Taylor's formulas, we equate the coefficients at the same powers k , we calculate the coefficient d3k 2 from (25) in explicit form:

d = —

d3k ,2 = 0

8п

j q(t )dt + j q(t) • cos((^(t) +1 )• ~ )• dt

_ 0 0

Formulas similar to formulas (25) - (26) are also valid in other sectors of the indicator diagram ABCD :

~ 3

k = k + —,k g N. (26)

2

—(m-2) (27)

skm = sk,2 ■ e 2 ,m = 3,4,1 d3km = d3k,2, m = 3,4,1. (2/)

Moreover, the eigenvalues of the functional differential operator (1) - (2) - (3) are found by the following formula:

= stm, m = 1,2,3,4; k e N. (28)

Conclusion

Using formulas (25) - (28), we can find the asymptotic behavior of the eigenfunctions of the operator (1) - (2) - (3) and solve the question of their completeness and basis property. The proposed method for studying functional differential operators can be transferred to sixth and eighth order operators with summable potential.

Конфликт интересов Conflict of Interest

Не указан. None declared.

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