Asymptotics of the eigenvalues of a fourth-order functional-differential operator with summable potential

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ASYMPTOTICS OF THE EIGENVALUES OF A FOURTH-ORDER FUNCTIONAL-DIFFERENTIAL OPERATOR WITH SUMMABLE POTENTIAL

Mitrokhin S.I.

Lomonosov Moscow State University, Moscow, Russia

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.

functional differential operator fourth order

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

(1)

with boundary conditions of the form

(2)

under the assumption of the summability of the potential :

(3)

almost everywhere on the segment .

We assume that in equation (1) the weight function is constant: and the delay value is a smooth and increasing function: there is a derivative .

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 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 , moreover, that branch of the root for which is fixed. Let be the various roots of the fourth degree of unity:

(4)

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:

(5)

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

(6)

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

(7)

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 from the formula (5), substitute the resulting expression in (5), and get:

(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:

(9)

where are arbitrary constants, and for the fundamental system of solutions of equation (1) for the following asymptotic estimates are valid:

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 . It follows from Kramer's method that such a system has a nonzero solution 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:

(17)

and from the formulas (14) it follows that

(18)

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

(19)

where the following notation is introduced:

(20)

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

(21)

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

(22)

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

(23)

where the notation is introduced:

(24)

Similarly, it is possible to write out functions 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: . 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 of the indicator diagram )) has the following form:

(25)

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 , we calculate the coefficient from (25) in explicit form:

(26)

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

(27)

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

(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.

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