MATHEMATICS

M. A. BARNOVSKA

ASYMPTOTIC REPRESENTATION OF THE SPECTRAL MATRIX OF A FOURTH-ORDER DIFFERENTIAL OPERATOR

(Presented by Academician I. G. Petrovskii, 13 I 1965)

In this note the spectral matrix of a fourth-order differential operator (L) on the half-axis ((0,\infty)) is found. The operator (L) is defined by the differential expression

[
l(x)=x^{\mathrm{IV}}+q(t)x
\tag{1}
]

and by the boundary conditions

[
x(0)\sin\alpha_1+x'''(0)\cos\alpha_1=0,
]

[
x'(0)\sin\alpha_2+x''(0)\cos\alpha_2=0,
\tag{2}
]

where (q(t)\geq 0), (\displaystyle \int_0^\infty q(t)\,dt<\infty), and (\alpha_1,\alpha_2) are real numbers. This problem, more general than that considered by the author in ((^1)), is also solved by the method used in ((^2)). In this general case the investigation is carried out in greater detail, and the resulting transcendental equation for the eigenvalues is solved in a more natural way.

For the spectral matrix of the stated problem one can formulate the following theorem:

Theorem. For the spectral matrix (\rho=|\rho_{ij}|), (i,j=1,2), of the operator (L) defined above, the following asymptotic formulas hold:

[
\rho_{11}(\widetilde{s}2)-\rho_1)}(\widetilde{s
=
\frac{4}{\pi h_1^2}(\widetilde{s}_2-\widetilde{s}_1)(1+o(1)),
]

[
\rho_{12}(\widetilde{s}2)-\rho}(\widetilde{s1)
=
\rho}(\widetilde{s2)-\rho_1)}(\widetilde{s
=
-\frac{4}{\pi h_1h_2}
\left(
\frac{\widetilde{s}_2^{\,2}-\widetilde{s}_1^{\,2}}{2}
-
\frac{\widetilde{s}_2-\widetilde{s}_1}{h_2}
\right)(1+o(1)),
]

[
\rho_{22}(\widetilde{s}2)-\rho_1)}(\widetilde{s
=
\frac{4}{\pi h_2^2}
\left
\frac{\widetilde{s}_2^{\,3}-\widetilde{s}_1^{\,3}}{3}
-
\frac{\widetilde{s}_2^{\,2}-\widetilde{s}_1^{\,2}}{h_2}
+
\frac{4(\widetilde{s}_2-\widetilde{s}_1)}{h_2^2}
\right.
]

To solve the problem, we first consider the boundary-value problem on the finite interval ((0,l)), defined by the differential equation

[
x^{\mathrm{IV}}+q(t)x=\lambda x,
\tag{3}
]

the boundary conditions (2), and the boundary conditions

[
x(l)\sin\beta_1+x'''(l)\cos\beta_1=0,
]

[
x'(l)\sin\beta_2+x''(l)\cos\beta_2=0,
\tag{4}
]

where (\beta_1,\beta_2) are real numbers and (\lambda) is a sufficiently large parameter.

Linearly independent solutions of equation (3) are found by the method presented in ((^3)). Next we seek solutions (w_1(t,\lambda)) and (w_2(t,\lambda)) of equation (3),

satisfying the initial conditions

[
w_1(0,\lambda)=h_1,\quad w_1'(0,\lambda)=0,\quad w_1''(0,\lambda)=0,\quad w_1'''(0,\lambda)=1;
]

[
w_2(0,\lambda)=0,\quad w_2'(0,\lambda)=h_2,\quad w_2''(0,\lambda)=1,\quad w_2'''(0,\lambda)=0,
]

where (h_1=-\operatorname{ctg}\alpha_1,\ h_2=-\operatorname{ctg}\alpha_2).

Obviously, the linear combination (C_1 w_1(t,\lambda)+C_2 w_2(t,\lambda)) satisfies the boundary conditions (2). Therefore, from the requirement that it also satisfy the boundary conditions (4), we find the transcendental equation for determining the eigenvalues of the given boundary-value problem, i.e.

[
\left(p_1+\frac{p_2}{s}\right)\cos sl+
\left(\frac{p_2}{s}+\frac{g_1}{s^2}\right)\sin sl+
o\left(\frac{1}{s^2}\right)=0,
\tag{5}
]

where (p_1=H_1H_2h_1h_2,\ p_2=H_1h_1(H_2+h_2),\ g_1=2H_1h_1). Here (H_1=\operatorname{ctg}\beta_1;)

[
H_2=\operatorname{ctg}\beta_2;
]

(h_1,h_2) are the notation mentioned above; (s=\sqrt[4]{\lambda};\ \lambda>0).

Obviously, for large (s) the roots of equation (5) are close to the numbers ((2n+1)\pi/2). It can be shown that they are simple.

We find the roots of equation (5) by putting

[
s_n l=(2n+1)\pi/2+\delta_n,\qquad n=0,1,2,3,\ldots,
]

where (\delta_n) is a sufficiently small number. Then we obtain that the difference between two consecutive eigenvalues is

[
\Delta s_n=s_{n+1}-s_n=\frac{\pi}{l}(1+o(1)).
]

We seek the eigenfunctions in the form of the linear combination

[
C_1 w_1(t,\lambda_n)+C_2 w_2(t,\lambda_n).
]

In determining (C_1) and (C_2) we use the definition of an eigenfunction. After this we obtain the asymptotic formula

[
w_n(t)=C\left[\frac{h_2}{s_n}(\cos s_n t-\sin s_n t)+\frac{2}{s_n^2}\cos s_n t+o\left(\frac{1}{s_n^2}\right)\right],
\tag{6}
]

where (C=-\frac{1}{2}C_2). Using the way in which we found the solutions (w_1(t,\lambda)), (w_2(t,\lambda)) and the eigenfunctions themselves, it is easy to prove the boundedness of (w_n(t)) for sufficiently large (s).

From the normalization condition for the eigenfunctions we have

[
C=\frac{1}{a_n},\qquad
a_n^2=\int_0^l \widetilde w_n^{\,2}(t)\,dt,
]

where the function (\widetilde w_n(t)) is defined by equality (6) with (C=1).

Since the boundary-value problem (3)—(2)—(4) is self-adjoint and has simple eigenvalues, the system of its eigenfunctions is complete and orthonormal (we can always perform the normalization). Therefore we can write Parseval’s equality for the finite interval. If (f(t)\in L_2(0,\infty)), then

[
\int_0^l f^2(t)\,dt
=
\sum_n \left{
C_1^2(\lambda_n)F_1^2(\lambda_n)
+
2C_1(\lambda_n)C_2(\lambda_n)F_1(\lambda_n)F_2(\lambda_n)
+
C_2^2(\lambda_n)F_2^2(\lambda_n)
\right},
\tag{7}
]

where

[
F_k(\lambda)=\int_0^\xi f(t)w_k(t,\lambda)\,dt,\qquad k=1,2.
]

Finding (C_1) and (C_2) from equality (7), we obtain the following asymptotic formulas for the spectral matrix in the case of a finite interval:

[
\rho_{11}^{(l)}(s)
=
\sum_{\widetilde{s}_1<s_n\leq \widetilde{s}_2}^{n}
\frac{4}{h_1^2 l}\,(1+o(1)),
]

[
\rho_{12}^{(l)}(s)=\rho_{21}^{(l)}(s)
=
\sum_{\widetilde{s}_1<s_n\leq \widetilde{s}_2}^{n}
\frac{4}{h_1 h_2 l}
\left(s_n-\frac{1}{h_2}+o(1)\right),
]

[
\rho_{22}^{(l)}(s)
=
\sum_{\widetilde{s}_1<s_n\leq \widetilde{s}_2}^{n}
\frac{4}{h_2^2 l}
\left(s_n^2-\frac{2s_n}{h_2}+\frac{4}{h_2^2}+o(1)\right).
]

Multiplying and dividing the right-hand sides of the last equalities by (\Delta s_n) and passing in them to the limit as (l\to\infty) (i.e. (\Delta s_n\to 0)), we obtain the asymptotic formulas of the theorem. In doing so, we use the rules for integrating asymptotic formulas (see at the end of book ((^4))).

In this case it is assumed that (\sin\alpha_k\sin\beta_k\ne 0,\ k=1,2).

REFERENCES

(^1) M. O. Barnovska, Reports of the Academy of Sciences of the Ukrainian SSR, No. 6, 713 (1961).
(^2) B. M. Levitan, Expansion in Eigenfunctions of a Second-Order Differential Equation, Moscow—Leningrad, 1950.
(^3) I. M. Rapoport, On Some Asymptotic Methods in the Theory of Differential Equations, Kyiv, 1954.
(^4) R. Bellman, Stability Theory of Solutions of Differential Equations, IL, 1954.