MATHEMATICS

M. V. FEDORYUK

ASYMPTOTICS OF SOLUTIONS OF ORDINARY LINEAR DIFFERENTIAL EQUATIONS OF THE \(n\)-TH ORDER

(Presented by Academician I. G. Petrovskii on 12 IV 1965)

1. Consider the equation

\[ ly=\sum_{k=0}^{n}\varepsilon^k q_k(x)y^{(k)}=0,\qquad q_n(x)\equiv 1, \tag{1} \]

on the interval \([0,\infty)\), where \(q_k(x)\) are continuous complex-valued functions of \(x\). Denote by \(\lambda_j(x)\) the roots of the equation

\[ f(\lambda,x)=\sum_{k=0}^{n}q_k(x)\lambda^k=0 \]

and put

\[ \varphi_j(x)=-\frac{\lambda'}{2}\frac{\partial}{\partial\lambda} \left(\ln \frac{\partial f(\lambda,x)}{\partial\lambda}\right)\bigg|_{\lambda=\lambda_j(x)}, \qquad \tau_0(x)=q_0^{1/n}(x). \]

Let \(q_k(x)\) satisfy the conditions:

1) \(q_0(x)\ne 0\) for sufficiently large \(x\), the limits

\[ \lim_{x\to\infty} q_k\tau_0^{-n+k}=c_k \]

exist and are finite;

2) the equation

\[ g(\xi)=\sum_{k=0}^{n}c_k\xi^k=0 \tag{2} \]

has no multiple roots;

3) \(f_{ij}(x)=\operatorname{Re}\bigl((\xi_i-\xi_j)\tau_0(x)\bigr)\ne 0\) for \(i\ne j\) and for sufficiently large \(x\), and

\[ \int^{\infty} f_{ij}\,dx=\infty \]

(here \(\xi_j\) are the roots of equation (2));

4) the functions \(q_k''q_0^{-2+(2k-1)/n}\), \(q_k''q_0^{-1+(k-1)/n}\) are summable on the interval \([0,\infty)\);

5)

\[ q_k'q_0^{-1+k/n}=o\left(\min_{i\ne j}|f_{ij}(x)|\right), \qquad x\to\infty. \]

Theorem 1. Let conditions 1)–5) be satisfied. Then there exists \(x_0>0\) such that for \(x_0\le x<\infty\) and \(0<\varepsilon\le 1\), equation (1) has \(n\) linearly independent solutions such that

\[ y_j(x)=\exp\left[\int_{x_0}^{x}\bigl(\varepsilon^{-1}\lambda_j(t)+\varphi_j(t)\bigr)\,dt\right] \bigl(1+\varepsilon\psi_j(x,\varepsilon)\bigr), \tag{3} \]

where \(|\psi_j(x,\varepsilon)|<C\varphi(x)\) and \(\varphi(x)\to 0\) as \(x\to\infty\), \(C\) does not depend on \(\varepsilon\). Analogous formulas are obtained for \(y_j^{(k)}(x)\) for \(1\le k\le n\). We also note that \(\lambda_j(x)\sim \xi_j\tau_0(x)\) as \(x\to\infty\).

1. Consider the equation

\[ Ly=\sum_{k=0}^{n}(-1)^k\varepsilon^{2k}\bigl(p_{n-k}(x)y^{(k)}\bigr)^{(k)}=0 \tag{4} \]

On the interval \(I=[0,\infty]\), where \(p_k(x)\) are continuous complex-valued functions of \(x\) and \(p_0(x)\ne0\) for \(x\in I\). Let \(\Lambda_j(x)\) be the roots of the equation

\[ F(\Lambda,x)=\sum_{k=0}^{n}(-1)^k p_{n-k}(x)\Lambda^{2k}=0. \]

Let the \(p_k(x)\) satisfy the conditions:

1) \(p_n(x)\ne0\) for sufficiently large \(x\); the limits

\[ \lim_{\infty} p_k p_0^{-1}\tau^{-2k}=c_k \]

exist and are finite, where \(\tau(x)=[p_n(x)p_0^{-1}(x)]^{1/2n}\);

2) the equation

\[ G(\xi)=\sum_{k=0}^{n}(-1)^k c_{n-k}\xi^{2k}=0 \tag{5} \]

has no multiple roots;

3) condition 3) of item 1 is fulfilled for the functions
\(F_{ij}(x)=\operatorname{Re}((\xi_i-\xi_j)\tau(x))\), where \(\xi_j\) are the roots of equation (5);

4) the functions

\[ p_k^{\prime 2}p_0^{-4}\tau^{-4k-1},\qquad p_k''p_0^{-2}\tau^{-2k-1} \]

are summable on the interval \([0,\infty)\);

5) \(p_k'p_0^{-2}\tau^{-2k-1}=o\bigl(\min_{i\ne j}|F_{ij}(x)|\bigr)\), \(x\to\infty\).

Theorem 2. Let conditions 1)—5) be fulfilled. Then there exists \(x_0>0\) such that, for \(x_0\le x<\infty\) and \(0<\varepsilon\le1\), equation (4) has \(2n\) linearly independent solutions \(y_j\) such that

\[ y_j(x)=\left[\partial F(\Lambda,x)/\partial\Lambda\right]_{\Lambda=\Lambda_j(x)}^{-1/2} \times \]

\[ \times \exp\left(\varepsilon^{-1}\int_{x_0}^{x}\Lambda_j(t)\,dt\right) \bigl(1+\varepsilon\Psi_j(x,\varepsilon)\bigr). \tag{6} \]

The functions \(\Psi_j\) have the same properties as the functions \(\psi_j\) in Theorem 1.

Analogous formulas hold for the quasi-derivatives \(y^{[k]}\) for \(1\le k\le 2n-1\). As \(x\to\infty\) we have \(\Lambda_j(x)\sim \xi_j\tau(x)\).

Formula (6) is new also for binomial equations of the form (4) and coincides with the formulas obtained in \((1^{-3})\).

1. Let all \(p_k(x)\) be real, \(\varepsilon=1\), and let \(L_0\) be the closed symmetric operator generated by the operation \(L\) of the form (4) and considered on the interval \([0,\infty)\) (see (2), § 17, item 5). Let the \(p_k(x)\) satisfy the conditions:

1′) conditions 1), 2), 4) of item 2 are fulfilled;

2′)

\[ \lim_{x\to\infty}p_0(x)=1,\qquad \lim_{x\to\infty}p_n(x)=\infty; \]

3′)

\[ \lim_{x\to\infty}p_k'p_n^{-(2k-1)/2n}=0. \]

Put \(\xi_i'=\xi_i\), if \(p_n(x)\to+\infty\) as \(x\to\infty\), and \(\xi_i'=e^{i\pi/2n}\xi_i\), if \(p_n(x)\to-\infty\) as \(x\to\infty\).

Theorem 3. Let \(p_k(x)\) satisfy conditions 1′)—3′), and suppose that when \(\operatorname{Re}\xi_i'=0\) and \(i\ne j\), either \(\operatorname{Re}\xi_i'\ne \operatorname{Re}\xi_j'\), or \(\xi_i'=\xi_j'\). Let \(\operatorname{Im}G'(\xi_i')\ne0\) in the latter case. Then:

\(1^\circ\). If \(\operatorname{Re}\xi_i'\ne0\) for all \(i\), then the deficiency index of the operator \(L_0\) is equal to \((n,n)\).

2°. Let \(\operatorname{Re}\xi_i'=0\); \(G'(\xi_i')\ne G'(\xi_j')\) for \(1\le i,j\le 2k\) and \(i\ne j\); \(\operatorname{Re}\xi_i'\ne0\) for the remaining \(i\). Then the defect index of the operator \(L_0\) is equal to \((n+k,n+k)\) or \((n,n)\), depending on whether the integral
\[ J=\int^\infty p_n-\frac{1}{2n}\,dx \]
converges or diverges.

It is known that the defect index of the operator \(L_0\) is equal to \((m,m)\), where \(n\le m\le 2n\). An example of an operator \(L_0\) with any possible defect index was first constructed by I. M. Glazman \((^4)\). For the operators studied in \((^2)\), \(m=n, n+1\), or \(2n\). S. A. Orlov \((^5)\) constructed operators \(L_0\) with any possible defect index; in the case considered by him the equation \(Ly=\mu y\) has a regular singular point at \(x=+\infty\).

Theorem 3 gives a new broad class of operators \(L_0\) having any possible defect number \(m\), while the point \(x=+\infty\) is an irregular singular point for the equation \(Ly=\mu y\). Let us also note that for \(p_0(x)\equiv1\) the principal restriction on the order of growth of the function \(p_n(x)\) in Theorem 2 is as follows: the integral
\[ \int^\infty p_n^{-1/2n}\,dx \]
diverges.

Theorem 4. Let \(L_u\) be an arbitrary self-adjoint extension of the operator \(L_0\), and suppose that the conditions of Theorem 3 are satisfied. Then in case \(1^\circ\) of Theorem 3 the spectrum of \(L_u\) is discrete; in case \(2^\circ\) and \(J<\infty\) the spectrum of \(L_u\) is discrete and
\[ R_\mu=(L_u-\mu E)^{-1} \]
is an integral operator with a Hilbert–Schmidt kernel at all regular points \(\mu\); in case \(2^\circ\) and \(J=\infty\), the continuous part of the spectrum of \(L_u\) fills the entire real axis.

Received
28 III 1965

REFERENCES

\(^1\) I. M. Rapoport, On some asymptotic methods in the theory of differential equations, Kiev, 1954.
\(^2\) M. A. Naimark, Linear Differential Operators, Moscow, 1954.
\(^3\) F. G. Maksudov, DAN, 153, No. 4 (1963).
\(^4\) I. M. Glazman, UMN, 5, No. 6 (1950).
\(^5\) S. A. Orlov, DAN, 92, No. 3 (1953).