UDC 517.94 : 517.948.35

ASYMPTOTIC METHODS IN THE THEORY OF ONE-DIMENSIONAL SINGULAR DIFFERENTIAL OPERATORS OF ODD ORDER

R. A. SHIRIKYAN

§ 1. Introduction

In the present paper we consider ordinary linear differential equations of self-adjoint type on the half-axis \(x \geqslant 0\). Equations of even order (asymptotics of solutions, qualitative investigations of the spectrum, and deficiency indices of the corresponding differential operators) were studied earlier in the papers [1—7] and by other authors.

In the present article an equation of odd order of self-adjoint type is considered,

\[ ly \equiv \sum_{k=0}^{n}(-1)^k \varepsilon^{2k+1} \left[ -\frac{d^{k+1}}{dx^{k+1}}\left(i p_{k+1}y^{(k)}\right) + \frac{d^k}{dx^k}\left(i p_{k+1}y^{(k+1)}\right) \right]=0,\quad x \geqslant 0, \tag{1.1} \]

where \(\varepsilon>0\). Asymptotic formulas have been obtained for solutions \(y_j\) of equation (1.1) as \(x\to+\infty\),

\[ y_j\sim \exp\left\{\int_{x_0}^{x}\left[\varepsilon^{-1}\lambda_j(t)+\lambda_j^{(1)}(t)\right]dt\right\}. \tag{1.2} \]

Here \(\lambda_j\) are the solutions of the equation

\[ f(\lambda,x)\equiv \sum_{k=0}^{n}(-1)^{n+k+1} \left(2p_{k+1}\lambda^{2k+1}+p_{k+1}'\lambda^{2k}\right)=0, \tag{1.3} \]

which we shall call the characteristic equation for equation (1.1).

\[ \lambda_j^{(1)} = -\frac12\,\frac{d}{dx} \left(\ln\frac{\partial f(\lambda_j,x)}{\partial\lambda}\right) - \left(f_\lambda(\lambda_j,x)\right)^{-1} \sum_{l=0}^{n}(-1)^{n+l}p_{l+1}'\lambda_j^{2l}. \tag{1.4} \]

Formula (1.2) is also asymptotic with respect to \(\varepsilon\), as \(\varepsilon\to+0\). The main conditions on the coefficients of equation (1.1) are the following: \(p_k\) are three-times differentiable complex-valued functions of \(x\) in \((0,\infty)\), satisfying the conditions:

\[ \begin{aligned} &1)\qquad \text{a)}\quad \lim_{x\to\infty} p_{n-k+1}'\,p_{n+1}^{-1}\tau^{-(2k+1)}=0,\\ &\hspace{3.1cm} \text{b)}\quad \lim_{x\to\infty} p_{n-k+1}p_{n+1}^{-1}\tau^{-2k}=c_{2k+1}; \end{aligned} \]

2) the equation

\[ g(\xi)=\sum_{k=0}^{n}(-1)^{k+1}c_{2k+1}\xi^{2n-2k+1}=0 \tag{1.5} \]

has no multiple roots;

3) \(\displaystyle \lim_{x\to\infty} p_{n-k+1}^{\prime\prime}p_{n+1}^{-1}\tau^{-(2k+2)}=0\);

4) the function

\[ F=\sum_{k=0}^{n}\left\{\left(\left|p_{n-k+1}^{\prime}p_{n+1}^{-1}\tau^{-\frac{4k+1}{2}}\right|+ \left|p_{n-k+1}^{\prime\prime}p_{n+1}^{-1}\tau^{-\frac{4k+3}{2}}\right|\right)^{2}+\right. \]

\[ \left.\quad+\left|p_{n-k+1}^{\prime\prime}p_{n+1}^{-1}\tau^{-(2k+1)}\right|+ \left|p_{n-k+1}^{\prime\prime\prime}p_{n+1}^{-1}\tau^{-(2k+2)}\right|\right\} \tag{1.6} \]

is summable on the interval \((0,\infty)\). Here

\[ \tau=(p_1p_{n+1}^{-1})^{\frac{1}{2n}}. \tag{1.7} \]

In what follows we shall call these conditions conditions 1)—4). These conditions are analogous to the conditions in [5—7]. Condition 2) means that the roots of the characteristic equation (except one) have the same order of growth as \(x\to\infty\).

In § 2 equation (1.1) is transformed into a \(z\)-diagonal system. In § 3 the asymptotic formula (1.2) is proved. In § 4 the deficiency indices of the corresponding differential operators on the half-axis \(x\geqslant 0\) are investigated.

An essential difference from equations of even order (see [5—7]) is that a solution \(y_0\) appears, corresponding to the root \(\xi_0=0\) of equation (1.5), whose behavior differs strongly from that of the remaining solutions.

§ 2. Transformation of the Equation

We transform equation (1.1) into a first-order system in the following way. Put

\[ \left. \begin{gathered} \widetilde y_1=y,\quad \varepsilon\,\frac{d\widetilde y_1}{dx}=\widetilde y_2,\ \ldots,\ \varepsilon\,\frac{d\widetilde y_n}{dx}=\widetilde y_{n+1}, \\[4pt] \varepsilon\,\frac{d\widetilde y_{n+1}}{dx} =\frac{1}{2p_{n+1}}\widetilde y_{n+2} -\frac{p_{n+1}^{\prime}}{2p_{n+1}}\widetilde y_{n+1},\quad \varepsilon\,\frac{d\widetilde y_{n+2}}{dx}=p_n^{\prime}\widetilde y_n+ \\[4pt] +2p_n\widetilde y_{n+1}-\widetilde y_{n+3},\ \ldots,\ \varepsilon\,\frac{d\widetilde y_{2n}}{dx} =p_2^{\prime}\widetilde y_2+2p_2\widetilde y_3-\widetilde y_{2n+1}, \\[4pt] \varepsilon\,\frac{d\widetilde y_{2n+1}}{dx} =p_1^{\prime}\widetilde y_1+2p_1\widetilde y_2 . \end{gathered} \right\} \tag{2.1} \]

For \(\varepsilon=1\) we denote \(\widetilde y_{k+1}=y^{[k]}\), which, following I. M. Rapoport [1], we shall call the \(k\)-th quasiderivative of the function \(y\).

Introduce the vector-valued function

\[ Y=(\widetilde y_1,\ \widetilde y_2,\ \ldots,\ \widetilde y_{2n+1}) \tag{2.2} \]

and the matrix

\[ A= \left\| \begin{array}{ccccccccc} 0&1&&&&&&0\\ &\cdot&\cdot&&&&\cdot&\\ &&\cdot&\cdot&&\cdot&&0\\ &&&0&1&\cdot&&&\\ &&&-\dfrac{p_{n+1}}{2p_{n+1}}&\dfrac{1}{2p_{n+1}}&&&&\\ &&&p_n'&2p_n&0&-1&&\\ &&\cdot&\cdot&&\cdot&\cdot&&\\ &\cdot&\cdot&&&&\cdot&\cdot&\\ p_1'&2p_1&&&&&&-1\\ &&&&&&&&0 \end{array} \right\| \]

\[ \left(\frac{1}{2p_{n+1}}\ \text{stands at the intersection of the }(n+2)\text{-nd column and the }(n+1)\text{-st row.}\right. \]
All unindicated elements are equal to zero.) If \(p_{n+1}\ne 0\) on the interval in which equation (1.1) is considered, then system (2.1) and equation (1.1) are equivalent. System (2.1) is written in the form

\[ \varepsilon Y'=AY. \tag{2.3} \]

Introduce diagonal matrices \(\Lambda_0\) and \(\Lambda_1\) with elements

\[ (\Lambda_0)_{ii}=\lambda_i,\qquad (\Lambda_1)_{ii}=\lambda_i^{(1)}, \tag{2.4} \]

where \(\lambda_i\) are the roots of equation (1.3), and \(\lambda_i^{(1)}\) are determined below. Introduce the matrix \(T_0\), which is determined by the condition

\[ T_0^{-1}AT_0=\Lambda_0, \tag{2.5} \]

and the matrix \(T_1\) with elements

\[ (T_1)_{ij}=(\lambda_i-\lambda_j)^{-1}(T_0^{-1}T_0')_{ij},\qquad (T_1)_{ii}=0. \tag{2.6} \]

Denote

\[ -(T_0^{-1}T_0')_{ii}=\lambda_i^{(1)}. \tag{2.7} \]

Then

\[ \Lambda_1=[\Lambda_0,T_1]-T_0^{-1}T_0'. \tag{2.8} \]

Make the transformation (analogously to [7])

\[ Y=T_0(E+\varepsilon T_1)Z. \tag{2.9} \]

It brings system (2.3) to the form

\[ \varepsilon Z'=\bigl(\Lambda_0(x)+\varepsilon\Lambda_1(x)+\varepsilon^2B(x,\varepsilon)\bigr)Z, \tag{2.10} \]

where \(\Lambda_0\) and \(\Lambda_1\) are determined by formulas (2.4) and (2.8), and

\[ \begin{aligned} B(x,\varepsilon)&=(E+\varepsilon T_1)^{-1}\bigl[(T_1^2\Lambda_0-T_1T_0^{-1}T_0')(E+\varepsilon T_1)-{}\\ &\qquad{}-T_1'\bigr]-T_1\Lambda_0T_1-T_0^{-1}T_0'T_1. \end{aligned} \tag{2.11} \]

For the proof of (2.10), see [7].
Let us compute the matrices \(T_0\) and \(T_0^{-1}\).

Lemma 2.1. The matrix \(T_0\) with elements

\[ \left. \begin{aligned} (T_0)_{i,k}&=\lambda_k^{\,i-1},\qquad 1\le i\le n+1,\quad 1\le k\le 2n+1,\\ (T_0)_{n+i,k}&=\lambda_k^{\,n-i+2}\sum_{m=0}^{i-2}(-1)^m\bigl(2p_{n-i+m+3}\lambda_k^{\,2m+1} +p'_{n-i+m+3}\lambda_k^{\,2m}\bigr),\\ &\hspace{45mm}2\le i\le n+1,\quad 1\le k\le 2n+1 \end{aligned} \right\} \tag{2.12} \]

satisfies relation (2.9). The inverse matrix has the form

\[ \left. \begin{aligned} (T_0^{-1})_{k,n+2-j} &=\Bigl\{(-1)^{j-1}\lambda_k^{\,n-j+1} \sum_{m=0}^{j-1}(-1)^m 2p_{n-j+m+2}\lambda_k^{\,2m}\\ &\quad+(-1)^j\lambda_k^{\,n-j+2} \sum_{m=0}^{j-2}(-1)^m p'_{n-j+m+3}\lambda_k^{\,2m}\Bigr\} [f_\lambda(\lambda_k,x)]^{-1},\\ &\hspace{39mm}1\le k\le 2n+1,\quad 1\le j\le n+1,\\ (T_0^{-1})_{k,n+2+j} &=(-1)^j\lambda_k^{\,n-j-1}[f_\lambda(\lambda_k,x)]^{-1},\\ &\hspace{39mm}1\le k\le 2n+1,\quad 0\le j\le n-1 . \end{aligned} \right\} \tag{2.13} \]

Proof. From (2.5) it follows that

\[ AT_0=T_0\Lambda_0. \tag{2.5'} \]

We find the elements of the matrix \(T_0\) from condition (2.5′).

Denote \((T_0)_{ij}=x_{ij}\). We obtain a system of equations with respect to \(x_{ij}\):

\[ \left. \begin{aligned} x_{i,k}&=\lambda_k x_{i-1,k},\qquad 1\le i\le n+1,\quad 1\le k\le 2n+1,\\ -p'_{n+1}x_{n+1,k}+x_{n+2,k}&=2\lambda_k p_{n+1}x_{n+1,k},\\ p'_{n-i}x_{n-i,k}+2p_i x_{n-i+1,k}-x_{n+i+3,k} &=\lambda_k x_{n+i+2,k},\\ &\hspace{31mm}0\le i\le n-2,\quad 1\le k\le 2n+1 . \end{aligned} \right\} \tag{2.14} \]

Putting \(x_{1,k}=1,\ 1\le k\le 2n+1\), and solving system (2.14), we obtain (2.12). The elements of the matrix \(T_0^{-1}\) are determined analogously from the equation
\[ T_0^{-1}A=\Lambda T_0^{-1}. \]

Lemma 2.2. The matrix \(T_0^{-1}T_0'\) has the form

\[ (T_0^{-1}T_0')_{ii} =\frac12\,\frac{d}{dx}\bigl(\ln f_\lambda(\lambda_i,x)\bigr) +(f_\lambda(\lambda_i,x))^{-1} \sum_{l=1}^{n+1}(-1)^{n+l+1}p_l'\lambda_i^{\,2l-2}. \tag{2.15} \]

\[ \begin{aligned} (T_0^{-1}T_0')_{ik} &=\frac{(-1)^{n+1}(f_\lambda(\lambda_i,x))^{-1}}{\lambda_i-\lambda_k} \Bigl[p_1\lambda_k' +2p_1'\lambda_k+p_1''\\ &\quad+\sum_{l=2}^{n+1}(-1)^{l+1}\bigl(2p_l'\lambda_i^{\,l-1}\lambda_k^{\,l} +p_l''\lambda_i^{\,l-1}\lambda_k^{\,l-1}\bigr)\Bigr], \end{aligned} \tag{2.15'} \]

\[ 1\le i,k\le 2n+1. \]

Proof. Denote \((T_0^{-1}T_0')_{i,k}=z_{i,k}\). Then

\[ z_{i,k}=\sum_{p=1}^{2n+1}y_{ip}x'_{pk} =\lambda_k'\bigl(z^{(1)}_{i,k}-z^{(2)}_{i,k}+z^{(3)}_{i,k}+z^{(4)}_{i,k}\bigr), \tag{2.16} \]

\(z_{i,k}^{(1)}\) contains only the functions \(p_k\); \(z_{i,k}^{(2)}\), \(z_{i,k}^{(3)}\)—only \(p_k'\); \(z_{i,k}^{(4)}\)—only the functions \(p_k''\). Let us compute separately \(z_{i,k}^{(m)}\), \(m=1,2,3,4\):

\[ \begin{aligned} z_{i,k}^{(1)}={}&\Bigg\{ \sum_{p=1}^{n+1}(-1)^{n+p+1}(p-1)\lambda_i^{p-1}\lambda_k^{p-2} \sum_{m=0}^{\,n-p+1}2p_{p+m}\lambda_i^{2m} \\ &\quad+\sum_{p=n+2}^{2n+1}(-1)^{p-n}\lambda_i^{2n-p+1} \sum_{l=0}^{\,p-n-2}(-1)^l(2n+2l-p+3)\times \\ &\qquad\qquad\times p_{2n-p+l+3}\lambda_k^{2l+1} \Bigg\}(f_\lambda(\lambda_i,x))^{-1}. \end{aligned} \tag{2.17} \]

\[ \begin{aligned} z_{i,k}^{(2)}={}&\Bigg\{ \sum_{p=1}^{n}(-1)^{n-p}(p-1)\lambda_i^p\lambda_k^{p-2} \sum_{m=0}^{\,n-p}(-1)^m p_{p+m+1}\lambda_i^{2m} \\ &\quad+\sum_{p=n+2}^{2n+1}(-1)^{p-n}\lambda_i^{2n-p+1}\lambda_k^{2n-p+1} \sum_{l=0}^{\,p-n-2}(-1)^l(2n+2l-p+2)\times \\ &\qquad\qquad\times p'_{2n-p+l+3}\lambda_k^{2l} \Bigg\}(f_\lambda(\lambda_i,x))^{-1}. \end{aligned} \tag{2.18} \]

\[ \begin{aligned} z_{i,k}^{(3)}={}&(f_\lambda(\lambda_i,x))^{-1} \sum_{p=n+2}^{2n+1}(-1)^{p-n-1}\lambda_k^{2n-p+1}\lambda_i^{2n-p+1}\times \\ &\qquad\qquad\times \sum_{l=0}^{\,p-n-2}(-1)^l\,2p'_{2n-p+l+3}\lambda_k^{2l+1}, \end{aligned} \tag{2.19} \]

\[ \begin{aligned} z_{i,k}^{(4)}={}&(f_\lambda(\lambda_i,x))^{-1} \sum_{p=n+2}^{2n+1}(-1)^{p-n-1}\lambda_k^{2n-p+1}\lambda_i^{2n-p+1}\times \\ &\qquad\qquad\times \sum_{l=0}^{\,p-n-2}(-1)^l p''_{2n-p+l+3}\lambda_k^{2l}. \end{aligned} \tag{2.20} \]

First compute \(z_{ii}\). The coefficient of \(p_l\) in the sum (2.17) is equal to

\[ \begin{aligned} a_l^{(1)} &=2(-1)^{n+l+1}\lambda_i^{2l-3} \left\{ \sum_{p=2}^{l}(p-1)+ \sum_{p=n-l+3}^{n+1}(2l-n+p-3) \right\} \\ &=2(-1)^{n+l+1}\lambda_i^{2l-3}(2l-1)(l-1). \end{aligned} \tag{2.21} \]

The coefficient of \(p'_{l+1}\) in the sum (2.18) is equal to

\[ \begin{aligned} a_l^{(2)} &=(-1)^{n+l}\lambda_i^{2l-2} \left\{ \sum_{p=2}^{l}(p-1)+ \sum_{p=n-l+2}^{n+1}(2l-n+p+1) \right\} \\ &=(-1)^{n+l}l(2l-1)\lambda_i^{2l-2}. \end{aligned} \tag{2.22} \]

The coefficient of \(p_l'\) in the sum (2.19) is equal to

\[ a_l^{(3)} =2(-1)^{n+l}\lambda_i^{2l-2} \sum_{p=n-l+3}^{n+1}1 =(-1)^{n+l}(2l-2)\lambda_i^{2l-2}. \tag{2.23} \]

The coefficient of \(p_l''\) in the sum (2.20) is equal to

\[ a_l^{(4)}=(-1)^{n+l}\lambda_i^{2l-3} \sum_{p=n-l+3}^{n+1}1 = (-1)^{n+l}(l-1)\lambda_i^{2l-3}. \tag{2.24} \]

From (2.17)—(2.23) we obtain (2.15). Formula (2.15′) is obtained similarly.

§ 3. Asymptotics of the solution of equation (1.1)

1. Some estimates. We shall obtain estimates of the matrices \(T_0^{-1}T_0'\) and \(B(x,\varepsilon)\).

Denote

\[ a(x)=\max_k \left| \frac{p_{n-k+1}'}{p_{n+1}} \right| \left| \frac{p_{n+1}}{p_1} \right|^{\frac{k}{n}}, \tag{3.1} \]

\[ b(x)=\max_k \left| \frac{p_{n-k+1}''}{p_{n+1}} \right| \left| \frac{p_{n+1}}{p_1} \right|^{\frac{2k+1}{2n}}, \]

\[ a_1(x)=\max_k \left| \frac{p_{n-k+1}''}{p_{n+1}} \right| \left| \frac{p_{n+1}}{p_1} \right|^{\frac{k}{n}}, \tag{3.2} \]

\[ b_1(x)=\max_k \left| \frac{p_{n-k+1}''}{p_{n+1}} \right| \left| \frac{p_{n+1}}{p_1} \right|^{\frac{2k+1}{2n}} . \]

Lemma 3.1. Let \(p_k\) be continuous on the interval \((0,\infty)\) complex-valued functions satisfying conditions 1)—3). Then for the matrices \(T_0^{-1}T_0'\) and \(B(x,\varepsilon)\) the following estimates hold:

\[ \|T_0^{-1}T_0'\|\le C\bigl(a(x)+b(x)\bigr), \tag{3.3} \]

\[ \|B(x,\varepsilon)\| \le C\left[\bigl(a(x)+b(x)\bigr)^2+\bigl(a_1(x)+b_1(x)\bigr)\right]|\tau|^{-1}, \tag{3.4} \]

where

\[ \tau=(p_1p_{n+1}^{-1})^{\frac{1}{2n}}. \]

Proof. Choose \(x_0\) such that \(p_{n+1}\ne 0\) for \(x>x_0\), and fix a branch of the function

\[ \tau=(p_1p_{n+1}^{-1})^{\frac{1}{2n}}. \tag{3.5} \]

Put

\[ \lambda(x)=\tau(x)\xi(x) \tag{3.6} \]

in equation (1.3). Then for \(\xi(x)\) we obtain the equation

\[ g(x,\xi)\equiv \sum_{k=0}^{n}(-1)^{k+1} \left(c_{2k+1}(x)\xi^{2n-2k+1} + c_{2k+2}(x)\xi^{2n-2k}\right)=0, \tag{3.7} \]

where

\[ c_{2k+1}(x)=p_{n-k+1}p_1^{-\frac{k}{n}}p_{n+1}^{-1+\frac{k}{n}}, \tag{3.8} \]

\[ c_{2k+2}(x)=p_{n-k+1}'p_1^{-\frac{2k+1}{2n}}p_{n+1}^{-1+\frac{2k+1}{2n}}. \tag{3.9} \]

Equation (3.7) as \(x \to \infty\) coincides with equation (1.5). Consequently, \(\xi_j(x) \to \xi_j\) as \(x \to \infty\), so that, as \(x \to \infty\),

\[ \lambda_j(x)\sim \xi_j \tau(x),\qquad \xi_j\ne 0,\qquad 1\leq j\leq 2n. \]

In particular,

\[ 0<c_1|\tau|\leq |\lambda_j|\leq c_2|\tau|\quad \text{for } x\gg 1,\quad 1\leq j\leq 2n. \tag{3.10} \]

Denote by \(\lambda_0(x)\) the root of the characteristic equation (1.3) corresponding to \(\xi_0=0\). Then, taking into account that

\[ \lambda_0\lambda_1\cdots \lambda_{2n}=p_1', \]

we obtain that, for \(x\gg 1\),

\[ 0<c_1\left|\frac{p_1'}{p_1}\right|\leq |\lambda_0|\leq c_2\left|\frac{p_1'}{p_1}\right|. \tag{3.11} \]

From (3.10) and (3.11) it follows that, for \(x\gg 1\),

\[ 0\leq c_1|\tau|\leq |\lambda_i-\lambda_j|\leq c_2|\tau|,\qquad i\ne j,\quad 0\leq i,j\leq 2n. \tag{3.12} \]

Since

\[ f(\lambda,x)=-2p_{n+1}\prod_{k=0}^{n}(\lambda-\lambda_k), \]

we have

\[ 0<c_1|p_{n+1}|\,|\tau|^{2n}\leq |f_\lambda(\lambda_j,x)|\leq c_2|p_{n+1}|\,|\tau|^{2n},\qquad 0\leq j\leq 2n, \tag{3.13} \]

\[ |f_{\lambda\lambda}(\lambda_j,x)|\leq c\,|p_{n+1}|\,|\tau|^{2n-1},\qquad 0\leq j\leq 2n. \tag{3.14} \]

Let us estimate \(f_x(\lambda_j,x)\), \(f_{\lambda x}(\lambda_j,x)\), and \(f_{xx}(\lambda_j,x)\). We shall carry out the estimate for \(f_x(\lambda_j,x)\) in detail; the others are obtained analogously. All estimates given below hold for \(x\gg 1\).

From (1.3), (3.10), and (3.1) it follows that

\[ |f_x(\lambda_j,x)|\leq \sum_{k=0}^{n}\left(2|p_{k+1}'|\,|\lambda_j|^{2k+1}+|p_{k+1}''|\,|\lambda_j|^{2k}\right)\leq \]

\[ \leq c\,|p_{n+1}|\,|\tau|^{2n+1}\bigl(a(x)+b(x)\bigr). \tag{3.15} \]

It is proved similarly that

\[ |f_{\lambda x}(\lambda_j,x)|\leq c\,|p_{n+1}|\,|\tau|^{2n}\bigl(a(x)+b(x)\bigr), \tag{3.16} \]

\[ |f_{xx}(\lambda_j,x)|\leq c\,|p_{n+1}|\,|\tau|^{2n+1}\bigl(a_1(x)+b_1(x)\bigr). \tag{3.17} \]

Taking into account (3.13)—(3.17) and

\[ \lambda_j'=-\frac{f_x(\lambda_j,x)}{f_\lambda(\lambda_j,x)},\qquad \lambda_j''=-\frac{f_{xx}f_\lambda^2-2f_x f_\lambda f_{\lambda x}+f_x^2 f_{\lambda\lambda}}{f_\lambda^3}, \]

we obtain

\[ |\lambda_j'|\leq c\bigl(a(x)+b(x)\bigr), \tag{3.18} \]

\[ |\lambda_j''|\leq c\left[(a(x)+b(x))^2+a_1(x)+b_1(x)\right]. \tag{3.19} \]

From (3.1) and (3.2) the following estimates are easily obtained:

\[ |p_1'|\leq c\,|p_{n+1}|\,a(x)\,|\tau|^{2n},\qquad |p_1'|\leq c\,|p_{n+1}|\,b(x)\,|\tau|^{2n+1}, \]

\[ |p_1''|\leq c\,|p_{n+1}|\,a_1(x)\,|\tau|^{2n},\qquad |p_1''|\leq c\,|p_{n+1}|\,b(x)\,|\tau|^{2n+1}. \]

From formulas (2.6), (2.15), (2.15′) and from the estimates (3.13), (3.18), (3.19), taking the latter estimates into account, in the same way as above, we obtain

\[ \|T_0^{-1}T_0'\|\leq c\bigl(a(x)+b(x)\bigr), \]

\[ \|T_1\|\leq c\bigl(a(x)+b(x)\bigr)|\tau|^{-1}, \tag{3.20} \]

\[ \|T_1'\|\leq c\bigl[(a(x)+b(x))+a_1(x)+b_1(x)\bigr]|\tau|^{-1}. \tag{3.21} \]

From estimate (3.21) and from conditions 1, a) and 3) it follows that \(\lim_{x\to\infty}\|T_1\|=0\). Consequently, there exists \(x(\varepsilon_0)\) (for fixed \(\varepsilon_0\)) such that, for \(\varepsilon\geq \varepsilon_0,\ x\geq x(\varepsilon_0)\),

\[ \|(E+\varepsilon T_1)^{-1}\|\leq \frac12 . \]

Taking this into account, from formulas (2.11) we obtain that, for \(\varepsilon\geq \varepsilon_0,\ x\geq x(\varepsilon_0)\),

\[ \|B(x,\varepsilon)\|\leq c\bigl(\|T_1\|^2|\Lambda_0|+\|T_1\|\|T_0^{-1}T_0'\|+\|T_1'\|\bigr)\leq \]

\[ \leq c\bigl[(a(x)+b(x))^2+a_1(x)+b_1(x)\bigr]. \]

Lemma (3.1) is proved.

2. Asymptotics of solutions of equation (1.1). We shall obtain asymptotic formulas for solutions of equation (1.1).

Denote

\[ y_{j0}(x_0,x,\varepsilon)= \exp\left\{\int_{x_0}^{x}\bigl[\varepsilon^{-1}\lambda_j(t)+\lambda_j^{(1)}(t)\bigr]\,dt\right\}, \quad 0\leq j\leq 2n. \tag{3.22} \]

Here \(\lambda_j\) are the roots of the characteristic equation (1.3), while \(\lambda_j^{(1)}\) is determined by formula (1.4).

Theorem 3.1. Let \(p_k\) be three times continuously differentiable complex-valued functions for \(x\geq 0\), and let \(p_{n+1}\neq 0\) for \(x\geq 0\). Suppose that conditions 1)—4) are satisfied and that

\[ f_{ij}(x)=\operatorname{Re}\left[(\xi_i-\xi_j)(p_1p_{n+1}^{-1})^{\frac{1}{2n}}\right]\neq 0 \tag{3.23} \]

\[ \int^{\infty}\left|p_1p_{n+1}^{-1}\right|^{\frac{1}{2n}}\,dx=\infty. \tag{3.24} \]

Then for every \(\varepsilon_0\) there exists \(x(\varepsilon_0)\) such that, for \(\varepsilon\geq \varepsilon_0,\ x\geq x(\varepsilon_0)\), equation (1.1) has \(2n+1\) linearly independent solutions \(y_j(x_0,x,\varepsilon)\), for which, when \(0<\varepsilon_0\leq 1,\ x\geq x(\varepsilon_0)\), and \(x(\varepsilon_0)\gg 1\), the following asymptotic formulas hold:

\[ y_j^{(k)}= \varepsilon^{-k}\lambda_j^k y_{j0}(x_0,x,\varepsilon) \bigl(1+\varepsilon\psi_{jk}(x,\varepsilon)\bigr), \quad 0\leq k\leq n, \tag{3.25} \]

\[ y_j^{\{k,\varepsilon\}}= (-1)^{k-n}\lambda_j^{\,k-1}\,2p_{n+1}\,y_{j0}(x_0,x,\varepsilon)\times \]

\[ \times \sum_{m=0}^{k-n-2}(-1)^m \bigl(c_{2m+1}\xi_j^{-2m}+o(1)\bigr) \bigl(1+\varepsilon\psi_{jk}(x,\varepsilon)\bigr), \tag{3.26} \]

\[ n+1\leq k\leq 2n+1,\qquad 0\leq j\leq 2n, \]

where

\[ |\psi_{jk}(x,\varepsilon)|\leq \psi(x),\qquad \lim_{x\to\infty}\psi(x)=0,\qquad 0<\varepsilon\leq 1. \tag{3.27} \]

Proof. By means of the transformation (2.1) we reduce equation (1.1) to the form (2.3), where the matrix \(A\) is determined by the formu-

(2.2). Then, with the aid of the transformation (2.9), we reduce equation (2.3) to the form (2.10); \(\Lambda_0\), \(\Lambda_1\), and \(B(x,\varepsilon)\) are determined, respectively, by formulas (2.5), (2.8), and (2.11). Since for \(B(x,\varepsilon)\) the estimate (3.4) holds, the further proof of the theorem is analogous to the proof of Theorem (1.3.1) from [7].

Corollary 1. Let all the conditions of Theorem 3.1 be satisfied, with the exception of condition (3.23), which is satisfied for some \(j\) and for all \(i \ne j\). Then there exist solutions \(y_j\) for which the asymptotic formulas (3.25) and (3.26) hold.

Corollary 2. Let conditions 1)—4) and (3.24) be satisfied, and let \(\varepsilon=1\). Then the asymptotic formulas (3.25), (3.26) hold for fixed \(i\), if for some \(i \ne j\) condition (3.23) is satisfied, and for the remaining \(j\) one of the conditions

\[ \text{a)}\quad \left|\int_{x_0}^{x} \operatorname{Re}(\lambda_i-\lambda_j)\,dt\right|\le \tilde c, \]

\[ \text{b)}\quad \int_{0}^{\infty}\operatorname{Re}(\lambda_i-\lambda_j)\,dt=\infty . \]

holds.

For the proof see [7] (Corollary 2, Theorem (1.6.1)).

§ 4. The case when \(p_{n+1}\to 1,\quad |p_1|\to\infty\) as \(x\to\infty\).

Symmetric differential operators. We investigate the asymptotics of the solutions of equation (1.1) when

\[ \lim_{x\to\infty} p_{n+1}(x)=p\;(\ne 0,\ \infty),\qquad \lim_{x\to\infty}|p_1(x)|=\infty . \tag{4.1} \]

For brevity we put \(\varepsilon=1\) (all conclusions are valid for any positive \(0<\varepsilon\le 1\)). In addition, we normalize \(p_{n+1}\) by the condition

\[ \lim_{x\to\infty} p_{n+1}(x)=1. \tag{4.1′} \]

Theorem 4.1. Let conditions (4.1), (4.1′) and the conditions of Theorem (3.1) be satisfied. Let

\[ \lim_{x\to\infty}\arg \left[p_1p_{n+1}^{-1}\right]^{\frac{1}{2n}} =\varphi_0,\qquad \operatorname{Re}(\xi_j e^{i\varphi_0})\ne 0, \tag{4.2} \]

\[ 1\le j\le 2n. \]

Number the \(\xi_j\) in such a way that
\[ \operatorname{Re}(\xi_j e^{i\varphi_0})< \operatorname{Re}(\xi_{j+1} e^{i\varphi_0}),\qquad 1\le j\le 2n. \]
Then the solutions \(y_1,y_2,\ldots,y_n\in L_2=L_2(0,\infty)\), while the solutions \(y_0,y_{n+1},\ldots,y_{2n}\) and no nontrivial linear combination of them belong to \(L_2\).

To the solution \(y_0\) corresponds the root
\[ \lambda_0=\frac{p_1'}{p_1}(1+o(1)). \]
In what follows we shall call this solution the “exceptional solution.”

Proof. From Theorem 3.1 it follows that equation (1.1) has \(2n+1\) linearly independent solutions for which, as \(x\to\infty\), the asymptotic formulas

\[ y_j(x)=\exp\left\{\int_{x_0}^{x}\left(\lambda_j(t)+\lambda_j^{(1)}(t)\right)\,dt\right\}(1+o(1)),\qquad 0\le j\le 2n. \tag{4.3} \]

From (3.13) and (1.4) it follows that for \(x \gg 1\)

\[ f_\lambda(\lambda_j,x)=A_j |p_1|(1+o(1)), \qquad 0\leq j\leq 2n, \tag{4.4} \]

where

\[ A_j=2e^{2ni\varphi_0}g'(\xi_j)\ne 0, \]

\[ \lambda_j^{(1)}=-\frac{2n+1}{4n}\frac{p_1'}{p_1}(1+o(1)), \qquad j\ne 0, \tag{4.5} \]

\[ \lambda_0^{(1)}=-\frac{1}{2}\frac{p_1'}{p_1}(1+o(1)). \]

Denote

\[ f(x_0,x)=\int_{x_0}^{x}|p_1|^{\frac{1}{2n}}\,dx. \tag{4.6} \]

From condition 1, a) it follows that, as \(x\to\infty\),

\[ \ln |p_1|=o\bigl(f(x_0,x)\bigr). \tag{4.7} \]

Let \(1\leq j\leq n\). Then from formulas (4.3)—(4.7) and from the fact that, as \(x\to\infty\),

\[ \lambda_j=\xi_j e^{i\varphi_0}|p_1|^{\frac{1}{2n}}(1+o(1)), \qquad 1\leq j\leq 2n, \]

it follows that for every \(\delta\) \((0<\delta<1)\) there exists \(x_0>0\) such that for \(x>x_0\)

\[ |y_j|\leq c\exp\left[\operatorname{Re}(\xi_j e^{i\varphi_0})(1-\delta)f(x_0,x)\right]. \tag{4.8} \]

Since \(|p_1|\to\infty\) as \(x\to\infty\), it follows from (4.8) that \(y_j\in L_2\), \(1\leq j\leq n\). It is proved in the same way that \(y_j\in L_2\) for \(n+1\leq j\leq 2n\). Let us examine the exceptional solution. From (3.11), (4.3), and (4.5) we obtain that for \(x\gg 1\)

\[ y_0=(p_1)^{-\frac{1}{2}+o(1)}, \tag{4.9} \]

and since \(|p_1|\to\infty\) as \(x\to\infty\), \(y_0\in L_2\). It is not hard to prove that for \(n+1\leq j\leq 2n-1\), \(y_j=o(y_{j+1})\), and for \(n+1\leq j\leq 2n\), \(y_0=o(y_j)\), and hence no nontrivial combination of the form \(y=c_0y_0+\sum c_jy_j\in L_2\).

Theorem 4.1 is proved.

Remark. Consider the equation

\[ ly=\mu y. \tag{4.10} \]

Suppose that the conditions of Theorem 3.1 are satisfied. Then equation (4.10) has \(2n+1\) linearly independent solutions, and moreover:

1) If \(|p_1'|\to\infty\) as \(x\to\infty\), then for all complex \(\mu\) the maximal number of linearly independent solutions in \(L_2\) is \(n\).

2) If \(p_1'\to c\) \((0\leq c<\infty)\) as \(x\to\infty\), then the maximal number of linearly independent solutions in \(L_2\) depends on \(\mu\) and is equal to \(n\) or \(n+1\). In this case the exceptional solution may or may not belong to \(L_2\), depending on \(\mu\).

Consider the example: \(p_1=x^\alpha\), \(\alpha\leq 1\). For equation (4.10)

\[ \lambda_0=(\alpha x^{\alpha-1}-\mu)x^{-\alpha}(1+o(1)), \qquad \lambda_0^{(1)}=-\frac{1}{2}(\alpha x^{\alpha-1}-\mu)x^{-\alpha}(1+o(1)). \]

Substituting these values into (4.3), for \(\alpha<1\) we obtain

\[ y_0=x^{-\frac{\alpha}{2}+o(1)} \exp\left\{-\frac{\mu x^{1-\alpha}}{2(1-\alpha)}(1+o(1))\right\}, \qquad x\to\infty . \tag{*} \]

For \(\alpha=1\) we obtain

\[ y_0=x^{\frac12-\frac{\mu}{2}+o(1)},\qquad x\to\infty . \tag{**} \]

From \((*)\) and \((**)\) it is easy to see that, for \(\alpha\leq 1\), \(y_0\) may or may not belong to \(L_2\), depending on \(\mu\).

The case of real \(p_k\). Consider equation (4.14). Denote by \(\lambda_j(x,\mu)\) the solutions of the equation

\[ f(\lambda,x)=(-1)^{n+1}\mu, \tag{4.11} \]

and by \(\xi_j(x)\) and \(\xi_j(x,\mu)\) the solutions, respectively, of equation (3.7) and

\[ g(x,\xi,\mu)= \sum_{k=0}^{n}(-1)^{k+1} \left(c_{2k+1}(x,\mu)\xi^{2n-2k+1} +c_{2k+2}(x,\mu)\xi^{2n-2k}\right)=0 \tag{4.12} \]

where

\[ c_k(x,\mu)=c_k(x),\qquad 1\leq k\leq 2n+1, \]

\[ c_{2n+2}(x,\mu)=\frac12(p_1'-\mu)p_1^{-\frac{2n+1}{2n}}p_{n+1}^{\frac{1}{2n}}, \tag{4.13} \]

and \(c_{2k+1}(x)\) and \(c_{2k+2}(x)\) are determined by formulas (3.8) and (3.9).

Obviously, \(\xi_j(x)\), \(\xi_j(x,\mu)\to \xi_j\) as \(x\to\infty\). Moreover, \(\lambda_j(x)\) is determined by formula (3.6), and

\[ \lambda_j(x,\mu)=\left(p_1p_{n+1}^{-1}\right)^{\frac{1}{2n}}\xi_j(x,\mu). \tag{4.14} \]

Let us study the behavior of the functions \(\lambda_j(x,\mu)\) as \(x\to\infty\). Recall that the \(p_k\) are real.

Lemma 4.1. Suppose that conditions 1), 2), and (4.1), (4.1′) are fulfilled. Then for any fixed \(\mu\) and as \(x\to\infty\),

\[ \lambda_j(x,\mu)=p_1^{\frac{1}{2n}} \left( \xi_j(x)+ \frac{(-1)^{n+1}\mu} {p^{1+\frac{1}{2n}}\,g_\xi'(x,\xi_j(x))} +O(p_1^{-2}) \right)(1+o(1)), \tag{4.15} \]

where \(o(1)\) is real and does not depend on \(\mu\).

This lemma is proved analogously to Lemma (1.6.3) in [7].

In the subsequent Theorems 4.2 and 4.3 we study the behavior only of the solutions \(y_1,y_2,\ldots,y_{2n}\). For the exceptional solution see the remark to Theorem 4.1.

Theorem 4.2. Suppose that all \(p_k\) are real and

\[ \lim_{x\to\infty}p_{n+1}=1,\qquad \lim_{x\to\infty}p_1=+\infty . \tag{4.16} \]

Suppose that conditions 1)—4) are fulfilled, and if \(\operatorname{Re}\xi_i=\operatorname{Re}\xi_j\), then either \(i=j\), or \(\xi_i=\overline{\xi_j}\) and \(\operatorname{Im}\xi_j\ne0\). Suppose that

\[ \operatorname{Im} g'(\xi_j)\ne0 . \tag{4.17} \]

in the latter case. Then equation (4.10), for any \(\mu\), has \(2n+1\) linearly independent solutions \(y_j\), for which the asymptotic formulas (3.25), (3.26) hold. Moreover \(y_1,\ldots,y_n\in L_2\), while the solutions \(y_{n+1},\ldots,y_{2n}\) and no nontrivial linear combination of them belong to \(L_2\).

Proof. If \(\xi_j\) is real, then by condition 2) \(\operatorname{Re}(\xi_i-\xi_j)\ne 0\) for \(i\ne j\), and the proof of the theorem follows from Theorem 4.1. Let \(\xi_j\) be complex and \(\mu\) purely imaginary. Then equation (1.5) has no purely imaginary roots, and if, for some \(j\), \(\xi_{j+1}=\overline{\xi}_j\), then \(\xi_{j+1}(x)=\overline{\xi}_j(x)\), \(\xi_{j+1}(x,\mu)=\overline{\xi}_j(x,\mu)\), and \(\operatorname{Re}\lambda_{j+1}(x,\mu)=\operatorname{Re}\lambda_j(x,\mu)\) for \(x\gg1\). Thus condition a) of Corollary 2 of Theorem 3.1 is satisfied. This proves the existence of a solution \(y_j\) of the form (3.25), (3.26) of equation (4.10) for purely imaginary \(\mu\).

Let \(\operatorname{Re}\mu\ne0\). Since \(g(x,\overline{\xi})=\overline{g(x,\xi)}\), from (4.15) we obtain that, for \(x\gg1\),

\[ \operatorname{Re}\bigl(\lambda_{j+1}(x,\mu)-\lambda_j(x,\mu)\bigr) = \operatorname{Re}\mu\,\alpha_j p_1^{-1}(1+o(1)), \tag{4.18} \]

where, by (4.17), \(\alpha_j\ne0\). Denote

\[ J=\int^\infty p_1^{-1}\,dx . \tag{4.19} \]

If \(J=\infty\), then for \(i=j+1\) condition b) of Corollary 2 of Theorem 3.1 is fulfilled, while if \(J<\infty\), condition d) is fulfilled. Consequently, equation (4.10), for any \(\mu\), has \(2n+1\) linearly independent solutions for which the asymptotic formulas (3.25), (3.26) hold. It can be shown that \(y_1,y_2,\ldots,y_n\in L_2\), while \(y_{n+1},\ldots,y_{2n}\notin L_2\), and

\[ \sum_{j=n+1}^{2n} c_j y_j \notin L_2 \]

if not all \(c_j=0\). (The proof is analogous to the proof of Theorem (1.6.3) in [7].)

Theorem (4.2) is proved.

Corollary 1. Let \(\lim_{x\to\infty}p_1(x)=-\infty\). Put \(\xi_j^*=e^{\frac{i\pi}{2n}}\xi_j\). If all the conditions of Theorem 4.2 are satisfied for the numbers \(\xi_j^*\), then the theorem remains valid for this case.

Introduce the operator \(l_\Omega\), whose domain of definition \(D_\Omega\) is the set of functions \(y(x)\) such that \(y\in L_2\), \(ly\in L_2\), and

\[ \sum_{j=0}^{2n+1}\omega_{ij}y^{(j)}(0)=0,\qquad i=1,2,\ldots,n . \]

Here \(\omega_{ij}\) are complex numbers such that the rank of the matrix \(\Omega=(\omega_{ij})\) is equal to \(n\). If \(y\in D_\Omega\), then

\[ l_\Omega y=ly . \]

Corollary 2. Let all the conditions of Theorem 3.3 be satisfied and \(p_1'\to\infty\). Then the spectrum of the operator \(l_\Omega\) is purely discrete and has no finite limit points. If \(\mu\) is not an eigenvalue of the operator \(l_\Omega\), then the resolvent \(R_\mu\) of the operator \(l_\Omega\) is an integral operator with kernel \(G(x,\xi,\mu)\), satisfying the conditions:

\[ \int_0^\infty |G|^2\,dx<\infty,\qquad \int_0^\infty |G|^2\,d\xi<\infty . \tag{4.20} \]

The proof follows from Lemma (1.41) of [7].

Let \(l(y)\) be the differential expression considered on the interval \((0,\infty)\) and defined from (1.1). Denote by \(D\) the totality of all functions \(y(x)\in L_2\), all of whose quasiderivatives up to order \(2n\) inclusive are absolutely continuous, while the quasiderivative \(y^{\{2n+1\}}\) belongs to \(L_2\). Define the operator \(L\) in \(L_2\) as follows. The domain of the operator \(L\) is \(D\), and for \(y\in D\)
\[ Ly=l(y). \]

Denote by \(D'_0\) the totality of all finite functions and by \(L'_0\) the restriction of the operator \(L\) to \(D'_0\), i.e., \(L'_0\) is the operator whose domain is \(D'_0\), and for \(y\in D'_0\)
\[ L'_0y=Ly=l(y), \]
\(L'_0\) is a symmetric operator. The closure of the operator \(L'_0\) will be denoted by \(L_0\).

Corollary 3. If the conditions of Corollary 2 are satisfied, then the defect index of the operator \(L_0\) is equal to \((n,n)\).

Let \(\xi_j\) be a purely imaginary root of equation (1.5). Then
\[ g'(\xi_j)=(-1)^{n+1}b_j,\qquad \operatorname{Im} b_j=0. \tag{4.21} \]
Denote
\[ B_j=\left(\frac{1}{b_j}+\frac{1}{4n}\right)\operatorname{Im}\mu, \tag{4.22} \]
\[ J_1=\int^\infty p_1^{-1-\frac{1}{2n}} \exp\left\{c\int_{x_0}^x p_1^{-1}\,dt\right\}\,dx, \tag{4.23} \]
\[ f(x_0,x)=\int_{x_0}^x p_1^{-1}\,dt. \tag{4.24} \]

Consider the linear subspace \(M\) of solutions of equation (4.10) with basis \(y_1,\ldots,y_{2n}\); denote by \(m\) the dimension of the subspace \(M\cap L_2\). The following holds.

Theorem 4.3. Let all \(p_k\) be real and satisfy conditions 1)—4), (4.16), and, for \(1\le i,j\le 2k\),
\[ \operatorname{Re}\xi_j=0,\qquad g'(\xi_j)\ne g'(\xi_i),\qquad i\ne j. \]
Let the remaining \(\operatorname{Re}\xi_j\ne0\), and from the equality \(\operatorname{Re}\xi_i=\operatorname{Re}\xi_j\) let it follow either that \(i=j\), or that \(\xi_i=\xi_j\), and in the latter case condition (4.17) is satisfied. Let \(J_1=\infty\) if \(J=\infty\) and \(c>0\). Then equation (4.10), for \(\operatorname{Im}\mu\ne0\), has \(2n+1\) linearly independent solutions \(y_j\) for which the asymptotic formulas (3.25), (3.26) hold, and moreover:

1) \(n-k\le m\le n+k\), if \(J=\infty\) \((y_j\in L_2\), if \(B_j<0\), where \(1\le j\le 2k)\);

2) \(m=n+k\), if \(J<\infty\), where \(J\), \(B\), and \(J_1\) are determined by formulas (4.19), (4.22), and (4.23), respectively.

Proof. We have \(g(\xi)=\xi g_1(\xi)\), where \(g_1(\xi)\) is an even function of \(\xi\); consequently, the \(\xi_j\) can be numbered so that \(\operatorname{Re}\xi_j=0\), \(1\le j\le 2k\), \(\operatorname{Re}\xi_j<0\), \(2k+1\le j\le n+k\), \(\operatorname{Re}\xi_j>0\), \(n+k+1\le j\le 2n\). From the proofs of Theorems 4.1 and 4.2 it follows that for arbitrary \(\mu\) there exist solutions \(y_j\), \(2k+1\le j\le 2n\), having the form (3.25), (3.26), and moreover \(y_{2k+1},\ldots,y_{n+k}\in L_2\), while
\[ \sum_{n+k+1}^{2n} c_j y_j\in L_2, \]
if not all \(c_j\) are zero. We investigate the solutions \(y_1,\ldots,y_{2k}\) corresponding to purely imaginary \(\xi_j\).

Two cases are possible:

I) \(J=\infty\), if \(1\leq j\leq 2k\), \(\xi_j=-i\eta_j\), where \(\operatorname{Im}\eta_j=0\) and (4.24) is satisfied. Then it is easy to prove that for \(x\gg 1\)

\[ \operatorname{Re}\xi_j(x)=0,\quad 1\leq j\leq 2k . \tag{4.25} \]

From (4.15), (4.21), and (4.25) it follows that, for \(x\gg 1\),

\[ \operatorname{Re}\lambda_j(x,\mu)=\operatorname{Im}\mu\, b_j^{-1}p_1^{-1}(1+o(1)). \tag{4.26} \]

Therefore, as \(x\to\infty\), \(1\leq i,j\leq 2k\), \(i\ne j\),

\[ \operatorname{Re}\bigl(\lambda_i(x,\mu)-\lambda_j(x,\mu)\bigr) =(b_i^{-1}-b_j^{-1})\operatorname{Im}\mu\, p_1^{-1}(1+o(1)). \tag{4.27} \]

Since \(J=\infty\) and \(b_i\ne b_j\) for \(i\ne j\), condition b) of Corollary 2 of Theorem 3.1 is satisfied for fixed \(j\), \(1\leq j\leq 2k\), and for all \(i\ne j\), \(1\leq i,j\leq 2k\). For the remaining \(i\), condition b) is also satisfied, since from (4.15) it follows that, as \(x\to\infty\),

\[ |\operatorname{Re}\lambda_i(x,\mu)|\leq c\,p_1^{\frac{1}{2n}}|\operatorname{Re}\xi_i|, \]

and for these \(i\), \(\operatorname{Re}\xi_i\ne 0\). Consequently, there exist solutions \(y_1,y_2,\ldots,y_{2k}\) for which the asymptotic formulas (3.25), (3.26) are valid. It remains to determine which of these solutions belong to \(L_2\).

Taking into account (4.4), (4.5), (4.26), and replacing \(p_1'\) by \(p_1'-\mu\), from (4.3) we obtain that, as \(x\to\infty\), \(1\leq j\leq 2k\),

\[ c_1p_1^{-\frac12-\frac{1}{4n}}\exp\{(B_j-\delta)f(x_0,x)\} <|y_j|< \]

\[ <c_2p_1^{-\frac12-\frac{1}{4n}}\exp\{(B_j+\delta)f(x_0,x)\}, \tag{4.28} \]

where \(\delta>0\) will be chosen so small that the signs of \(B_j+\delta\), \(B_j-\delta\) coincide with the signs of \(B_j\). Next, for \(1\leq j\leq 2k\), from the right-hand side of inequality (4.28) we obtain that

\[ \int_{x_0}^{x}|y_j|^2\,dt <c\exp\{2(B_j+\delta)f(x_0,t)\}\bigg|_{t=x_0}^{t=x}. \]

If for some \(j\), \(B_j<0\), then \(y_j\in L_2\) (since \(J=\infty\)). From the left-hand side of inequality (4.28) we obtain that

\[ \int_{x_0}^{x}|y_j|^2\,dt >c\int_{x_0}^{x}p_1^{-1-\frac{1}{2n}} \exp\{2(B_j-\delta)f(x_0,t)\}\,dt. \]

If for some \(j\), \(B_j>0\), then \(J_1=\infty\), and consequently \(y_j\notin L_2\). Thus we have proved that if \(J=\infty\), then \(n-k\leq m\leq n+k\).

II) \(J<\infty\). From the right-hand side of inequality (4.28) we obtain that for \(x\gg 1\)

\[ |y_j|\leq c\,p_1^{-\frac12-\frac{1}{4n}} . \]

Hence we obtain that

\[ \int^{\infty}|y_j|^2\,dx \leq c\int^{\infty}p_1^{-1-\frac{1}{2n}}\,dx \leq c\int^{\infty}p_1^{-1}\,dx<\infty, \]

i.e. \(y_j \in L_2,\ 1 \le j \le 2k\). Thus, if \(J < \infty\), then \(y_1, y_2, \ldots, y_{n+k} \in L_2\), and

\[ \sum_{n+k+1}^{2n} c_j y_j \in L_2, \]

if not all \(c_j\) are zero. Consequently, \(m = n + k\). Theorem 4.3 is proved.

Remark. The condition \(J_1 = \infty\), if \(J = \infty\) and \(c > 0\), is satisfied, for example, when \(p_1 = x^\alpha,\ 0 < \alpha < 1\).

Corollary. If all the conditions of Theorem (4.3) are satisfied, then the defect index of the operator \(L_0\) is equal to \((m, m)\), where \(n - k - 1 \le m \le n + k + 1\).

The main results were reported at the International Congress of Mathematicians in Moscow.

I take this opportunity to express my deep gratitude to M. V. Fedoryuk for posing the problem and for his constant attention to the work.

References

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3. Maksudov F. G. Dokl. Akad. Nauk SSSR, 133, No. 4, 758—761, 1963.
4. Coddington E. A., Levinson N. Theory of Ordinary Differential Equations. Moscow, IL, 1958.
5. Fedoryuk M. V. Dokl. Akad. Nauk SSSR, 165, No. 4, 777—769, 1965.
6. Fedoryuk M. V. Trudy Moskovskogo Matematicheskogo Obshchestva, 15, 296—345, 1966.
7. Fedoryuk M. V. Doctoral dissertation. IPM, Academy of Sciences of the USSR, 1966.
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Received by the editors
30 November 1966

Moscow Institute of Physics and Technology