UDC 517.941.92

MATHEMATICS

M. V. FEDORYUK

ASYMPTOTICS OF EIGENVALUES AND EIGENFUNCTIONS OF ONE-DIMENSIONAL SINGULAR DIFFERENTIAL OPERATORS

(Presented by Academician I. M. Vinogradov on 29 X 1965)

I. Consider the system

\[ y'=\mu A(x)y, \tag{1} \]

where \(A(x)\) is a square matrix of order \(n\), \(y\) is a vector with \(n\) components, and \(\mu\) is a parameter. Everywhere in what follows it is assumed that, for \(x\ge 0\), the matrix \(A''(x)\) is continuous (the prime denotes differentiation with respect to \(x\); the matrix \(A(x)\) is complex-valued). In this part of the article we study the asymptotics of solutions of system (1) as \(x\to +\infty\).

We describe the form of the matrix \(A(x)\).

\(1^\circ.\) \(A(x)=q(x)Q(x)B(x)Q^{-1}(x)\), where \(q(x)\ne 0\) for \(x\ge 0\) and \(Q\) is a diagonal matrix; \((Q(x))_{ii}=q^{\alpha_i}(x)\); \(\alpha_i\) are complex numbers.

\(2^\circ.\) The matrix \(B(+\infty)\) exists, is finite, nondegenerate, and has distinct eigenvalues \(\eta_j\).

\(3^\circ.\)

\[ \lim_{x\to+\infty}\left(\left|q'(x)\right|\left|\bar q(x)\right|^{-2} +\left|\bar q(x)\right|^{-1}\|B'(x)\|\right)=0, \]

where

\[ \|B(x)\|=\max_{i,j}|b_{ij}(x)|. \]

\(4^\circ.\) The function

\[ F(x)=|q''q^{-3}|+\left|q''q^{-2}+\left(|q|^{-1}+|q'q^{-2}|\right)\|B'\|+|q|^{-1}\left(\|B'\|^2+\|B''\|\right)\right| \]

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

Denote by \(\lambda_j(x)\) the eigenvalues of the matrix \(A(x)\), by \(e_j(x), e_j^*(x)\) the right and left eigenvectors of this matrix corresponding to the eigenvalue \(\lambda_j(x)\), and by \(T(x)\) the matrix whose \(j\)-th column is \(e_j(x)\), \(1\le j\le n\). Put

\[ \lambda_j^{(1)}(x)=-(T^{-1}(x)T'(x))_{jj} \tag{2} \]

and introduce the diagonal matrices \(\Lambda(x),\Lambda_1(x)\):

\[ (\Lambda)_{jj}=\lambda_j(x),\qquad (\Lambda_1)_{jj}=\lambda_j^{(1)}(x). \tag{3} \]

All these notations, by virtue of \(2^\circ\), have meaning for sufficiently large \(x\).

Theorem 1. Suppose that conditions \(1^\circ\)—\(4^\circ\) are satisfied,

\[ \int_0^\infty |q(x)|\,dx=\infty, \tag{4} \]

and, for sufficiently large \(x\), \(i\ne j\),

\[ \left|\operatorname{Re}(\lambda_i(x)-\lambda_j(x))\right|\ge C|q(x)|,\qquad C>0. \tag{5} \]

Then for every \(\mu_0>0\) there exists \(x(\mu_0)<\infty\) such that, for \(x\ge x(\mu_0)\), \(\mu\ge\mu_0\), there exists a fundamental matrix \(Y(x,\mu)\) of system (1) having the form

\[ Y=T(x)(E+\mu^{-1}U(x,\mu))\exp\left(\int_{x^0}^{x}\bigl(\mu\Lambda(t)+\Lambda_1(t)\bigr)\,dt\right). \tag{6} \]

\[ \|U(x,\mu)\|\le u(x),\qquad \lim_{x\to+\infty}u(x)=0. \tag{7} \]

Let

\[ \lim_{x\to+\infty}|q(x)|=+\infty,\qquad \lim_{x\to+\infty}\arg q(x)=\varphi_0 . \tag{8} \]

Denote

\[ \eta_j'=\eta_j e^{i\varphi_0} \tag{9} \]

where \(\eta_j\) are the eigenvalues of the matrix \(B(+\infty)\), and \(m^-\), \(m^+\) are the numbers of points \(\eta_j'\) lying respectively in the left and right half-planes, and introduce the operator

\[ l=d/dx-A(x). \]

From Theorem 1 it follows:

Theorem 2. Suppose that the conditions of Theorem 1 and condition (8) are satisfied and

\[ \operatorname{Re}\eta_j'\ne 0,\quad 1\le j\le n;\qquad \operatorname{Re}(\eta_i'-\eta_j')\ne 0,\quad i\ne j. \tag{10} \]

Then, for any complex \(\mu\), the maximal number of linearly independent solutions of the system

\[ ly=\mu y, \tag{11} \]

belonging to \(L_2[0,+\infty)\), is equal to \(m^-\).

Denote by \(D\) the collection of all vectors \(y\in L_2[0,+\infty)\), all components of which are absolutely continuous on every finite interval \([0,a]\), \(a>0\), and such that \(ly\in L_2[0,+\infty)\). By \(D_\Omega\) we denote the collection of all vectors \(y\in D\) such that

\[ \Omega y(0)=0, \tag{12} \]

where \(\Omega\) is a constant matrix of rank \(m^-\). By \(L_\Omega\) we denote the operator in \(L_2[0,+\infty)\) with domain of definition \(D_\Omega\), and \(L_\Omega y=ly\) for \(y\in D_\Omega\).

Theorem 3. If the conditions of Theorem 2 are satisfied, then the spectrum of the operator \(L_\Omega\) is purely discrete and has no finite limit points.

Let us discuss the conditions of Theorem 1. If one requires that the matrix \(A(x)\) be nondegenerate in a certain sense (see, for example, (1)) and that the \(\lambda_j(x)\) have the same order of growth as \(x\to+\infty\), then \(A(x)\) has the form \(1^\circ, 2^\circ\). In particular, for the scalar equation \(y^{(n)}+\mu^n p(x)y=0\) these conditions are automatically satisfied. If the characteristic roots of the scalar equation

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

have the same order of growth as \(x\to+\infty\), then the matrix \(A(x)\) has the form \(1^\circ\), and \(B(+\infty)\) exists and is finite; moreover \(a_j=j-1\). Therefore the main results of \((^3)\) are contained in Theorem 1. Theorem 1 also remains valid in the case where condition \(2^\circ\) is replaced by

\(2^{0'}\). The matrix \(B(+\infty)\) has no multiple roots.

Condition \(3^\circ\) is a restriction on the rate of decrease of \(q(x)\) as \(x\to+\infty\), and condition \(4^\circ\) is a restriction on the regularity of the behavior of \(q(x)\) and \(B(x)\) as \(x\to+\infty\). For example, if \(\|B^{(j)}(x)\|=O(x^{-j})\), \(x\to+\infty\), \(j=1,2\), and \(q(x)=x^\alpha\), \(\operatorname{Re}\alpha>-1\), then conditions \(3^\circ,4^\circ\) are satisfied. These conditions are also satisfied if \(q(x)\) grows like \((\ln x)^\beta\) or like \(\exp(Cx^\gamma)\), \(\gamma>0\), \(\operatorname{Re}C>0\), with arbitrary \(\beta\), and this asymptotic expression can be differentiated twice. Condition (4) is satisfied in these examples. Finally, condition (5) can be weakened. In particular, if \(A(x)\) is real, then it is sufficient to require (5) only for those pairs \(\lambda_i(x),\lambda_j(x)\) which are not complex conjugates. G. Birkhoff \((^2)\) investigated the case where \(\mu=1\), \(Q(x)=E\), \(q(x)=x^r\), where \(r\ge 0\) is an integer and \(B(+\infty)\) has distinct eigenvalues; these results are contained in Theorem 1.

II. Consider the eigenvalue problem on the whole axis \((-\infty,+\infty)\) for system (1). A solution \(y(x,\mu_0)\ne0\) belonging to \(L_2(-\infty,+\infty)\) is called an eigenfunction, and \(\mu_0\) is called an eigenvalue.

value. A point \(x_0\) is called a turning point of system (1) if the matrix \(A(x_0)\) has multiple eigenvalues. A turning point \(x_0\) is called simple if \(A(x_0)\) has a double eigenvalue \(\lambda_0\), the remaining eigenvalues are simple, and \(f_x(\lambda_0,x_0)\ne 0\), where \(f(\lambda,x)\) is the characteristic polynomial of the matrix \(A(x)\).

Introduce the conditions:

1. System (1) has on the real axis exactly two, and moreover simple, turning points \(x_1<x_2\), and \(\lambda_m(x_j)=\lambda_{m+1}(x_j)\), \(j=1,2\).

2. \(\operatorname{Re}(\lambda_m(x)-\lambda_{m+1}(x))\equiv 0\) for \(x_1\le x\le x_2\), \(\operatorname{Re}\lambda_m(x)-\operatorname{Re}\lambda_{m+1}(x)<0\) for \(x>x_2,\ x<x_1\).

3. For all real \(x\),
   \[ \operatorname{Re}(\lambda_j(x)-\lambda_{j+1}(x))<0,\quad j\ne m,n. \]

In particular, for the scalar equation \(y''-\mu^2 q(x)y=0\), conditions 1–3 are satisfied if \(q(x)\) is real for real \(x\), has exactly two, and moreover simple, zeros \(x_1<x_2\) on the real axis, and \(q(x)>0\) for \(x>x_2\).

Denote by \(S(\rho,\alpha,\beta)\) the sector \(|\mu|\ge \rho,\ \alpha<\arg\mu<\beta\).

Theorem 4. Suppose the conditions of Theorem 1 and conditions 1–3 are satisfied on the real axis, \(1\le m\le n-1\), and the matrix \(A(x)\) is regular on the interval \([x_1,x_2]\). Then there exist \(\rho>0,\ \alpha<0<\beta\) such that all eigenvalues of system (1) (for the problem on the whole axis) lying in \(S(\rho,\alpha,\beta)\) have the form
\[ \mu_k=\xi_0^{-1}(2\pi k i+\xi_1)+O(k^{-1}),\quad k_0\le k<+\infty. \tag{13} \]

Here
\[ \xi_0=\frac12\int_C(\lambda_{m+1}-\lambda_m)\,dt,\qquad \xi_1=\frac12\int_C(\lambda_{m+1}^{(1)}-\lambda_m^{(1)})\,dt, \tag{14} \]
where \(C\) is a closed contour in the complex \(x\)-plane which contains the interval \([x_1,x_2]\) in its interior, traverses it in the positive direction, and contains no other turning points of system (1) in its interior. The branches of the functions \(\lambda_{m+1},\lambda_m,\lambda_{m+1}^{(1)},\lambda_m^{(1)}\) are chosen for \(x\in C,\ x>x_2\).

We note that \(\xi_0\) and \(\xi_1\) are purely imaginary quantities and \(\operatorname{Im}\xi_0>0\). In addition, the matrices \(B(+\infty)\) and \(B(-\infty)\) may be different, and \(q(x)\) may have different orders of growth as \(x\to\pm\infty\).

Corollary. Let \(\gamma_0\) be a smooth curve in the complex \(x\)-plane connecting two real points \(x_3<x_4\), and let
\[ \gamma=(-\infty,x_3]\cup\gamma_0\cup[x_4,+\infty). \]
Suppose that on \(\gamma\) the conditions of Theorem 1, condition 1, and the following conditions hold:

\[ 2'.\quad \operatorname{Re}\left(\int_{x_1}^{x}(\lambda_{m+1}-\lambda_m)\,dt\right)\equiv 0, \quad \text{if } x\in\gamma \text{ and lies between } x_1 \text{ and } x_2; \]
\[ \operatorname{Re}\left(\int_{x_1}^{x}(\lambda_{m+1}-\lambda_m)\,dt\right)>0 \quad \text{for } x\in\gamma \text{ to the right of } x_2,\text{ and is less than zero for } \]
\[ x\in\gamma \text{ to the left of } x_1. \]

\[ 3'.\quad \operatorname{Re}\left(\int_{x_2}^{x}(\lambda_j-\lambda_{j+1})\,dt\right)<0 \quad \text{for } j\ne m,\ x\in\gamma \text{ and lying to the right of } x_2, \]
and this function is positive for \(x\in\gamma\) to the left of \(x_2\).

Suppose, moreover, that the matrix \(A(x)\) is regular for \(x\in\overline{\gamma_0}\). Then all conclusions of Theorem 4 remain valid.

For the scalar equation \(y''-\mu^2 p(x)y=0\), this question has been investigated more fully in \((^{4,5})\).

Our methods also make it possible to find the asymptotics of the eigenfunctions as \(\mu\to+\infty\) (in the case of Theorem 4) on the whole real axis, except for certain neighborhoods of the turning points.

Introduce the conditions:

1) System (1) has turning points on the real axis

\(x_1 < x_2 < \cdots < x_{2l}\), all of them are simple, and \(\lambda_{m-j+1}(x_j) = \lambda_{m+j}(x_{2l-j+1})\), \(j = 1, 2, \ldots, l\).

2) \(\operatorname{Re}\bigl(\lambda_{m-j+1}(x) - \lambda_{m+j}(x)\bigr) \equiv 0\) for \(x \in [x_j, x_{2l-j+1}]\).

3) \(\operatorname{Re}\bigl(\lambda_j(x) - \lambda_{j+1}(x)\bigr) < 0\) for all \(x, j\), with the exception of those indicated in 2), and \(\operatorname{Re}\lambda_{m+1}(x) > 0\), \(\operatorname{Re}\lambda_m(x) < 0\) for \(x \in [x_1, x_{2l}]\).

Theorem 5. Suppose that the hypotheses of Theorem 1 and conditions 1)–3) are satisfied and that the matrix \(A(x)\) is regular for \(x \in [x_1, x_{2l}]\). Then there exist \(\rho > 0\), \(\alpha < 0 < \beta\) such that all eigenvalues of system (1) lying in \(S(\rho,\alpha,\beta)\) have the form
\[ \mu_{kj} = \xi_{0j}^{-1}(2\pi k i + \xi_{1j}) + O(k^{-1}), \qquad j = 1,2,\ldots,l;\ k \ge k_0 . \tag{15} \]
Here \(\xi_{0j}, \xi_{1j}\) have the form (14), where the integral is taken over the contour \(C_j\) enclosing the segment \([x_j, x_{2l-j+1}]\).

The quantities \(\xi_{0j}\) and \(\xi_{1j}\) are purely imaginary, and \(\operatorname{Im}\xi_{0j} > 0\). The corollary from Theorem 4 remains valid. Similar formulas are obtained for the asymptotics of the eigenvalues for a problem on a half-line (say, on the half-line \(x \ge x_0\), \(x_1 < x_0 < x_2\), under the hypotheses of Theorem 4).

Moscow Institute of Physics and Technology

Received
18 X 1965

REFERENCES

\(^{1}\) L. R. Volevich, Uspekhi Mat. Nauk, 18, No. 3, 155 (1963).
\(^{2}\) G. Birkhoff, Trans. Am. Math. Soc., 10, 436 (1909).
\(^{3}\) M. V. Fedoruk, Dokl. Akad. Nauk, 165, No. 4 (1965).
\(^{4}\) Yu. N. Dnestrovsky, L. P. Kostomarov, Zh. Vychisl. Mat. Mat. Fiz., 4, No. 2, 267 (1964).
\(^{5}\) M. V. Fedoruk, Mat. Sb., 68 (110), No. 1, 68 (1965).