Reports of the Academy of Sciences of the USSR
1958. Volume 122, No. 6

MATHEMATICS

I. S. KAC

SOME GENERAL THEOREMS ON THE BEHAVIOR OF SPECTRAL FUNCTIONS OF SECOND-ORDER DIFFERENTIAL SYSTEMS

(Presented by Academician S. L. Sobolev on 5 VI 1958)

1. In this article we give two theorems on the asymptotic behavior, as (\lambda \to +\infty), of the spectral functions of the differential system

[
-\frac{d}{dx}\left(p(x)\frac{d}{dx}y(x)\right)+q(x)y(x)-\lambda\rho(x)y(x)=0
\qquad (0\leq x<L\leq \infty),
]

[
y(0)=n,\qquad
p(x)\frac{d}{dx}y(x)\Big|_{x=0}=m,
\tag{1}
]

where (m) and (n) ((m^{2}+n^{2}>0)) are real constants; (\lambda) is a complex parameter; (\rho(x)\geq 0), (p(x)>0), and (q(x)) ((0\leq x<L)) are real measurable functions such that for every (l\in(0,L))

[
0<\int_{0}^{l}\rho(x)\,dx<\infty,\qquad
\int_{0}^{l}\frac{1}{p(x)}\,dx<\infty,\qquad
\int_{0}^{l}|q(x)|\,dx<\infty.
]

Let us recall the definition of the spectral functions of system (1). Let (u(x;\lambda)) be a solution of system (1). Denote by (M(x)) the function defined by the equality

[
M(x)=\int_{0}^{x}\rho(s)\,ds.
]

As is known, a nondecreasing function (\tau(\lambda)=\tau(\lambda-0)) ((-\infty<\lambda<\infty;\ \tau(0)=0)) is called a spectral function of system (1) if, for every (M)-measurable function (f(x)) ((0\leq x<L)) having an (M)-summable square on ([0,L)) and vanishing identically in some left neighborhood of the point (x=L), the equality

[
\int_{-\infty}^{\infty}
\left|\int_{0}^{L} f(x)u(x;\lambda)\,dM(x)\right|^{2}\,d\tau(\lambda)
=
\int_{0}^{L}|f(x)|^{2}\,dM(x)
\quad(<\infty)
]

holds.

1. We shall assign a nondecreasing function (\omega(\lambda)) to the class ((K_{\nu})) if it is defined for (1\leq \lambda<\infty) (or on a wider set), (\omega(\lambda)\to\infty) as (\lambda\to+\infty), and there exist a number (\gamma<\nu) and a sufficiently large number (N>1) such that, for (\eta>\lambda>N),

[
\frac{\omega(\eta)}{\omega(\lambda)}<\left(\frac{\eta}{\lambda}\right)^{\gamma}.
]

We shall assign a nondecreasing function (\theta(\lambda)), defined for (1\leq \lambda<\infty), to—

belong to the class ((\overline{K_\nu})), if there exists a function (\omega(\lambda)\in (K_\nu)) such that

[
\lim_{\lambda\to\infty}\frac{\theta(\lambda)}{\omega(\lambda)}=1.
]

It is obvious that, for any positive (\nu), ((K_\nu)\subset(\overline{K_\nu})), and for (\mu>\nu), ((K_\mu)\supset(K_\nu)), ((\overline{K_\mu})\supset(\overline{K_\nu})).

In addition to the differential system (1), let us consider one more differential system

[
-\frac{d}{dx}\left(p_0(x)\frac{d}{dx}y(x)\right)+q_0(x)y(x)-\lambda\rho_0(x)y(x)=0
\quad (0\leq x<L_0\leq\infty),
]

[
y(0)=n_0,\qquad
p_0(x)\frac{d}{dx}y(x)\bigg|_{x=0}=m_0
\tag{2}
]

of the same type as system (1).

Theorem 1. If (n=n_0\ne0),

[
\lim_{x\to\infty}\frac{p(x)}{p_0(x)}=1,\qquad
\lim_{x\to0}\frac{\rho(x)}{\rho_0(x)}=1
\tag{3}
]

and at least one spectral function (\tau_0(\lambda)) of system (2) belongs to the class ((\overline{K_1})), then for any spectral function (\tau(\lambda)) of system (1) (and, consequently, of system (2)) the equality

[
\lim_{\lambda\to\infty}\frac{\tau(\lambda)}{\tau_0(\lambda)}=1
]

holds.

Let us give an example. If (L_0=\infty), (m_0=0), (n_0=n\ne0), (\rho_0(x)=Sx^\beta), (p_0(x)=Rx^\alpha), (q_0(x)=0) ((0\leq x0), (R>0), (\beta>-1), and (\alpha<1), then the differential system (2) has a unique spectral function (\tau_0(\lambda)), with (\tau_0(\lambda)=0) for (\lambda<0), while for (\lambda\geq0)

[
\tau_0(\lambda)=
n^{-2}S^{-\frac{1-\alpha}{\beta-\alpha+2}}
R^{-\frac{\beta+1}{\beta-\alpha+2}}
(1-\alpha)^{-\frac{\alpha+\beta}{\beta-\alpha+2}}
T\left(\frac{\beta+\alpha}{1-\alpha}\right)
\lambda^{\frac{\beta+1}{\beta-\alpha+2}},
]

where

[
T(\zeta)=(\zeta+2)^{-\frac{2(\zeta+1)}{\zeta+2}}(\zeta+1)\Gamma^{-2}\left(\frac{2\zeta+3}{\zeta+2}\right),
\tag{4}
]

(\Gamma(z)) is Euler’s gamma function.

Since (\dfrac{\beta+1}{\beta-\alpha+2}<1), in this case the function (\tau_0(\lambda)) belongs to the class ((K_1)) and, consequently, to the class ((\overline{K_1})). Thus, with the choice indicated here of the functions (\rho_0(x)), (p_0(x)), and (q_0(x)), and of the number (n_0), system (2) satisfies the condition of the theorem. Therefore, if (n=0), and

[
\lim_{x\to\infty}\rho(x)x^{-\beta}=S,\qquad
\lim_{x\to0}p(x)x^{-\alpha}=R
\quad (\alpha<1;\ \beta>-1),
\tag{5}
]

then for any spectral function of system (1), as (\lambda\to+\infty), the following asymptotic equality holds:

[
\tau(\lambda)=
n^{-2}S^{-\frac{1-\alpha}{\beta-\alpha+2}}
R^{-\frac{\beta+1}{\beta-\alpha+2}}
(1-\alpha)^{-\frac{\alpha+\beta}{\beta-\alpha+2}}
T\left(\frac{\beta+\alpha}{1-\alpha}\right)
\lambda^{\frac{\beta+1}{\beta-\alpha+2}}
+
O\left(\lambda^{\frac{\beta+1}{\beta-\alpha+2}}\right),
]

where (T(\zeta)) is defined by equality (4).

Putting, in particular, (\alpha=\beta=0) and (R=S=1), we obtain that when (n=1) and

[
\lim_{x\to\infty}\rho(x)=1,\qquad \lim_{x\to 0}p(x)=1,
]

for any spectral function (\tau(\lambda)) of system (1), as (\lambda\to+\infty) the asymptotic equality

[
\tau(\lambda)=\frac{2}{\pi}\sqrt{\lambda}+O(\sqrt{\lambda})
]

holds.

In the case when (p(x)\equiv 1) and (\rho(x)\equiv 1) ((0\le x<\infty)), the last equality was first obtained by V. A. Marchenko ((^6)) and was subsequently refined more than once ((^{5,7})).

For the case when (n=0), the following proposition holds.

Theorem 2. Let (n=n_0=0,\ m=m_0\ne0), let conditions (3) be satisfied, and let at least one spectral function (\tau_0(\lambda)) of system (2) belong to the class ((K_2)); furthermore, let the function (\sigma_0(\lambda)), connected with (\tau_0(\lambda)) by the equality

[
\sigma_0(\lambda)=\int_1^\lambda \frac{d\tau(\xi)}{\xi}\qquad(\lambda>1),
]

belong to the class ((\overline{K}_1)), and

[
\lim_{\lambda\to\infty}\lambda\tau_0^{-1}(\lambda)\sigma(\lambda)<\infty .
]

Then for every spectral function (\tau(\lambda)) of system (1) (and, consequently, of system (2)) the equality

[
\lim_{\lambda\to\infty}\tau(\lambda)/\tau_0(\lambda)=1
]

holds.

In the case when (L_0=\infty,\ m_0=m\ne0,\ n_0=0,\ \rho_0(x)=Sx^\beta,\ p_0(x)=Rx^\alpha), and (q_0(x)=0) ((0\le x0,\ R>0,\ \beta>-1) and (\alpha<1), the differential system (2) has the unique spectral function (\tau_0(\lambda)):

[
\tau_0(\lambda)=\left[SR^{\frac{\beta+1}{1-\alpha}}(1-\alpha)^{\frac{\alpha+\beta}{1-\alpha}}\right]^{\frac{1-\alpha}{\beta-\alpha+2}}
T_1!\left(\frac{\alpha+\beta}{1-\alpha}\right)
\lambda^{\frac{\beta-2\alpha+3}{\beta-\alpha+2}}
\qquad(\lambda>0),
]

where

[
T_1(\zeta)=(\zeta+2)^{-\frac{2}{\zeta+2}}(\zeta+3)^{-1}
\Gamma^{-2}!\left(\frac{\zeta+3}{\zeta+2}\right).
]

It is easy to see that in this case (\tau_0(\lambda)) satisfies the condition of Theorem 2. Therefore, if (n=0) and conditions (5) are satisfied, then for any spectral function (\tau(\lambda)) of system (1), as (\lambda\to+\infty), the asymptotic equality

[
\tau(\lambda)=\left[SR^{\frac{\beta+1}{1-\alpha}}(1-\alpha)^{\frac{\alpha+\beta}{1-\alpha}}\right]^{\frac{1-\alpha}{\beta-\alpha+2}}
T_1!\left(\frac{\alpha+\beta}{1-\alpha}\right)
\lambda^{\frac{\beta-2\alpha+3}{\beta-\alpha+2}}
+o!\left(\lambda^{\frac{\beta-2\alpha+3}{\beta-\alpha+2}}\right).
]

1. Theorems 1 and 2 also extend to the case when (\rho(x)), (\dfrac{1}{p(x)}), (q(x)), (\rho_0(x)), (\dfrac{1}{p_0(x)}), and (q_0(x)) are generalized derivatives of the functions (M(x)), (N(x)), (Q(x)), (M_0(x)), (N_0(x)), and (Q_0(x)), respectively, where the function (M_0(x)) ((M(0)=0)) is nondecreasing on ([0,L_0)); (N_0(x)) ((N(0)=0)) is continuous and monotonically increasing on ([0,L_0)); (Q_0(x)) is real-valued and

having bounded variation on each interval ([0,l)), where (0M(+0)=M(0)\) and \(M_0(x)>M_0(+0)=M(0)) for (x>0), and replace conditions (3) by the conditions

[
\lim_{x,s\to 0}\frac{M(x)-M(s)}{M_0(x)-M_0(s)}=1,\qquad
\lim_{x,s\to 0}\frac{N(x)-N(s)}{N_0(x)-N_0(s)}=1.
\tag{6}
]

In this form, Theorems 1 and 2 are generalizations of propositions previously proved by the author (see ({}^{1}), Theorems 3 and 4). This is easily verified if Theorems 1 and 2, in this generalized form, are applied to the case where (M_0(x)=(\beta+1)^{-1}x^{\beta+1}) and (N_0(x)=x).

Theorems 1 and 2 also admit a further generalization. Namely, if conditions (6) are not satisfied, but on a sufficiently small interval ([0,b]) there exists a monotone continuous function (x(t)) such that

[
\lim_{t,r\to 0}\frac{M(x(t))-M(x(r))}{M_0(t)-M_0(r)}=1,\qquad
\lim_{t,r\to 0}\frac{N(x(t))-N(x(r))}{N_0(t)-N_0(r)}=1,
]

then, with all the other conditions retained, the assertions of Theorems 1 and 2 hold.

In proving the propositions presented in the present article, use was made of works of M. G. Krein (({}^{2,3})), which give a description of the set of spectral functions of second-order differential systems, and of one general Tauberian theorem of B. I. Korenblum (({}^{4})).

Izmail State
Pedagogical Institute

Received
5 III 1958

REFERENCES

({}^{1}) I. S. Kats, DAN, 106, No. 2, 183 (1956).
({}^{2}) M. G. Krein, DAN, 87, No. 6, 881 (1952).
({}^{3}) M. G. Krein, DAN, 89, No. 1, 5 (1958).
({}^{4}) B. I. Korenblum, DAN, 88, No. 5, 745 (1953).
({}^{5}) B. M. Levitan, Izv. AN SSSR, ser. matem., 19, No. 1, 33 (1955).
({}^{6}) V. A. Marchenko, Tr. Moskovsk. matem. obshch., 1, 327 (1952).
({}^{7}) V. A. Marchenko, Izv. AN SSSR, ser. matem., 19, No. 6, 381 (1955).