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 \(L_0=\infty\), \(m_0=0\), \(n_0=n\ne0\), \(\rho_0(x)=Sx^\beta\), \(p_0(x)=Rx^\alpha\), \(q_0(x)=0\) \((0\leq x<L_0)\), where \(S>0\), \(R>0\), \(\beta>-1\), 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 \(q_0(x)=0\) \((0\le x<L)\), where \(S>0,\ R>0,\ \beta>-1\) 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 \(0<l<L_0\), while the functions \(\hat M(x)\), \(N(x)\), and \(Q(x)\) satisfy analogous conditions on \([0,L)\). In addition, one must assume that \(M(x)>M(+0)=M(0)\) 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).
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