UDC 517.94

MATHEMATICS

V. A. YAVRYAN

ON THE REGULARIZED TRACE OF THE DIFFERENCE OF TWO SINGULAR STURM–LIOUVILLE OPERATORS

(Presented by Academician N. N. Bogolyubov on 27 IX 1965)

As is known, I. M. Gel'fand and B. M. Levitan \((^{1})\) were the first to obtain a formula for the sum of the differences of the eigenvalues of two regular Sturm–Liouville operators. Subsequently, in \((^{2-4})\), analogues of these formulas were obtained in the case of certain singular Sturm–Liouville operators. In the present note these questions are investigated in connection with the works of M. G. Krein \((^{5,6})\) from a unified point of view.

1. In the space \(\mathscr L_2(0,\infty)\) we consider the operators \(H_0\) and \(H_1\) generated by the equalities

\[ H_0y=-y''+q_0(x)y, \tag{1} \]

\[ H_1y=-y''+q_0(x)y+q(x)y=-y''+q_1(x)y \tag{2} \]

and by the boundary condition

\[ y'(0)-hy(0)=0. \tag{3} \]

It is assumed that \(q_0(x)\) is a real locally integrable function such that \(H_0\) is a self-adjoint operator; \(q(x)\) is a real finite integrable function; \(h\) is a real number. For \(h=\infty\), condition (3) is understood as \(y(0)=0\). It is easy to show that the difference of the resolvents of the operators \(H_0\) and \(H_1\) is a nuclear operator.

Consequently, from the theorem of M. G. Krein \((^{6})\) it follows that there exists a real function \(\xi(t)=\xi(t,H_0,H_1)\) such that, for any non-real \(z\),

\[ \operatorname{Sp}\{R_z(H_1)-R_z(H_0)\} = -\int_{-\infty}^{\infty}\frac{\xi(t)}{(t-z)^2}\,dt, \tag{4} \]

where \(R_z(H)=(H-zI)^{-1}\).

From equality (4) the function \(\xi(t)\) is determined almost everywhere up to a constant term.

Theorem 1. The function \(\xi(t)\) in equality (4) can be chosen so that, for any \(a\ge 0\),

\[ \int_{-\infty}^{0} e^{a\sqrt{|t|}}|\xi(t)|\,dt<+\infty. \tag{5} \]

Moreover, if the Fourier cosine transform of the finite function \(q(x)\) converges to \(q(0)\) at the point \(x=0\), then

\[ \int_{-\infty}^{0}\xi(t)\,dt + \int_{0}^{\infty} \left( \xi(t)-\frac{1}{2\pi\sqrt t}\int_{0}^{\infty}q(x)\,dx \right)\,dt = \frac14 q(0) \quad \text{for } h\ne\infty, \tag{6} \]

\[ \int_{-\infty}^{0}\xi(t)\,dt + \int_{0}^{\infty} \left( \xi(t)-\frac{1}{2\pi\sqrt t}\int_{0}^{\infty}q(x)\,dx \right)\,dt = -\frac14 q(0) \quad \text{for } h=\infty. \tag{6'} \]

In \((^{7})\) this theorem was proved under the condition that \(H_0\) is a semibounded operator (in this case inequality (5) is trivial).

For the proof of Theorem 1 the following inequality is used, valid in the case \(q(x)\ge 0\):

\[ \int_0^\infty q(x)\theta_1(x,x,\lambda)\,dx \le \int_{-\infty}^{\lambda}\xi(t)\,dt \le \int_0^\infty q(x)\theta_0(x,x,\lambda)\,dx, \tag{7} \]

where \(\theta_0(x,s,\lambda)\) and \(\theta_1(x,s,\lambda)\) are the spectral kernels (see (8)) of the operators \(H_0\) and \(H_1\).

Inequality (5) is obtained if inequality (7) is multiplied by \(ae^{a\sqrt{|\lambda|}}/2\sqrt{|\lambda|}\), where \(a>0\), and integrated with respect to \(\lambda\) from \(-\infty\) to \(0\). In doing so one must use the known estimate (9) for the spectral function.

From the asymptotics for the spectral kernels \(\theta_0(x,s,\lambda)\) and \(\theta_1(x,s,\lambda)\) (see (8)) and from (13) it follows that, as \(\lambda\to +\infty\),

\[ \int_{-\infty}^{\lambda}\xi(t)\,dt -\frac{1}{\pi}\sqrt{\lambda}\int_0^\infty q(x)\,dx = \frac{1}{2\pi}\int_0^\infty q(x)\frac{\sin 2\sqrt{\lambda}x}{x}\,dx+o(1) \quad \text{for } h\ne \infty, \tag{8} \]

\[ \int_{-\infty}^{\lambda}\xi(t)\,dt -\frac{1}{\pi}\sqrt{\lambda}\int_0^\infty q(x)\,dx = -\frac{1}{2\pi}\int_0^\infty q(x)\frac{\sin 2\sqrt{\lambda}x}{x}\,dx+o(1) \quad \text{for } h=\infty. \tag{8'} \]

These relations were obtained under the assumption \(q(x)\ge 0\). However, this assumption is easily removed. Theorem 1 follows from (8) and (8′) and from the properties of Dirichlet integrals. In the case where the operator \(H_0\) has a discrete spectrum bounded below, this theorem coincides with the result of M. G. Gasymov \((^3)\).

1. Consider the special case where \(q_0(x)\equiv 0\) and \(h=\infty\), requiring, instead of the finiteness of the function \(q(x)\), only that \((1+x)q(x)\in \mathscr L_1(0,\infty)\).

Theorem 2. If \((1+x)q(x)\in \mathscr L_1(0,\infty)\) and the Fourier cosine transform of the function \(q(x)\) at the point \(x=0\) converges to \(q(0)\), then relation (6′) holds.

In the proof of this theorem the following lemma is used.

Lemma. If \((1+x)q(x)\in \mathscr L_1(0,\infty)\), then there exists a constant \(C\) such that for all \(0\le x<+\infty\) and \(\lambda<+\infty\) the inequality

\[ \left| \theta_1(x,x,\lambda) -\frac{1}{\pi}\int_0^\lambda \frac{\sin^2\sqrt{t}\,x}{\sqrt{t}}\,dt \right| \le C \]

is valid.

As is known, under the condition \(q(x)\in \mathscr L_1(0,\infty)\) the asymptotic formula

\[ \psi_1(x,\lambda)=\frac{A(k)}{k}\sin(kx-\eta(k))+o(1), \qquad k^2=\lambda,\quad x\to +\infty \]

holds. The equality (see \((^{10,11})\)) is valid

\[ \xi(\lambda)= \begin{cases} \dfrac{1}{\pi}\eta\left(\sqrt{\lambda}\right), & \lambda>0,\\[6pt] -\displaystyle\int_{-\infty}^{\lambda}\delta(t-\lambda_k)\,dt, & \lambda<0, \end{cases} \]

where \(\{\lambda_k\}\) is the sequence of all negative eigenvalues of the operator \(H_1\), and \(\delta(t)\) is the Dirac function.

This relation, together with Theorem 2, gives

Theorem 3. Under the assumptions of Theorem 2, the relation

\[ \sum_k \lambda_k +\frac{2}{\pi}\int_0^\infty \lambda\left( \eta(\lambda)-\frac{1}{2\lambda}\int_0^\infty q(x)\,dx \right)d\lambda = -\frac14 q(0). \tag{9} \]

Formula (9), for a twice continuously differentiable function \(q(x)\) and under the condition \((1+x^2)q(x)\in \mathscr L_1(0,\infty)\), was first rigorously proved

by L. D. Faddeev (see (2), and also (10, 12, 13)). Let us note that in the formulation of Theorem 1 of (2) there is an inaccuracy, namely, it is asserted there that formula (9) holds whenever \(q(x)\) is continuous at the point \(x=0\) and \(xq(x)\in \mathscr L_1(0,\infty)\). The incorrectness of this assertion follows from the fact that, as (8′) shows, for equality (9) to hold under the condition \((1+x)q(x)\in \mathscr L_1(0,\infty)\), it is necessary and sufficient that the Fourier cosine transform of the function \(q(x)\) converge at the point \(x=0\) to \(q(0)\).

1. In this section we shall consider two singular Sturm—Liouville operators which are defined by one and the same differential expression (1), but by different conditions at zero. In the case when these operators have a discrete spectrum, M. G. Gasymov and B. M. Levitan (4) obtained a formula for the sum of the differences of the eigenvalues of these operators. Here this formula is generalized (Theorem 5) to arbitrary Sturm—Liouville operators (not necessarily with discrete spectrum). For semibounded operators this generalization was obtained by the author in another way in (7). Here we also establish formulas which connect the spectral shift function \(\xi(t,H_{h_1},H_{h_2})\) and the spectral function.

Let the operator \(H_h\) be defined by equalities (1) and (3). Since the resolvents of the operators \(H_{h_1}\) and \(H_{h_2}\) differ by a one-dimensional operator, formula (4) is valid for these operators.

From equality (4) and \((1+|t|)^{-1}\xi(t,H_{h_1},H_{h_2})\in \mathscr L_1(0,\infty)\), the function \(\xi(t,h_1,h_2)=\xi(t,H_{h_1},H_{h_2})\) is determined uniquely. Denote by \(\sigma_h(\lambda)\) the spectral function of the operator \(H_h\), normalized by the condition \(\sigma_h(-\infty)=0\). The following is valid.

Theorem 4. The following formulas hold \(*\):

\[ \int_{-\infty}^{\infty}\frac{\xi(t,h_1,h_2)}{t-\lambda}\,dt = \ln\left(1+(h_2-h_1)\int_{-\infty}^{\infty}\frac{d\sigma_{h_1}(t)}{t-\lambda}\right), \tag{10} \]

\[ \int_{-\infty}^{\lambda}\xi(t,h_1,h_2)\,dt = \int_{h_1}^{h_2}\sigma_h(\lambda)\,dh. \tag{11} \]

From the asymptotic formula for the spectral function (see (8, 14)) and from (11) there follows the following

Theorem 5. The formula holds

\[ \int_{-\infty}^{0}\xi(t,h_1,h_2)\,dt + \int_{0}^{\infty}\left(\xi(t,h_1,h_2)-\frac{1}{\pi\sqrt t}(h_2-h_1)\right)\,dt = -\frac{h_2^2-h_1^2}{2}. \]

For every \(a\geq 0\)

\[ \int_{-\infty}^{0} e^{a\sqrt{|t|}}|\xi(t)|\,dt<+\infty . \]

The author expresses his gratitude to Prof. M. G. Krein for posing the problems and for valuable advice.

Yerevan
State University

Received
10 IX 1965

CITED LITERATURE

1. I. M. Gel'fand, B. M. Levitan, DAN, 88, No. 4 (1953).
2. L. D. Faddeev, DAN 115, No. 5 (1957).
3. M. G. Gasymov, DAN, 150, No. 6 (1963).
4. M. G. Gasymov, B. M. Levitan, DAN, 151, No. 5 (1963).
5. M. G. Krein, Matem. sborn., 33 (75), 3 (1953).
6. M. G. Krein, DAN, 144, No. 2 (1962).
7. V. A. Yavryan, Dokl. ArmSSR, 41, No. 1 (1965).
8. B. M. Levitan, Izv. AN SSSR, ser. matem., 19, 33 (1955).
9. V. A. Marchenko, Tr. Moskovsk. matem. obshch., 1 (1952).
10. B. S. Buslaev, L. D. Faddeev, DAN, 132, No. 1 (1960).
11. V. A. Yavryan, Dokl. ArmSSR, 33 (75), 3 (1953).
12. M. G. Krein, DAN, 144, No. 2 (1962).
13. V. A. Yavryan, Dokl. ArmSSR, 33 (75), 3 (1953).
14. [[unclear: author initials]], [[unclear: surname]]berg, V. A. Kramer, Duke Math. J., 30, No. 2 (1963).
15. V. A. Marchenko, Izv. AN SSSR, ser. matem., 19, 381 (1955).

* Formula (10) was communicated to the author by M. G. Krein.