Full Text
M. G. GASYMOV
ON THE SUM OF THE DIFFERENCES OF THE EIGENVALUES OF TWO SELF-ADJOINT OPERATORS
(Presented by Academician A. A. Dorodnitsyn on 16 I 1963)
- The formula for the sum of the differences of the eigenvalues of two regular Sturm–Liouville operators was first obtained in the work of I. M. Gelfand and B. M. Levitan \((^1)\), and then by other methods in the works of L. A. Dikii \((^2)\) and C. Halberg and V. Kramer \((^3)\). In the present paper an analogue of the aforementioned formula is derived for the sum of the differences of the eigenvalues of two singular Sturm–Liouville operators differing from one another by a finite perturbation. The proposed method is based on certain inequalities obtained in the present work for the sums
\[ \sum_{n=1}^{N}(\mu_n-\lambda_n), \]
where \(\mu_1 \leqslant \mu_2 \leqslant \cdots\) and \(\lambda_1 \leqslant \lambda_2 \leqslant \cdots\) are the eigenvalues of self-adjoint operators \(C\) and \(A\), bounded below, in a Hilbert space \(H\), with the same domain of definition \(D_A\), and on the asymptotics of the spectral function of a Sturm–Liouville operator found by B. M. Levitan \((^4)\). In addition, with the aid of the inequalities found, some abstract theorems are proved. In particular, it is shown that if the difference
\[ C-A=B \]
is a nuclear operator, then the series
\[ \sum_{n=1}^{\infty}(\mu_n-\lambda_n) \]
converges to the trace of the operator \(B\).
- Let us consider in a Hilbert space \(H\) a self-adjoint operator \(A\) with domain of definition \(D_A\), bounded below, with eigenvalues \(\lambda_1 \leqslant \lambda_2 \leqslant \cdots\) and the corresponding orthonormal eigenfunctions \(\varphi_1,\varphi_2,\ldots\), and any orthonormal basis \(f_n \in D_A,\ n=1,2,\ldots\). Then the following holds:
Lemma 1.
\[ \sum_{n=1}^{N}(Af_n,f_n)\geqslant \sum_{n=1}^{N}\lambda_n. \tag{1} \]
Proof. It is obvious that
\[ f_n=\sum_{k=1}^{\infty}u_{nk}\varphi_k, \]
where \(\|u_{nk}\|\) is a unitary matrix. Then
\[ Af_n=\sum_{k=1}^{\infty}\lambda_k u_{nk}\varphi_k. \]
Hence
\[ (Af_n,f_n)=\sum_{k=1}^{\infty}\lambda_k|u_{nk}|^2. \]
Thus,
\[ \begin{aligned} \sum_{n=1}^{N} (A f_n, f_n) &= \sum_{n=1}^{N} \sum_{k=1}^{\infty} \lambda_k |u_{nk}|^2 \\ &= \sum_{n=1}^{N} \sum_{k=1}^{N} \lambda_k |u_{nk}|^2 + \sum_{n=1}^{N} \sum_{k=N+1}^{\infty} \lambda_k |u_{nk}|^2 \\ &= \sum_{n=1}^{\infty} \sum_{k=1}^{N} \lambda_k |u_{nk}|^2 - \sum_{n=N+1}^{\infty} \sum_{k=1}^{N} \lambda_k |u_{nk}|^2 + \sum_{n=1}^{N} \sum_{k=N+1}^{\infty} \lambda_k |u_{nk}|^2 \\ &\geq \sum_{k=1}^{N} \lambda_k + \lambda_{N+1} \left\{ \sum_{n=1}^{N} \sum_{k=N+1}^{\infty} |u_{nk}|^2 - \sum_{n=N+1}^{\infty} \sum_{k=1}^{N} |u_{nk}|^2 \right\} = \sum_{k=1}^{N} \lambda_k . \end{aligned} \]
Thus, the lemma is proved.
Let us note that an inequality similar to inequality (1), for finite matrices, was proved by Ky Fan \((^5)\).
Corollary 1. If \(\omega(t)\) is a continuous nondecreasing function, then
\[ \sum_{n=1}^{N} (\omega(A) f_n, f_n) \geq \sum_{n=1}^{N} \omega(\lambda_n). \tag{2} \]
This corollary is proved analogously to Lemma 1.
Let \(C\) be a self-adjoint operator in \(H\), bounded from below, with domain of definition \(D_C = D_A\), with eigenvalues \(\mu_1 \leq \mu_2 \leq \cdots\) and corresponding orthonormal eigenfunctions \(\psi_1, \psi_2, \ldots\). From Lemma 1 the inequalities follow
\[ \sum_{n=1}^{N} (A\psi_n, \psi_n) \geq \sum_{n=1}^{N} \lambda_n, \qquad \sum_{n=1}^{N} (C\varphi_n, \varphi_n) \geq \sum_{n=1}^{N} \mu_n. \tag{3} \]
Theorem 1. Let \(B = C - A\). Then
\[ \sum_{n=1}^{N} (B\psi_n, \psi_n) \leq \sum_{n=1}^{N} (\mu_n - \lambda_n) \leq \sum_{n=1}^{N} (B\varphi_n, \varphi_n). \tag{4} \]
Proof. Using inequalities (3), we obtain
\[ \sum_{n=1}^{N} (B\psi_n, \psi_n) = \sum_{n=1}^{N} \{(C\psi_n, \psi_n) - (A\psi_n, \psi_n)\} \leq \sum_{n=1}^{N} (\mu_n - \lambda_n) \leq \]
\[ \leq \sum_{n=1}^{N} \{(C\varphi_n, \varphi_n) - (A\varphi_n, \varphi_n)\} = \sum_{n=1}^{N} (B\varphi_n, \varphi_n), \]
which proves the theorem.
Using Corollary 1, one can prove the following theorem.
Theorem 2. Let \(\omega(t)\) be a continuous nondecreasing function and \(B_\omega = \omega(C) - \omega(A)\). Then
\[ \sum_{n=1}^{N} (B_\omega \psi_n, \psi_n) \leq \sum_{n=1}^{N} \{\omega(\mu_n) - \omega(\lambda_n)\} \leq \sum_{n=1}^{N} (B_\omega \varphi_n, \varphi_n). \tag{5} \]
Definition. If for an orthonormal basis \(\{f_n\}\) there exists the limit
\(\lim\limits_{N\to\infty} \sum_{n=1}^{N} (A f_n, f_n)\), then this limit is called the trace of the operator \(A\) in the basis \(\{f_n\}\).
Using inequalities (4) and (5), we obtain the following theorem.
Theorem 3. If the traces of the operator \(B=C-A\) in the bases \(\{\varphi_n\}\) and \(\{\psi_n\}\) are equal, then
\[ \lim_{N\to\infty}\sum_{n=1}^{N}(\mu_n-\lambda_n) = \lim_{N\to\infty}\sum_{n=1}^{N}(B\varphi_n,\varphi_n). \tag{6} \]
Theorem 4. If the traces of the operator \(B_\omega=\omega(C)-\omega(A)\), where \(\omega(t)\) is a continuous nondecreasing function, in the bases \(\{\varphi_n\}\) and \(\{\psi_n\}\) are equal, then
\[ \lim_{N\to\infty}\sum_{n=1}^{N}\{\omega(\mu_n)-\omega(\lambda_n)\} = \lim_{N\to\infty}\sum_{n=1}^{N}(B_\omega\varphi_n,\varphi_n). \tag{7} \]
It is known that a nuclear operator has equal traces in any basis. Therefore the following theorems hold:
Theorem 5. Let \(A\) be any self-adjoint operator in \(H\), bounded below and with discrete spectrum, and let \(B\) be a nuclear self-adjoint operator. Then the series of differences of the eigenvalues of the operators \(C=A+B\) and \(A\) converges to the trace of the operator \(B\).
Theorem 6. If \(\omega(t)\) is a continuous nondecreasing function and \(\omega(C)-\omega(A)=B_\omega\) is a nuclear operator, then
\[ \lim_{N\to\infty}\sum_{n=1}^{N}\{\omega(\mu_n)-\omega(\lambda_n)\} \]
exists and is equal to the trace of the operator \(B_\omega\).
We note that in Theorem 6 it is not required that the operator \(B=C-A\) be nuclear. For example, if \(A\) and \(C\) are of Hilbert–Schmidt type and \(\omega(t)\ge t^2\), then the conditions of the theorem are satisfied.
- With the aid of the results obtained above, one can compute the sum of differences of eigenvalues of the operators \(L_0\) and \(L\) generated by the differential equations
\[ y''+\{\lambda-q(x)\}y=0, \tag{8} \]
\[ y''+\{\lambda-q(x)-p(x)\}y=0, \tag{9} \]
respectively. Let us consider the following examples.
A. Let \(-\infty<x<\infty\), let the continuous function \(q(x)\to\infty\) as \(|x|\to\infty\), and let \(p(x)\) be a finite continuous function. Then the spectra of the operators \(L_0\) and \(L\) are discrete. Denote the eigenvalues of the operators \(L_0\) and \(L\) by \(\lambda_1<\lambda_2<\cdots\) and \(\mu_1<\mu_2<\cdots\), and the eigenfunctions by \(\varphi_1(x),\varphi_2(x),\ldots\) and \(\psi_1(x),\psi_2(x),\ldots\), respectively. Then the following holds:
Theorem 7. If \(\int p(x)\,dx=0\), then the series of differences of the eigenvalues of the operators \(L\) and \(L_0\) converges and is equal to zero.
Proof. \(D_{L_0}=D_L\), therefore, according to Theorem 3, it suffices to show that
\[ \lim_{N\to\infty}\sum_{n=1}^{N}\int p(x)\psi_n^2(x)\,dx = \lim_{N\to\infty}\sum_{n=1}^{N}\int p(x)\varphi_n^2(x)\,dx. \]
We note that
\[ \lim_{N\to\infty}\sum_{n=1}^{N}\int p(x)\psi_n^2(x)\,dx = \lim_{\lambda\to\infty}\sum_{\mu_n<\lambda}\int p(x)\psi_n^2(x)\,dx. \]
On the other hand, it is known \((^4)\) that in any bounded interval of variation of \(x\) the formula
\[ \sum_{\mu_n<\lambda}\psi_n^2(x)=\frac{2}{\pi}\sqrt{\lambda}+o(1) \]
holds. Further, since \(p(x)\) is a finite function and \(\int p(x)\,dx=0\), we have
\[ \lim_{\lambda\to\infty}\sum_{\mu_n<\lambda}\int p(x)\psi_n^2(x)\,dx=0. \]
It is proved in a completely analogous way that
\[ \lim_{\lambda\to\infty}\sum_{\lambda_n<\lambda}\int p(x)\varphi_n^2(x)\,dx=0. \]
This also proves the theorem.
B. Let the operators \(L_0\) and \(L\) be generated by equations (8) and (9), respectively, given on the half-axis \(0\leq x<\infty\), with the boundary condition \(y'(0)-hy(0)=0\), where \(h\) is a real number. Then, if a continuous function \(q(x)\to\infty\) as \(x\to\infty\), and \(p(x)\) is a finite continuous function, the operators \(L_0\) and \(L\) have discrete spectra \(\lambda_1<\lambda_2<\ldots\) and \(\mu_1<\mu_2<\ldots\), and eigenfunctions \(\varphi_1(x),\varphi_2(x),\ldots\) and \(\psi_1(x),\psi_2(x),\ldots\), respectively.
Theorem 8. If the function \(p(x)\) has a first derivative in a small neighborhood of zero and
\[ \int p(x)\,dx=0, \]
then
\[ \lim_{N\to\infty}\sum_{n=1}^{N}(\mu_n-\lambda_n)=\frac{p(0)}{4}. \tag{10} \]
This formula is an analogue of the formula of I. M. Gelfand and B. M. Levitan \({}^{1}\) (they considered the regular case).
Proof. Obviously, it is sufficient to show that
\[ \lim_{\lambda\to\infty}\sum_{\mu_n<\lambda}\int p(x)\psi_n^2(x)\,dx - \lim_{\lambda\to\infty}\sum_{\lambda_n<\lambda}\int p(x)\varphi_n^2(x)\,dx = \frac{p(0)}{4}. \tag{11} \]
It is known \({}^{4}\) that for large \(\lambda\)
\[ \sum_{\lambda_n<\lambda}\varphi_n^2(x)-\theta(x,x,\lambda)=o(1), \qquad \text{where } \theta(x,x,\lambda)=\frac{2}{\pi}\int_0^{\sqrt{\lambda}}\cos^2 sx\,ds. \]
Therefore
\[ \lim_{\lambda\to\infty}\sum_{\lambda_n<\lambda}\int p(x)\varphi_n^2(x)\,dx = \lim_{\lambda\to\infty}\int p(x)\theta(x,x,\lambda)\,dx. \tag{12} \]
Let us compute the last limit. First note that
\[ \theta(x,x,\lambda)=\frac{1}{\pi}\sqrt{\lambda} +\frac{1}{2\pi}\frac{\sin 2\sqrt{\lambda}\,x}{x}. \tag{13} \]
Therefore
\[ \lim_{\lambda\to\infty}\int p(x)\theta(x,x,\lambda)\,dx = \frac{1}{2\pi}\lim_{\lambda\to\infty}\int p(x)\frac{\sin 2\sqrt{\lambda}\,x}{x}\,dx = \frac{p(0)}{4}. \tag{14} \]
Similarly one can show that
\[ \lim_{\lambda\to\infty}\sum_{\mu_n<\lambda}\int p(x)\psi_n^2(x)\,dx = \frac{p(0)}{4}. \tag{15} \]
The theorem is proved.
In proving Theorems 7 and 8, the following was proved along the way.
Theorem 9. In cases A and B, for the convergence of the series
\[ \sum_{n=1}^{N}(\mu_n-\lambda_n) \]
it is necessary and sufficient that
\[ \int p(x)\,dx=0. \]
The authors take this opportunity to express their sincere gratitude to Prof. B. M. Levitan, F. A. Berezin, and A. G. Kostyuchenko for their attention to the work and for discussion of the results.
Received
12 I 1963
References Cited
- I. M. Gelfand, B. M. Levitan, DAN, 88, No. 4, 593 (1953).
- L. A. Dikii, UMN, 8, No. 2, 119 (1953).
- C. Halberg, V. Kramer, Duke Math. J., 27, No. 4 (1960).
- B. M. Levitan, Izv. AN SSSR, Ser. Mat., 17, 331 (1953).
- Ku Fan, Proc. Nat. Acad. Sci. U. S. A., 35, No. 11 (1949).