R. F. SHEVCHENKO

ON THE TRACE OF A DIFFERENTIAL OPERATOR

(Presented by Academician I. G. Petrovskii, 12 III 1965)

Mathematics

Let two differential operators be given
\[ S_1=\frac{d^n}{dx^n},\qquad S_2=\frac{d^n}{dx^n}+q(x), \tag{1} \]
defined on the interval \([0,1]\) by identical regular boundary conditions (definition of regularity \((^5)\))
\[ U_j(y)=U_{j0}(y)+U_{j1}(y)=\alpha_j y^{(k_j)}(0)+\ldots+\beta_j y^{(k_j)}(1)+\ldots, \qquad j=1,\ldots,n, \tag{2} \]
where \(q(x)\) is a sufficiently smooth function and
\[ \int_0^1 q(x)\,dx=0. \tag{3} \]

In the present work conditions are obtained for the regularization (existence) of the trace of the operator \(S_2\)
\[ \sum_{k=1}^{\infty *}\lambda_{2,k} = \sum_{k=1}^{\infty}(\lambda_{2,k}-\lambda_{1,k}-\ldots), \tag{4} \]
where \(|\lambda_{2,1}|\leq |\lambda_{2,2}|\leq \ldots\) and \(|\lambda_{1,1}|\leq |\lambda_{1,k}|\leq \ldots\) are the eigenvalues of the operators \(S_2\) and \(S_1\).

For \(n=2\), first in the work of I. M. Gelfand and B. M. Levitan \((^{1,2})\), and then by other authors \((^{3,4})\), the formula
\[ \sum_{k=1}^{\infty *}\lambda_{2,k} = \sum_{k=1}^{\infty}(\lambda_{2,k}-\lambda_{1,k}) = -\frac{q(1)+q(0)}{4} \]
was obtained, but in all these works the operators \(S_1\) and \(S_2\) were self-adjoint. The method used in the present work is based on the relation between the resolvents \(R_{\lambda,1}\) and \(R_{\lambda,2}\) of the operators \(S_1\) and \(S_2\)
\[ R_{\lambda,2}=R_{\lambda,1}-R_{\lambda,1}\cdot(qR_{\lambda,1}), \tag{5} \]
which is also valid in the non-self-adjoint case. This method makes it possible to prove the following result.

Let \(\omega_i\) be the roots of the \(n\)-th degree of \(-1\); \(\rho^n=-\lambda\), and \(A_0\) and \(A_1\) constants defined as follows:

1. \(n=2\mu\).
   \[ A_0=\frac{1}{n\theta_1}\sum_{i=1}^{\mu}\sum_{j=\mu+1}^{\mu}\bar A_{ij}\frac{\omega_j}{\omega_i-\omega_j}; \tag{6} \]
   \[ A_1=\frac{1}{n\theta_1}\sum_{i=\mu+1}^{n}\sum_{j=1}^{\mu} A_{ij}\frac{\omega_j}{\omega_i-\omega_j}. \tag{7} \]

1. \(n=2\mu+1\).

\[ A_0=\frac{1}{2n}\left[\frac{1}{\theta_2}\sum_{i=1}^{\mu}\sum_{j=\mu+1}^{n} \frac{\overline{\overline{A}}_{ij}\omega_j}{\omega_i-\omega_j} + \frac{1}{\theta_3}\sum_{i=1}^{\mu+1}\sum_{j=\mu+2}^{n} \frac{\overline{\overline{A}}_{ij}\omega_j}{\omega_i-\omega_j}\right]; \tag{8} \]

\[ A_1=\frac{1}{2n}\left[\frac{1}{\theta_2}\sum_{i=\mu+1}^{n}\sum_{j=1}^{\mu} \frac{\overline{\overline{A}}_{ij}\omega_j}{\omega_i-\omega_j} + \frac{1}{\theta_3}\sum_{i=\mu+2}^{n}\sum_{i=1}^{\mu+1} \frac{\overline{\overline{A}}_{ij}\omega_j}{\omega_i-\omega_j}\right]. \tag{9} \]

Here

\[ \theta_1= \begin{vmatrix} \alpha_1\omega_1^{k_1}&\cdots&\alpha_1\omega_\mu^{k_1}&\beta_1\omega_{\mu+1}^{k_1}&\cdots&\beta_1\omega_n^{k_1}\\ \cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot\\ \alpha_n\omega_1^{k_n}&\cdots&\alpha_n\omega_\mu^{k_n}&\beta_n\omega_{\mu+1}^{k_n}&\cdots&\beta_n\omega_n^{k_n} \end{vmatrix}; \tag{10} \]

\[ \theta_2= \begin{vmatrix} \alpha_1\omega_1^{k_1}&\cdots&\alpha_1\omega_\mu^{k_1}&\beta_1\omega_{\mu+1}^{k_1}&\cdots&\beta_1\omega_n^{k_1}\\ \cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot\\ \alpha_n\omega_1^{k_n}&\cdots&\alpha_n\omega_\mu^{k_n}&\beta_n\omega_{\mu+1}^{k_n}&\cdots&\beta_n\omega_n^{k_n} \end{vmatrix}; \tag{11} \]

\[ \theta_3= \begin{vmatrix} \alpha_1\omega_1^{k_1}&\cdots&\alpha_1\omega_{\mu+1}^{k_1}&\beta_1\omega_{\mu+2}^{k_1}&\cdots&\beta_1\omega_n^{k_1}\\ \cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot\\ \alpha_n\omega_1^{k_n}&\cdots&\alpha_n\omega_{\mu+1}^{k_n}&\beta_n\omega_{\mu+2}^{k_n}&\cdots&\beta_n\omega_n^{k_n} \end{vmatrix}^{*}, \tag{12} \]

where \(\overline{A}^{\,i}_j\), \(\overline{\overline{A}}_{ij}\), and \(\overline{\overline{\overline{A}}}_{ij}\) differ from the determinants standing in the denominator in that, in the \(i\)-th column, \(\omega_i\) is replaced by \(\omega_j\). Then the following holds.

Theorem 1. For \(n\geqslant 2\) and regular boundary conditions, the trace \(S_2\) exists and is equal to

\[ \sum_{k=1}^{\infty}\lambda_{2,k} = \sum_{k=1}^{\infty}(\lambda_{2,k}-\lambda_{1,k}) = A_1q(1)-A_0q(0). \tag{13} \]

Proof. Since the resolvents \(R_{\lambda,1}\) and \(R_{\lambda,2}\) are Hilbert–Schmidt operators with kernels \(R_1(x,t;\lambda)\) and \(R_2(x,t;\lambda)\), applying Lidskii’s theorem (6)**, we obtain

\[ \operatorname{sp} R_{\lambda,1} = \sum_{k=1}^{\infty}\frac{1}{\lambda_{1,k}-\lambda} = \int_0^1 R_1(x,x;\lambda)\,dx; \tag{14} \]

\[ \operatorname{sp} R_{\lambda,2} = \sum_{k=1}^{\infty}\frac{1}{\lambda_{2,k}-\lambda} = \int_0^1 R_2(x,x;\lambda)\,dx. \tag{15} \]

Iterating (5) \(m\) times and taking \(\operatorname{sp}\), we obtain

\[ \operatorname{sp} R_{\lambda,2}-\operatorname{sp} R_{\lambda,1} = \frac{d}{d\lambda}\left[\sum_{l=1}^{m}J_l(\lambda)\right]+R_m(\lambda), \tag{16} \]

where

\[ J_l(\lambda)= \frac{(-1)^l}{l} \int_0^1\cdots\int_0^1 q(x_1)\cdots q(x_l)R_1(x_1,x_2;\lambda)\cdots \]

\[ \cdots R_1(x_l,x_1;\lambda)\,dx_1\cdots dx_l . \tag{17} \]

* Comparing (11), (12), and (13) with the formulas in (5), p. 51, we see that the non-vanishing of \(\theta_1,\theta_2\), and \(\theta_3\) is equivalent to the regularity of the boundary conditions.

** For differential operators of a more general type this was done in (7).

Bearing in mind the simple identity

\[ \frac{1}{2\pi i}\oint_{|\lambda|=r} \lambda\frac{d}{d\lambda}\left[\frac{1}{\pi(\lambda_i-\lambda)^{n_i}}\right]\,d\lambda = -\frac{1}{2\pi i}\oint \frac{d\lambda}{\prod_{i=1}^{l}(\lambda_i-\lambda)^{n_i}}, \tag{18} \]

multiplying (16) by \(\lambda/2\pi i\) and integrating over the contour \(|\lambda|=r\), we obtain

\[ \sum_{|\lambda|<r}(\lambda_{2,k}-\lambda_{1,k}) = -\frac{1}{2\pi i}\sum_{l=1}^{m} \oint_{|\lambda|=r}J_l\,d\lambda+\widetilde R_m(r) = -\sum I_l(r)+\widetilde R_m(r). \tag{19} \]

According to (5), p. 71, in the regular case we have

\[ |R_1(x,t;\lambda)|=O(1/|\rho|^{n-1}),\qquad |J_l(\lambda)|=O(1/\rho^{l(n-1)}). \tag{20} \]

But condition (3) and the smoothness of \(q(x)\) make it possible to increase the order of decrease by 1, i.e.

\[ |J_l(\lambda)|=O(1/\rho^{l(n-1)+1}) = O(1/\rho^{(l-1)(n-1)+n}), \tag{21} \]

from which it follows that in (19) all terms, beginning with the second, vanish\(^*\). Therefore it is necessary to investigate \(I_{1,1}(r)\), i.e. \(J_1(\lambda)\).

From the formula given in (5), p. 37, by simple algebraic operations we obtain

\[ R_1(x,t;\lambda)= \begin{cases} \displaystyle \frac{1}{n\rho^{\,n-1}\Delta(\rho)} \sum_{i,j=1}^{n} A_{ij}(\rho)\,\omega_j e^{\rho(\omega_i x-\omega_j t)}, & x>t,\\[1.2em] \displaystyle \frac{1}{n\rho^{\,n-1}\Delta(\rho)} \sum_{i,j=1}^{n} B_{ij}(\rho)\,\omega_j e^{\rho(\omega_i x-\omega_j t)}, & x<t, \end{cases} \tag{22} \]

where

\[ \Delta(\rho)=\det\|U_i(e^{\rho\omega_j x})\|_{i,j=1}^{n}, \]

and the determinants \(A_{ij}(\rho)\) and \(B_{ij}(\rho)\) differ from \(\Delta(\lambda)\) in that in place of the \(i\)-th column there stand, respectively, the columns

\[ U_{10}(e^{\rho\omega_j x})|_{l=1}^{n} \quad\text{and}\quad U_{11}(e^{\rho\omega_j x})|_{l=1}^{n}. \]

Substituting (22) into the expression for \(I_1(r)\), passing to the variable \(\lambda=-\rho^n\), and integrating by parts once in the inner integral, we obtain

\[ I_1(r)= \frac{1}{2\pi i}\oint_{\delta_n} \sum_{i\ne j}^{n} \frac{A_{ij}(\rho)}{\Delta(\rho)} \frac{\omega_j}{\omega_i-\omega_j} e^{(\omega_i-\omega_j)\rho}\,d\rho\cdot q(1) - \]

\[ -\frac{1}{2\pi i}\oint_{\delta_n} \sum_{i\ne j}^{n} \frac{A_{ij}(\rho)}{\Delta(\rho)} \frac{\omega_j}{\omega_i-\omega_j}\,d\rho\cdot q(0). \tag{23} \]

Here \(\delta_n\) is an arc on the circle \(|\rho|=r^{1/n}\) of angular measure \(2\pi/n\). We choose \(\delta_n\) so that the following conditions are satisfied:

1. If \(n=2\mu\), one can number the \(\omega_i\) so that

\[ \operatorname{Re}(\rho\omega_i)<0,\quad i=1,\ldots,\mu, \]

\[ \operatorname{Re}(\rho\omega_i)>0,\quad i=\mu+1,\ldots,n, \qquad \rho\in\delta_n. \tag{24} \]

\(^*\) By choosing \(m\) sufficiently large, one can, according to (20), obtain that \(|\widetilde R_m(r)|=o(1)\).

1. For \(n=2\mu+1\), one can divide \(\delta_n\) into two arcs \(\delta_n'\) and \(\delta_n''\) of the same angular measure \(\pi/n\), and number the \(\omega_i\) in such a way that

\[ \begin{aligned} \operatorname{Re}(\rho\omega_i)&<0,\quad i=1,\ldots,\mu,\\ \operatorname{Re}(\rho\omega_i)&>0,\quad i=\mu+1,\ldots,n, \end{aligned} \qquad \rho\in\delta_n', \tag{25} \]

\[ \begin{aligned} \operatorname{Re}(\rho\omega_i)&<0,\quad i=1,\ldots,\mu+1,\\ \operatorname{Re}(\rho\omega_i)&>0,\quad i=\mu+2,\ldots,n, \end{aligned} \qquad \rho\in\delta_n''. \tag{26} \]

Using (22), (24), (25), and (26) for large \(|\rho|\), we obtain, for example \((n=2\mu)\):

\[ \frac{A_{ij}(\rho)}{\rho\Delta(\rho)} = \frac{A_{ij}}{\theta_1(\rho)} + o\!\left(\frac{e^{\rho\gamma_1}}{\rho}\right), \quad \operatorname{Re}(\rho\gamma_1)<0,\ \rho\in\delta_n,\ i\leq\mu,\ j\geq\mu+1, \tag{27} \]

and in the opposite case

\[ \frac{A_{ij}(\rho)}{\rho\Delta(\rho)} = O\!\left(\frac{e^{\rho\gamma_2}}{\rho}\right), \quad \operatorname{Re}(\rho\gamma_2)<0,\ \rho\in\delta_n. \]

Repeating analogous computations for the remaining terms of (23), substituting the obtained expressions into (23), and applying Jordan’s lemma \((^8)\), we obtain (13).

Moscow State University
named after M. V. Lomonosov

Received
9 III 1965

REFERENCES

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