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N. I. KALENPEROV
A BOUNDARY-VALUE PROBLEM WITH A COMPLEX WEIGHT FUNCTION
(Presented by Academician P. S. Novikov, 2 VII 1964)
In the present note we investigate the completeness of the system of eigen- and associated elements of certain classes of non-self-adjoint differential operators.
- Consider in the Hilbert space \(L^2(-\infty,+\infty)\) the differential equation
\[ -y''+q(x)y=\lambda p(x)y, \tag{1} \]
where \(p(x)=q_1(x)+iq_2(x)\) is a complex function of the real parameter \(x\), and \(q(x), q_1(x), q_2(x)\) are real functions summable on every finite interval of the real axis.
Let us write equation (1) in the form
\[ Ay=\lambda p(x)y, \tag{2} \]
where
\[ A=-\frac{d^2}{dx^2}+q(x). \]
For \(q(x)\geq 1\), \(A^{-1}\) is a self-adjoint positive operator (see \((^3)\), p. 75).
Put \(y=A^{-1/2}z\), after which equation (2) is written as
\[ AA^{-1/2}z=\lambda\bigl(q_1(x)A^{-1/2}z+iq_2(x)A^{-1/2}z\bigr). \tag{3} \]
Acting on equation (3) with the operator \(A^{-1/2}\) and taking into account that \(A\) and \(A^{-1/2}\) commute, we obtain:
\[ z=\lambda\bigl(A^{-1/2}q_1(x)A^{-1/2}z+iA^{-1/2}q_2(x)A^{-1/2}z\bigr), \]
or
\[ z=\lambda Lz, \]
where we have put
\[ L=L_R+iL_I,\qquad L_R=A^{-1/2}q_1(x)A^{-1/2},\qquad L_I=A^{-1/2}q_2(x)A^{-1/2}. \]
It is easy to verify that if \(q_j(x)\geq 0\) \((j=1,2)\), then the real and imaginary components of the operator \(L\) are positive definite. We note that, for
\[ \lim_{|x|\to\infty}\frac{q(x)}{|x|^\alpha}\geq C>0 \quad\text{and}\quad \alpha\geq \frac{2}{3}, \]
\(A^{-1}\) is an operator of Hilbert–Schmidt type (see \((^6)\), p. 6).
Assume that \(q_1(x)\) is a bounded function. Then the operator \(q_1(x)A^{-1}\), being the product of the bounded \(q_1(x)\) and the Hilbert–Schmidt operator \(A^{-1}\), is an operator of Hilbert–Schmidt type. Using the inequality (see \((^4)\), p. 12)
\[ N^2\bigl(A^{-1/2}q_1(x)A^{-1/2}\bigr) \leq \frac{1}{2}N^2\bigl(A^{-1}q_1(x)\bigr) + \frac{1}{2}N^2\bigl(q_1(x)A^{-1}\bigr), \]
where \(N(A)\) is the absolute norm of the operator \(A\), we conclude that the operator \(A^{-1/2}q_1(x)A^{-1/2}\) is of Hilbert–Schmidt type. By an analogous argument we obtain that \(A^{-1/2}q_2(x)A^{-1/2}\) is a Hilbert–Schmidt operator under the condition that \(q_2(x)\) is bounded.
Consequently, \(L\) is an operator of Hilbert–Schmidt type.
Applying Theorem 2 of V. B. Lidskii \((^2)\), we arrive at the following result.
Theorem 1. If \(q(x) \ge 1\) and a) \(q_1(x)\), \(q_2(x)\) are nonnegative and bounded functions, b)
\[ \lim_{|x|\to\infty}\frac{q(x)}{|x|^\alpha}\ge C>0 \]
for \(\alpha \ge \dfrac{2}{3}\), then the eigenvectors and associated vectors of the operator \(L\), corresponding to nonzero points of the spectrum, form a system complete in the range of values of the operator \(L\).
Remark. If \(q_1(x)>0\) and \(q_2(x)>0\), then the eigenvectors and associated vectors of the operator \(L\), corresponding to nonzero points of the spectrum, form a system complete in \(L^2(-\infty,+\infty)\).
- We now consider an equation of the form
\[ -y''+q(x)y=\lambda p_1(x)y+\frac{p_2(x)}{\lambda}y+p_3(x)y, \tag{4} \]
where \(p_j(x)=q_j(x)+iq'_j(x)\), \(j=1,2,3\), are complex functions of the real parameter \(x\), and \(q(x)\), \(q^j(x)\), \(q'_j(x)\) are real functions summable on every finite interval of the real axis.
We shall show that, under certain restrictions on \(q(x)\) and \(p_j(x)\), the system of eigenvectors and associated vectors of equation (4) is complete in \(L^2(-\infty,+\infty)\).
In what follows we shall use a theorem of D. E. Allakhverdiev \((^5)\).
Let \(A,B,C\) be completely continuous operators in a Hilbert space \(\mathfrak H\); suppose \((E-A)^{-1}\) exists and is bounded; \(A\) and \(C\) have finite order, respectively \(\rho_1\) and \(\rho_2\). Suppose, moreover, that the resolvents of the operators \(A\) and \(C\) are bounded respectively on rays \(\alpha_1,\alpha_2,\ldots,\alpha_n\) and \(\beta_1,\beta_2,\ldots,\beta_n\), such that the angle between neighboring rays is respectively less than \(\pi/\rho_1\) and \(\pi/\rho_2\), and the operators \(A^*\) and \(C^*\) are invertible. Then the system of eigenvectors and associated vectors of the equation
\[ y=\lambda Ay+\frac{B}{\lambda}y+Cy \]
is 2-fold complete in the Hilbert space \(\mathfrak H\).
Recall that the order of an operator \(A\) is the lower bound of the numbers \(\alpha\) for which
\[ \sum_{n=0}^{\infty}\varepsilon_n^\alpha(A)<+\infty, \]
where
\[ \varepsilon_0(A)=\|A\|;\qquad \varepsilon_n(A)=\inf_{A^{(n)}}\|A-A^{(n)}\|\quad\text{for } n\ge 1; \]
\(A^{(n)}\) runs through the set of all \(n\)-dimensional operators acting in \(\mathfrak H\) (see \((^4)\); there D. E. Allakhverdiev also proves that \(\varepsilon_n(A)=1/\sqrt{\lambda_{n+1}}\), where \(\{\lambda_k\}\) are the eigenvalues of the operator \(AA^*\)). Thus, the order of the operator \(A\) (in the sense of \((^4)\)) coincides with the order of the operator \((AA^*)^{1/2}\) in the sense of M. V. Keldysh.
We transform equation (4) to the form:
\[ z=\lambda A^{-1/2}p_1(x)A^{-1/2}z+\frac{A^{-1/2}p_2(x)A^{-1/2}}{\lambda}z +A^{-1/2}p_3(x)A^{-1/2}z. \]
For \(\alpha>\dfrac{2}{3}\), \(A^{-1}\) is an operator of finite order less than 2. As we noted, the order of the operator \(A^{-1/2}p_j(x)A^{-1/2}\) coincides with the order of the operator
\[ \left(A^{-1/2}p_j(x)A^{-1/2}A^{-1/2}p_j(x)A^{-1/2}\right)^{1/2}, \]
and the order of the latter, according to the result—
there, by M. V. Keldysh, does not exceed the order of the operator \(A^{-1}\) (see (4)). Thus, \(A^{-1/2}p_j(x)A^{-1/2}\) is an operator of finite order less than 2.
Suppose that \(q_j(x)\) and \(q'_j(x)\) have a definite sign. Let, for example, \(q_j(x)\geq 0\) and \(q'_j(x)\leq 0\). Then the values of the quadratic form \((A^{-1/2}p_j(x)A^{-1/2}f,f)\) lie in the second quadrant, and for \(\rho_1<2\) and \(\rho_2<2\), on the basis of a lemma of V. B. Lidskii (see (7)), on rays issuing from the origin and not falling into the second quadrant, the resolvent \((E-A^{-1/2}p_1(x)A^{-1/2})^{-1}\) and the resolvent \((E-A^{-1/2}p_3(x)A^{-1/2})^{-1}\) exist and are bounded.
Thus, we have proved the following theorem.
Theorem 2. If \(q(x)\geq 1\) and
a) \(\displaystyle \lim_{|x|\to\infty}\frac{q(x)}{|x|^\alpha}\geq C>0\) for \(\alpha>\frac{2}{3}\);
b) \(\overline{p_j(x)}\ne 0\) almost everywhere and \(q_j(x)\), \(q'_j(x)\) are bounded functions of definite sign;
c) the equation \(Az=p_1(x)z\) has no solutions other than the zero solution,
then the system of eigen- and associated elements of equation (4) is complete in \(L^2(-\infty,+\infty)\).
The author expresses deep gratitude to Prof. M. A. Naimark and D. E. Allakhverdiev for their attention to the present work.
Institute of Mathematics and Mechanics
Academy of Sciences of the Azerbaijan SSR
Received
21 VI 1964
REFERENCES
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