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MATHEMATICS
E. M. DZHABRAILOVA
ON THE SPECTRUM OF A SINGULAR NON-SELF-ADJOINT DIFFERENTIAL EQUATION OF SECOND ORDER WITH RESPECT TO A COMPLEX WEIGHT FUNCTION
(Presented by Academician L. S. Pontryagin on 17 III 1964)
Consider the following boundary-value problem in the class \(L^2(0,\infty)\):
\[ -y''+p(x)y-\lambda q(x)y=0, \tag{1} \]
\[ y'(0)-\theta y(0)=0,\qquad 0\leq x<\infty; \]
here \(p(x), q(x)\) are complex-valued functions summable on every finite interval \([0,a]\), \(a>0\); \(\theta\) is a complex number, \(\lambda\) is a parameter.
Denote by \(D_\theta\) the totality of all functions \(y(x)\) from \(L^2(0,\infty)\) satisfying the following conditions:
\(1^\circ.\) \(y\) and \(y'\) are absolutely continuous on every finite interval \([0,a]\), \(a>0\).
\(2^\circ.\) \(l(y)=-y''+p(x)y\in L^2(0,\infty)\).
\(3^\circ.\) \(y'(0)-\theta y(0)\).
By \(D_\theta^{0}\) denote the totality of functions \(y(x)\) from \(L^2(0,\infty)\) such that \(qy\in L^2(0,\infty)\) and \(y'(0)-\theta y(0)=0\).
Define a pair of operators \(L_\theta\) and \(Q\): the domain of definition of \(L_\theta\) is \(D_\theta\), and for \(y\in D_\theta\)
\[ L_\theta y=l(y), \]
while the domain of definition of the operator \(Q\) is \(D_\theta^{0}\), and for \(y\in D_\theta^{0}\)
\[ Qy=q(x)y. \]
The number \(\lambda\) is called an eigenvalue of the pair of operators \((L_\theta,Q)\) if in \(D_\theta\cap D_\theta^{0}\) there exists a function \(y\not\equiv 0\) such that
\[ L_\theta y=\lambda Qy; \]
in this case \(y\) is called an eigenfunction of the pair \((L_\theta,Q)\), corresponding to the eigenvalue \(\lambda\).
The number \(\lambda\) is called a regular point of the pair of operators \((L_\theta,Q)\) if the operator \(R_{\lambda Q}=(L_\theta-\lambda Q)^{-1}\) exists, is defined on the whole space \(L^2(0,\infty)\), and is bounded. In this case the operator \(R_{\lambda Q}\) is called the resolvent of the pair of operators \((L_\theta,Q)\). All nonregular points are called points of the spectrum of the pair of operators \((L_\theta,Q)\).
In the present note we set forth some results on the spectrum of the pair of operators \((L_\theta,Q)\) in the case \(p(x)\to\infty\) and under certain assumptions concerning \(p(x)\) and \(q(x)\). In deriving these results we used the methods presented by M. A. Naimark in the papers \((^1,^2)\).
Let \(p(x)\to\infty\) as \(x\to\infty\), and suppose, moreover, that the following conditions are satisfied:
a) \(|p'|=O(|p|^\alpha)\) as \(x\to\infty\), where \(0<\alpha<{}^3/{}_2\);
b) \(|p'|=O(|p|')\), \(|p''|=O(|p|'')\) as \(x\to\infty\);
c) \(|q|=O(|p|^{1/4})\) as \(x\to\infty\);
d) \(q'|p|^{-1}\) is a summable function.
Then the equation \(l(y)=\lambda q'(x)y\) has two linearly independent solutions \(y_1(x,\lambda)\), \(y_2(x,\lambda)\), analytic in \(\lambda\) in the entire \(\lambda\)-complex plane, such that, as \(x\to\infty\),
\[ y_1=\rho^{-1/2}e^\xi[1+o(1)], \qquad y_1'=\rho^{1/2}e^\xi[1+o(1)]; \tag{2a} \]
\[ y_2=\rho^{-1/2}e^{-\xi}[1+o(1)], \qquad y_2'=\rho^{1/2}e^{-\xi}[1+o(1)], \tag{2b} \]
where \(\rho=\sqrt{p-\lambda q}\), \(\xi=\displaystyle\int_{x_0}^{x}\rho\,dx\). For \(x\) sufficiently large, \(|p|>|\lambda||q|\), and \(\arg\sqrt{p-\lambda q}\) will be determined, by continuity, by a single-valued choice of its value for fixed \(x\).
Suppose that for \(x\) sufficiently large
\[ 0\leq \arg p\leq \gamma,\qquad \gamma<\pi . \]
Then, using the asymptotic formulas, it is not difficult to show that \(y_1\notin L_2(0,\infty)\), \(y_2\in L_2(0,\infty)\). Moreover, from conditions a) and c) it follows that \(qy_2\in L_2(0,\infty)\). Therefore an eigenfunction of the pair of operators \((L_\theta,Q)\) can only be \(cy_2\), and since it must be that \(y_2\in D_\theta\), the corresponding eigenvalues \(\lambda\) are determined from the equation
\[ y_2'(0,\lambda)-\theta y_2(0,\lambda)=0. \]
\(y_2'(0,\lambda)-\theta y_2(0,\lambda)\) is an analytic function in the entire \(\lambda\)-complex plane, and therefore the zeros of the equation either fill the entire \(\lambda\)-plane or else form a countable set having no finite limit points.
Next, using formulas (2a) and (2b), we prove that for those \(\lambda\) for which \(y_2'(0,\lambda)-y_2(0,\lambda)\neq0\), the operator \((L_\theta-\lambda Q)^{-1}\) is bounded, is defined on the whole space \(L^2(0,\infty)\), and is an operator of Carleman type.
Thus we arrive at the following result:
Theorem 1. Let \(\rho(x)\to\infty\) as \(x\to\infty\) and let conditions a), b), c), d) be fulfilled. Suppose, moreover, that for \(x\) sufficiently large \(0\leq \arg p\leq\gamma\), where \(\gamma<\pi\).
Then, if the number \(\lambda\) is not an eigenvalue, it does not belong to the spectrum of the pair of operators \((L_\theta,Q)\). The set of eigenvalues either fills the entire \(\lambda\)-plane or else forms a countable set having no finite limit points. For those \(\lambda\) that do not belong to the spectrum, the resolvent
\[ R_\lambda Q=(L_\theta-\lambda Q)^{-1} \]
is an integral operator with kernel \(K(x,y,\lambda)\) satisfying the conditions:
\[ \int_0^\infty |K(x,y,\lambda)|^2\,dy<\infty,\qquad \int_0^\infty |K(x,y,\lambda)|^2\,dx<\infty. \tag{3} \]
The author has not succeeded in constructing an example of a boundary-value problem (1) whose eigenvalues would fill the entire \(\lambda\)-plane; nevertheless, such a phenomenon probably does occur.
Suppose now that, in addition to conditions a), b), d), the following conditions are also fulfilled:
e) \(\operatorname{Re}(p^{1/2})=o(p^{-1/2})\) as \(x\to\infty\);
f) \(\displaystyle\int^\infty |p|^{-1/2}\,dx<\infty,\)
and instead of condition c) the condition
c′) \(q(x)\) is a bounded function.
Then from the asymptotic formulas (2) it follows that \(y_1 \in L^2(0,\infty)\) and \(y_2 \in L^2(0,\infty)\).
Instead of the pair of operators \((L_\theta,Q)\) one should consider another pair \((\hat L,\hat Q)\), defined as follows. Denote by \(\hat D\) the set of all functions \(y\) from \(L^2(0,\infty)\) satisfying conditions \(1^0\) and \(2^0\), and by \(\hat D^*\) the set of functions analogous to \(\hat D\), but constructed for the adjoint differential expression \(l^*(y)=-y''+\overline{p(x)}y\). If \(y\in\hat D,\ z\in\hat D^*\), then from Lagrange’s formula
\[ \int_\alpha^\beta l(y)\overline{z}\,dx-\int_\alpha^\beta y\,\overline{l^*(z)}\,dx=[y,z]_\alpha^\beta, \]
where
\[ [y,z]=y(x)\overline{z'(x)}-y'(x)\overline{z(x)}, \]
it follows that \([y,z]_0^\infty\) exists. Choose two functions \(z_1,z_2\in\hat D^*\) such that the determinant
\[ \Delta(\lambda)= \left| \begin{array}{cc} [y_2,z_1]_0^\infty & [y_1,z_2]_0^\infty\\ [y_2,z_1]_0^\infty & [y_2,z_2]_0^\infty \end{array} \right| \]
does not vanish identically. Denote by \(\hat{\hat D}\) the set of all functions \(y\in\hat D\) satisfying the conditions \([y,z_1]_0^\infty=0,\ [y,z_2]_0^\infty=0\), and by \(\hat L\) the operator with domain \(\hat{\hat D}\) such that \(Ly=l(y)\) for \(y\in\hat{\hat D}\). Denote by \(\hat D'\) the set of functions \(y\) from \(L^2(0,\infty)\) for which \(qy\in L^2(0,\infty)\) and \([y,z_1]_0^\infty=0,\ [y,z_2]_0^\infty=0\), and by \(\hat Q\) the operator with domain \(\hat D'\) and, for \(y\in\hat D'\), \(Qy=qy\).
Theorem 2. Suppose that conditions a), b), c′), d), e) are satisfied. Then the spectrum of the pair of operators \((\hat L,\hat Q)\) consists of no more than a countable number of eigenvalues having no finite limit points. For values of \(\lambda\) not belonging to the spectrum, the resolvent \((\hat L-\lambda\hat Q)^{-1}\) is an integral operator with a Hilbert–Schmidt kernel.
We now replace condition e) by the condition
\[ \text{e′)}\qquad \int^\infty |p|^{-1/2}\,dx=\infty. \]
We shall assume that \(q(x)\ge a\), where \(a\) is some positive number.
Theorem 3. Suppose that conditions a), b), c′), d), e′) are satisfied and \(q(x)\ge a,\ a>0\). Then all points of the real axis are points of the spectrum of the pair of operators \((L_\theta,Q)\). If \(\lambda,\ \operatorname{Im}\lambda\ne0\), is not an eigenvalue, then it does not belong to the spectrum of the pair of operators \((L_\theta,Q)\). The set of eigenvalues may fill one of the half-planes \(\operatorname{Im}\lambda>0,\ \operatorname{Im}\lambda<0\), or else form a countable set having no finite limit points. For all values of \(\lambda\) not belonging to the spectrum, the resolvent \(R_{\lambda Q}=(L_\theta-\lambda Q)^{-1}\) is an integral operator with a kernel satisfying condition (3).
I express my sincere gratitude to Prof. M. A. Naimark, under whose guidance this work was carried out.
Received
4 III 1964
References
- M. A. Naimark, DAN, 85, No. 1 (1952).
- M. A. Naimark, Tr. Moskovsk. matem. obshch., 3, 181 (1954).