Abstract Generated abstract
This note studies the spectrum of a singular second order non-self-adjoint boundary-value problem on the half-line with complex coefficients and a complex weight, formulated as a pair of operators \((L_\theta,Q)\) in \(L^2(0,\infty)\). Under growth and regularity assumptions on \(p\) and \(q\), asymptotic solutions of the differential equation are used to characterize eigenvalues through boundary conditions and to determine when non-eigenvalues are regular points. The results show that, in one class, the spectrum consists only of eigenvalues, either countable without finite limit points or possibly filling the plane, with a Carleman-type resolvent kernel. Further cases are treated by modified boundary conditions or by assumptions forcing the entire real axis into the spectrum, while nonreal regular points again have integral resolvents with square-integrability properties.
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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).