Abstract Generated abstract
This paper studies the spectrum of a self-adjoint first-order differential system on the real line with a constant diagonal principal part and continuous Hermitian, slowly varying coefficients, including a form related to the Dirac operator considered by Titchmarsh. Using splitting principles and localization of test functions, the authors reduce discreteness and boundedness questions for the continuous spectrum to estimates on unit intervals and then to a constant-coefficient problem analyzed by Fourier transform. For two-dimensional systems they obtain necessary and sufficient conditions for discreteness of the spectrum and for the continuous spectrum to be bounded above or below, expressed in terms of the matrix entries. In a symmetric special case these criteria are reformulated through the limiting behavior of the eigenvalues of the coefficient matrix, yielding a spectral alternative for the continuous part.
Full Text
UDC 517.925:517.948.35:513.88
MATHEMATICS
R. S. ISMAGILOV, V. V. MARTYNOV
A CRITERION FOR THE DISCRETENESS OF THE SPECTRUM OF A SELF-ADJOINT SYSTEM OF FIRST-ORDER DIFFERENTIAL EQUATIONS WITH SLOWLY VARYING COEFFICIENTS
(Presented by Academician I. M. Vinogradov on VII 24, 1965)
We consider the self-adjoint operator in \(L_2(-\infty,+\infty)\)
\[ D_0 y=i\Lambda_0 y'+Q_0(x)y(x) =i\begin{pmatrix}\lambda_1,&0\\[2pt]0,&\lambda_2\end{pmatrix}y' +\begin{pmatrix}p,&q\\[2pt]\bar q,&r\end{pmatrix}y, \tag{1} \]
where the real numbers \(\lambda_1\cdot\lambda_2\ne0\), and the matrix-function \(Q_0(x)\) is continuous everywhere on the axis and Hermitian in the unitary space \(E_2\). If \(\lambda_1=-\lambda_2=1\) and \(p(x)\equiv r(x)\), then, in essence, we are dealing with the operator
\[ T_0y=I_0y'+A_0(x)y(x) =\begin{pmatrix}0,&1\\[2pt]-1,&0\end{pmatrix}y' +\begin{pmatrix}a,&b\\[2pt]b,&c\end{pmatrix}y \tag{2} \]
in the Euclidean space \(R_2\), since \(T_0=UD_0U^{-1}\) for
\(U=\dfrac{1}{\sqrt2}\begin{pmatrix}-i,&i\\[2pt]1,&1\end{pmatrix}\).
The operator \(T_0\) was studied in a number of papers by E. C. Titchmarsh in connection with the relativistic Dirac equation.
In note \((^2)\) certain sufficient conditions were proposed for discreteness of the spectrum of the operator \(D_0\). The main content of the present note is the proof of a necessary and sufficient condition for discreteness of the spectrum, valid in the class of operators with slowly varying coefficients.
Theorem. 1) Let the matrix \(Q_0(x)\) vary slowly, i.e.
\[ \|Q_0(x)-Q_0(t)\|\le \mathrm{const}* \tag{3} \]
for all \(|x-t|\le1\). Then the spectrum of the operator is discrete if and only if
\[ \lim_{|x|\to\infty} \left[\,|q(x)|\sqrt{-4\lambda_1\lambda_2} -\left|\lambda_2p(x)-\lambda_1r(x)\right|\,\right]=+\infty . \tag{4} \]
2) If the matrix \(Q_0(x)\) varies slowly, then the continuous spectrum of the operator \(D_0\) is bounded above (it may also be empty) if and only if
\[ \lim_{\lambda\to+\infty}\ \lim_{|x|\to\infty} \left[\,|q(x)|\sqrt{-4\lambda_1\lambda_2} -\left|(\lambda_2p-\lambda_1r)+\lambda(\lambda_1-\lambda_2)\right|\,\right]=+\infty . \tag{5} \]
3) If the matrix \(Q_0(x)\) varies slowly, then the continuous spectrum of the operator \(D_0\) is bounded below (it may also be empty) if and only if
\[ \lim_{\lambda\to-\infty}\ \lim_{|x|\to\infty} \left[\,|q(x)|\sqrt{-4\lambda_1\lambda_2} -\left|(\lambda_2p-\lambda_1r)+\lambda(\lambda_1-\lambda_2)\right|\,\right]=+\infty . \tag{6} \]
Corollary. If \(\lambda_1=-\lambda_2=1\) and \(p(x)\equiv r(x)\), then conditions (4), (5), (6) admit a reformulation in terms of the eigenvalues of the matrix \(Q_0(x)\) (and hence also of the matrix \(A_0(x)\)):
\[ \text{* \(\|B\|\), \(\mu[B]\), and \(\nu[B]\) will denote, respectively, the Euclidean norm, the smallest and the largest eigenvalues of the Hermitian matrix \(B\).} \]
1′) the spectrum is discrete if and only if
\[ \lim_{|x|\to\infty}\nu[A_0(x)]=+\infty,\qquad \lim_{|x|\to\infty}\mu[A_0(x)]=-\infty; \tag{7} \]
2′) the continuous spectrum is bounded above if and only if
\[ \lim_{|x|\to\infty}\nu[A_0(x)]=+\infty \quad\text{and the function }\mu[A_0(x)]\text{ is bounded above;} \tag{8} \]
3′) the continuous spectrum is bounded below if and only if
\[ \lim_{|x|\to\infty}\mu[A_0(x)]=-\infty \quad\text{and the function }\nu[A_0(x)]\text{ is bounded below.} \tag{9} \]
Thus, in this case the spectral alternative is valid (cf. \((^1)\), p. 174): the continuous part of the spectrum of the operator \(D_0\) either is absent, or else is unbounded at least on one side.
Proceeding to the proof of the theorem, consider, as in \((^2)\), the more general operator
\[ Dy=i\Lambda y' + Q(x)y(x), \tag{10} \]
where the principal part contains a constant real diagonal invertible matrix \(\Lambda\), and the matrix function \(Q(x)\) is continuous and Hermitian in the unitary \(E_k\) \((k\geqslant 2)\).
Lemma 1 (splitting principle; cf. \((^1)\), §§ 1, 2). The distance from a real point \(\lambda\) to the continuous spectrum of the operator (10) is equal to
\[ \lim_{r\to+\infty}\left( \inf_{y\in K(r,\infty)} \frac{\|Dy-\lambda y\|}{\|y\|} \right), \tag{11} \]
where \(K(r,\infty)\) denotes the class of finite piecewise-smooth vector functions, the compact support of each of which is situated outside the interval \((-r,+r)\).
Corollary 1. In order that the spectrum of the operator \(D\) be discrete, it is necessary and sufficient that
\[ \lim_{r\to+\infty}\left( \inf_{y\in K(r,\infty)} \frac{\|Dy\|}{\|y\|} \right)=+\infty. \tag{12} \]
Corollary 2. In order that the continuous part of the spectrum of the operator \(D\) be bounded above, it is necessary and sufficient that
\[ \lim_{\lambda\to+\infty}\lim_{r\to+\infty}\left( \inf_{y\in K(r,\infty)} \frac{\|Dy-\lambda y\|}{\|y\|} \right)=+\infty. \tag{13} \]
The condition for boundedness of the continuous spectrum below and for its two-sided boundedness is equivalent, respectively, to the requirements that the expression (11) tend to infinity as \(\lambda\to-\infty\) and as \(|\lambda|\to\infty\).
Lemma 2 (splitting principle; cf. \((^1)\), p. 66). Put
\[ d(\lambda;\Delta)=d(\lambda;a,b)= \inf_{y\in N(\Delta)} \frac{\|Dy-\lambda y\|^2}{\|y\|^2}, \]
where the class \(N(\Delta)=N(a,b)\) consists of finite piecewise-smooth vector functions vanishing outside the interval \(\Delta=[a,b]\). For any number \(h\) \((0<h<b-a\leqslant\infty)\) there exists in \(\Delta\) an interval \(\Delta'\) of length \(h\) such that
\[ d(\lambda;\Delta)\leqslant d(\lambda;\Delta')\leqslant d(\lambda;\Delta)+\frac{32}{h^2}\|\Lambda\|^2. \tag{14} \]
Corollary 1′. The spectrum of the operator \(D\) is discrete if and only if
\[ \lim_{|x|\to\infty} d(0;x,x+1)=+\infty . \tag{15} \]
Corollary 2′. The continuous part of the spectrum of the operator \(D\) is bounded above if and only if
\[ \lim_{\lambda\to+\infty}\ \lim_{|x|\to\infty} d(\lambda;x,x+1)=+\infty . \tag{16} \]
The conditions for boundedness of the continuous spectrum below and for its two-sided boundedness coincide with (16) for \(\lambda\to-\infty\) and \(|\lambda|\to\infty\), respectively.
Suppose now that the matrix \(Q\) varies slowly. Then \(d(0;\alpha,\alpha+1)=m(\alpha)+O(1)\), where
\[ m(\alpha)= \inf_{y\in N(\alpha,\alpha+1)} \left[ \int_\alpha^{\alpha+1} \left|i\Lambda y'(t)+Q(\alpha)y(t)\right|^2\,dt \bigg/ \int_\alpha^{\alpha+1}|y(t)|^2\,dt \right]. \]
Consequently, the spectrum is discrete if \(\lim_{|\alpha|\to\infty} m(\alpha)=+\infty\). But, on the other hand, by virtue of inequality (14), \(m(\alpha)=n(\alpha)+O(1)\), where
\[ n(\alpha)= \inf_{y\in N(-\infty,+\infty)} \left[ \int_{-\infty}^{+\infty} \left|i\Lambda y'(t)+Q(\alpha)y(t)\right|^2dt \bigg/ \int_{-\infty}^{+\infty}|y(t)|^2dt \right]. \]
Thus the spectrum is discrete if and only if \(\lim_{|\alpha|\to\infty} n(\alpha)=+\infty\). To compute \(n(\alpha)\), we apply the Fourier transform:
\[ \begin{aligned} n(\alpha) &=\inf_z \left[ \int_{-\infty}^{+\infty} \left|Q(\alpha)z(t)-t\Lambda z(t)\right|^2dt \bigg/ \int_{-\infty}^{+\infty}|z(t)|^2dt \right] \\ &=\inf_{t\in(-\infty,+\infty)} \mu\left[(Q(\alpha)-t\Lambda)^2\right] \\ &=\inf_t\min_{|\xi|=1} \left[ (\Lambda^2\xi,\xi)t^2 -2(\operatorname{Re}\Lambda Q\xi,\xi)t +(Q^2(\alpha)\xi,\xi) \right]. \end{aligned} \]
Thus we finally obtain: the spectrum of the operator \(D\) is discrete if and only if
\[ \lim_{|\alpha|\to\infty} \left[\min_{|\xi|=1} X(\alpha;\xi)\right]=+\infty, \tag{17} \]
where
\[ X(\alpha;\xi)= (\Lambda^2\xi,\xi)(Q^2(\alpha)\xi,\xi) -\left[\operatorname{Re}(\Lambda Q\xi,\xi)\right]^2 . \]
It is not difficult to show that for \(k=2\) condition (17) is equivalent to condition (4) of the theorem.
Remark. The lemmas and condition (17) remain valid if in the principal part of the operator \(D\) there stands an arbitrary (not necessarily diagonal) constant Hermitian matrix \(\Lambda\).
In conclusion, the authors express their gratitude to Prof. M. A. Naimark.
Moscow Institute of Physics and Technology
Received
21 VII 1965
References Cited
- I. M. Glazman, Direct Methods of Qualitative Spectral Analysis of Singular Differential Operators, 1963.
- V. V. Martynov, DAN, 165, No. 5 (1965).