P. B. Naiman

ON THE THEORY OF PERIODIC AND LIMIT-PERIODIC JACOBI MATRICES

(Presented by Academician S. N. Bernstein on 2 XI 1961)

Periodic, as well as limit-periodic, Jacobi matrices and the continued fractions associated with them have for a long time been the subject of a number of investigations (see (¹) and the bibliography cited there). In the present note we develop one algebraic approach to such matrices, which, in combination with operator-theoretic methods (²,³), leads to some new results.

1. By a generalized Jacobi matrix of order \(2m\) we shall mean an infinite matrix
   \[ A=\|a_{jk}\| \quad (j,k=\pm 0,1,2,\ldots) \]
   with complex elements satisfying the conditions \(a_{jk}=0\) for \(|j-k|>m\), \(a_{jk}\ne 0\) for \(|j-k|=m\). For \(m=1\) the generalized Jacobi matrix is the ordinary Jacobi matrix. The following theorem is a discrete analogue of one theorem of Burchnall and Chaundy (⁴) on linear differential expressions.

Theorem 1. If generalized Jacobi matrices \(A\) and \(B\), of orders \(2m\) and \(2n\), respectively, commute, then they satisfy an algebraic relation \(D(A,B)=0\) of degree \(2n\) with respect to \(A\) and of degree \(2m\) with respect to \(B\).

Proof is analogous to that given in (⁴) for the continuous case.

Denote by \(E_n\) the matrix all of whose elements, with the exception of
\[ e_{k,k-n}=e_{k,k+n}=1 \quad (k=\pm 0,1,2,\ldots), \]
are equal to zero. It is easy to see that the spectrum \(S(E_n)\) is the interval \(-2\le \mu\le 2\) and each point \(\lambda\in S(E_n)\) has multiplicity \(2n\). If a generalized Jacobi matrix \(A\) has period \(n\), then it obviously commutes with \(E_n\), so that \(D(A,E_n)=0\). In the simplest case of an ordinary \(n\)-periodic complex-symmetric Jacobi matrix
\[ T=\|t_{jk}\| \quad (t_{jk}=t_{kj};\ j,k=\pm 0,1,2,\ldots) \]
the polynomial \(D(\lambda,\mu)\) is an exact square, so that the following holds:

Theorem 2. Every complex-symmetric \(n\)-periodic Jacobi matrix \(T\) satisfies an algebraic equation of the \(n\)-th degree
\[ P(T)=E_n . \tag{1} \]

If the matrix \(T\) is real, then the coefficients of the polynomial \(P(\lambda)\) are also real. Since the spectrum \(S(E_n)\) is known, relation (1) makes it possible to characterize completely the spectrum \(S(T)\). In particular, in the case of a real matrix one obtains the known result on the structure of the spectrum (¹).

Theorem 3. The spectrum \(S(T)\) of a complex-symmetric \(n\)-periodic Jacobi matrix \(T\) coincides with the full \(\lambda\)-preimage \(\Gamma\) of the interval \(-2\le \mu\le 2\) under the mapping
\[ P(\lambda)=\mu \tag{2} \]
and, consequently, consists of \(n\) algebraic arcs, which in special cases may have common endpoints.

If the matrix \(T\) is real, then for \(-2\le \mu\le 2\) all \(\lambda\)—the roots of equation (2)—are real, so that \(S(T)\) is a system of \(n\) intervals, which in special cases may have common endpoints. Moreover, each point \(\lambda\in S(T)\) has multiplicity two.

Proof. The relation \(S(T)\subset \Gamma\) follows from (1). If the matrix \(T\) is real, then the multiplicity of \(S(T)\) does not exceed two, so that the multiplicity of the spectrum of the left-hand side of (1) is not greater than \(2n\). Since, at the same time, the multiplicity of each point of the spectrum \(S(E_n)\) is \(2n\), all assertions of the theorem pertaining to the real case follow from equality (1).

In the general case of a complex matrix \(T\), it is not difficult, following the well-known Floquet scheme, to establish that a necessary and sufficient condition for the boundedness of at least one of the solutions \(y=\{y_k\}_{k=-\infty}^{\infty}\) of the finite-difference equation

\[ t_{k,k-1}y_{k-1}+t_{k,k}y_k+t_{k,k+1}y_{k+1}=\lambda y_k \tag{3} \]

for a given complex \(\lambda\) is the inequality \(-2\le P(\lambda)\le 2\), so that the set \(\Gamma\) is a system of \(n\) curvilinear stability zones of equation (3). Let now \(\lambda\in\Gamma\). If in this case \(\{y_k\}_{k=-\infty}^{\infty}\in l_2\), then, evidently, the number \(\lambda\) would be an eigenvalue of \(T\). If, however, \(\{y_k\}_{k=-\infty}^{\infty}\bar{\in} l_2\), then

\[ \sum_{k=-\infty}^{\infty}|y_k|^2=\infty \]

and for the sequence of vectors \(z_N\) with coordinates

\[ z_{N,k}= \begin{cases} y_k, & (|k|\le N),\\ 0, & (|k|>N) \end{cases} \]

we shall have

\[ \lim_{N\to\infty}\frac{\|Tz_N-\lambda z_N\|}{\|z_N\|}=0, \]

so that \(\lambda\in S(T)\). Thus, \(\Gamma\subset S(T)\), and the theorem is proved.

1. In the real case we number the gaps in \(S(T)\) by the indices \(1,2,\ldots,n+1\), beginning with the gap extending from \(-\infty\), and take \(\widetilde T\) to be obtained by adding a completely continuous real diagonal matrix \(K\). By the well-known theorem of H. Weyl on completely continuous perturbations, the spectrum \(S(T+K)\) will consist of the system of intervals described in Theorem 3 and of a bounded set of eigenvalues, which can accumulate only at the ends of the gaps. Evidently, the corresponding perturbation

\[ Q=P(T+K)-P(T) \]

of the matrix \(P(T)\) is a completely continuous generalized Jacobi matrix of order \(2n-2\). Thus, the polynomial \(P(\lambda)\) transforms the limit-periodic matrix \(T+K\) into the limit-constant matrix

\[ P(T+K)=E_n+Q. \tag{4} \]

In view of relation (4), the conditions for the finiteness or infiniteness of the set of points of the spectrum \(S(T+K)\) lying in the union of all odd or even gaps of the spectrum \(S(T)\) coincide with the conditions for the finiteness or infiniteness of the set of points of the spectrum of the limit-constant matrix \(E_n+Q\), situated to the left of the point \(\mu=-2\) or to the right of the point \(\mu=2\). The determination of these latter conditions is carried out as in (³) and leads to the following results.

Denote by \(\sigma_r\) the sum of the absolute values of all elements of the \(r\)-th row of the matrix \(Q\), except for its diagonal element \(q_r\), and put

\[ \omega'_r=\sigma_r-q_r,\qquad \omega''_r=\sigma_r+q_r. \]

Further, by \(S'\) and \(S''\) denote the set of spectral points introduced by the perturbation \(K\) into the union of all odd and, respectively, even gaps of the spectrum \(S(T)\). Without loss of generality, we shall assume that

\[ (-1)^n t_{12}\cdot t_{23}\cdot \ldots \cdot t_{n,n+1}>0. \]

Theorem 4. If

\[ \limsup_{|r|\to\infty} r^2\omega'_r < \frac{n^2}{4}, \]

then \(S'\) is finite. If

\[ \limsup_{|r|\to\infty} r^2\omega''_r < \frac{n^2}{4}, \]

then \(S''\) is finite. If one of the \(2n\) series

\[ \sum_{r=0}^{\pm\infty} q_{j+nr}\quad (j=1,2,\ldots,n) \tag{5} \]

diverges to \(+\infty\), then \(S'\) is infinite. If one of the \(2n\) series (5) diverges to \(-\infty\), then \(S''\) is infinite.

Using Theorem 4, one can obtain conditions expressed only in terms of the elements \(k_r\) of the perturbation \(K\). Thus, for example, when \(k_r=o\!\left(\frac{1}{r^2}\right)\), the set \(S' + S''\) will be finite. In the particular case \(n=2\), assuming for definiteness that \(t_{12}-t_{10}>0\), we arrive in this way at the following result.

Theorem 5. If one of the two series

\[ \sum_{-\infty}^{+\infty} k_{2r-1} \]

consists only of positive terms and diverges, then the perturbation \(K\) introduces into the internal gap of the 2-periodic matrix \(T\) an infinite set of eigenvalues. If, however, one of the two series

\[ \sum_{-\infty}^{+\infty} k_{2r} \]

consists only of positive terms and diverges, then the perturbation \(K\) introduces an infinite set of eigenvalues into the union of the external gaps.

An analogous result holds for arbitrary \(n\). In those cases when almost all \(k_r \geq 0\) (or \(\leq 0\)), the spectrum introduced by the perturbation \(K\) cannot accumulate at the left (right) ends of the gaps.

Received
21 X 1961

References Cited

1. Ya. L. Geronimus, Izv. AN SSSR, ser. matem., 5, 203 (1943).
2. I. M. Glazman, DAN, 118, 423 (1958).
3. P. B. Naĭman, Izv. Vyssh. uchebn. zaved., No. 1(8), 129 (1959).
4. E. L. Ince, Ordinary Differential Equations, 1939.