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MATHEMATICS
A. V. KUZHEL
ON THE SPECTRUM OF A REGULAR QUASI-DIFFERENTIAL OPERATOR
(Presented by Academician A. A. Dorodnitsyn, 16 I 1964)
In this note a connection is established between the eigenvalues of a (in general non-self-adjoint) regular quasi-differential operator and the zeros of a certain “normalized” Wronskian determinant. In addition, a condition is obtained for the completeness of the system of root subspaces of dissipative \(\left({}^{1}\right)\) quasi-differential operators.
1. Consider in the space \(L^{2}(a,b)\) a regular quasi-differential operator \(A\), defined by the self-adjoint quasi-differential expression
\[ l(y)=\sum_{k=0}^{n}(-1)^{k}\bigl(p_{n-k}y^{(k)}\bigr)^{(k)} \]
and by the boundary conditions
\[ \sum_{k=1}^{2n} a_{jk}y^{[k-1]}(a)+\sum_{k=1}^{2n} b_{jk}y^{[k-1]}(b)=0 \quad (j=1,2,\ldots,2n). \tag{1} \]
The equations of the system (1) are naturally assumed to be linearly independent.
Let \(r_a\) and \(r_b\) be the ranks, respectively, of the matrices \(\|a_{ik}\|\) and \(\|b_{ik}\|\). In what follows we consider the case when \(r_a r_b\ne 0\), \(r_a+r_b=2n\).
Since \(\operatorname{rang}\|a_{ik}\|=r_a\), there exist \(n_1=2n-r_a\) linearly independent solutions \(u_s(x,\lambda)\) \((s=1,2,\ldots,n_1)\) of the equation
\[ l(u)=\lambda u, \tag{2} \]
which satisfy the initial conditions
\[ \sum_{k=1}^{2n} a_{rk}u_s^{[k-1]}(a,\lambda)=0 \quad (r=1,2,\ldots,2n;\quad s=1,2,\ldots,n_1). \tag{3} \]
Analogously, there exist \(n_2=2n-r_b\;(=2n-n_1)\) linearly independent solutions \(u_s(x,\lambda)\) \((s=n_1+1,\ldots,2n)\) of equation (2), which satisfy the initial conditions
\[ \sum_{k=1}^{2n} b_{rk}u_s^{[k-1]}(b,\lambda)=0 \quad (r=1,2,\ldots,2n;\quad s=n_1+1,\ldots,2n). \tag{4} \]
We shall call the Wronskian determinant \(W(\lambda)\) of the solutions \(u_s(x,\lambda)\) \((s=1,2,\ldots,2n)\), which satisfy the initial conditions (3) and (4), normalized. (We note that here and below it is assumed that the Wronskian determinant \(W(\lambda)\) of the functions \(u_s(x,\lambda)\) is formed from the quasi-derivatives of these functions. In that case \(W(\lambda)\) does not depend on \(x\).)
Theorem 1. The zeros of the normalized Wronskian determinant are eigenvalues of the quasi-differential operator \(A\).
As for the converse assertion, it has been established only for \(n=1\). In the general case \((n>1)\) the converse assertion is proved for non-real eigenvalues (Theorem 2).
- Denote by \(W_k(x,\lambda)\) the Wronskian determinant of the functions
\(u_s(x,\lambda)\) \((s=1,\ldots,k-1,\ k+1,\ldots,2n)\), and let
\[ v_k(x,\lambda)=(-1)^k\frac{W_k(x,\lambda)}{W(\lambda)} \qquad (k=1,2,\ldots,2n) \]
be the adjoint system of solutions of equation (2). Then, under conditions (3) and (4), the Green function \(\mathscr{G}(x,t,\lambda)\) of the operator \(A\) has the form
\[ \mathscr{G}(x,t,\lambda)= \begin{cases} \displaystyle \sum_{k=1}^{n_1} u_k(x,\lambda)v_k(t,\lambda), & (t\ge x),\\[6pt] \displaystyle -\sum_{k=n_1+1}^{2n} u_k(x,\lambda)v_k(t,\lambda), & (t\le x). \end{cases} \]
Now put
\[ r_{mk}(\lambda)= \begin{cases} [\,u_m,u_k\,]_a, & (k=1,2,\ldots,n_1),\\ [\,u_m,u_k\,]_b, & (k=n_1+1,\ldots,2n); \end{cases} \]
\[ s_{mk}(\lambda)= \begin{cases} [\,v_k,v_m\,]_b, & (k=1,2,\ldots,n_1),\\ [\,v_k,v_m\,]_a, & (k=n_1+1,\ldots,2n), \end{cases} \]
where \(u_j=u_j(x,\lambda)\), \(v_j=v_j(x,\lambda)\), and
\[ [f,g]=\sum_{k=1}^{n}\{f^{[k-1]}\overline{g}^{[2n-k]}-f^{[2n-k]}\overline{g}^{[k-1]}\}. \]
Then the auxiliary transformation
\(B_\lambda=iR_\lambda-iR_\lambda^*+2\,\operatorname{Im}\lambda\,R_\lambda^*R_\lambda\)
of the operator \(A\) can be represented in the form
\[ B_\lambda=\sum_{k,i=1}^{2n}(\,\cdot,\overline{v}_k)J_{ik}\overline{v}_i \qquad \left((f,g)=\int_a^b f\overline{g}\,dx\right), \]
where the matrix \(J=\|J_{ki}\|\) is related to the matrices
\(R(\lambda)=\|r_{ik}(\lambda)\|\), \(S(\lambda)=\|s_{ki}(\lambda)\|\) and to the Gram matrix \(G\) of the functions \(v_k(x,\lambda)\) \((k=1,2,\ldots,2n)\) by the relation
\[ J=\frac{1}{2\,\operatorname{Im}\lambda}\,G^{-1}[S(\lambda)R(\lambda)-E] \]
(\(E\) is the identity matrix).
Using now the properties of the transformation \(B_\lambda\) (2) and the relation
\[ \operatorname{Im}(Af,f)=\tfrac12(B_\lambda\varphi,\varphi) \qquad (\varphi=(A-\lambda I)f), \]
we arrive at the following assertions:
I. Let, for some \(\lambda_0\) \((\operatorname{Im}\lambda_0\ne0)\), the matrix \(J\) be Hermitian nonnegative (Hermitian nonpositive). Then the spectrum of the operator \(A\) is situated in the half-plane \(\operatorname{Im}\lambda\ge0\) \((\operatorname{Im}\lambda\le0)\).
II. The rank of the matrix \(S(\lambda)R(\lambda)-E\) does not depend on \(\lambda\). Moreover, if the operator \(A\) is self-adjoint, then for every \(\lambda\) \((\operatorname{Im}\lambda\ne0)\)
\[ S(\lambda)R(\lambda)=E. \tag{5} \]
Conversely, if relation (5) holds for at least one nonreal \(\lambda\), then the operator \(A\) is self-adjoint.
- The preceding results make it possible to compute the characteristic matrix-function \(\chi_A(\lambda)\) of the operator \(A\), which for unbounded operators was introduced in (3). As a result we find that
\[ \chi_A(\lambda)=W(\lambda)F^{-1}(\lambda), \tag{6} \]
where \(W(\lambda)\) is the normalized Wronskian determinant, and \(F(\lambda)\) is a certain matrix function whose determinant is a bounded function depending on the values of \([u_k,u_i]_x\) at the ends of the interval \([a,b]\). Using the results of [3] and relation (6), we arrive at the following assertion:
Theorem 2. The nonreal eigenvalues of a regular quasi-differential operator \(A\) turn the normalized Wronskian determinant into zero.
- We now suppose that the operator \(A\) under consideration is simple [4] and dissipative [1]. This means that the operator \(A\) has no invariant subspaces on which \(A^*=A\), and \(\operatorname{Im}(Af,f)\geqslant 0\) for every \(f\in D_A\). In this case the operator \(A\) has no real eigenvalues, its spectrum \(\{\lambda_k\}_{k=1}^{\infty}\) is situated in the half-plane \(\operatorname{Im}\lambda>0\), and
\[ \sum_{k=1}^{\infty}\operatorname{Im}\lambda_k=\infty. \]
(We note that a simple but not dissipative operator may have real eigenvalues.) Moreover,
\[ \det(E-\tau GG^*)\leqslant \prod_{k=1}^{\infty}\left|\frac{\lambda_k-i}{\overline{\lambda}_k-i}\right|^2, \tag{7} \]
where \(G\) is the Gram matrix of the vectors \(v_k(x,-i)\) \((k=1,2,\ldots,2n)\), and \(\tau\) is the matrix of transition from the system of vectors \(\overline{v_k(x,-i)}\) \((k=1,2,\ldots,2n)\) to the \(\alpha\)-basis [3] of the operator \(A\). The matrix \(\tau\) is a rectangular matrix with \(2n\) columns and \(r\) rows, where \(r=\operatorname{rang}[S(-i)R(-i)-E]\).
Theorem 3. The system of root subspaces of a simple dissipative quasi-differential operator \(A\) is complete in the space \(L^2(a,b)\) if and only if the equality sign holds in relation (7).
It is clear that the indicated criterion is applicable only to non-self-adjoint operators.
Uman
Pedagogical Institute
Received
22 VII 1963
CITED LITERATURE
- I. M. Glazman, UMN, 13, no. 3 (81) (1958).
- A. V. Kuzhel, UMN, 16, no. 3 (99) (1961).
- A. V. Kuzhel, DAN, 125, No. 1 (1959).
- A. V. Kuzhel, DAN, 119, No. 5 (1958).