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This note studies the spectral properties of regular quasi-differential operators in L2(a,b), generally non-self-adjoint, defined by a self-adjoint quasi-differential expression and linearly independent boundary conditions. It introduces a normalized Wronskian determinant formed from solutions satisfying endpoint boundary conditions, and proves that its zeros are eigenvalues, with the converse established for nonreal eigenvalues through the characteristic matrix function. The paper also derives representations of the Green function and an auxiliary operator transformation, yielding criteria related to spectral half-plane localization and self-adjointness. For simple dissipative quasi-differential operators, it gives a necessary and sufficient condition for completeness of the system of root subspaces in terms of equality in a determinant inequality involving the Gram matrix and spectral data.
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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).