Full Text
MATHEMATICS
M. G. GIMADISLAMOV
ON EXPANSION IN EIGENFUNCTIONS OF A NON-SELF-ADJOINT DIFFERENTIAL OPERATOR OF EVEN ORDER IN THE SPACE OF VECTOR FUNCTIONS
(Presented by Academician P. S. Aleksandrov on 28 X 1961)
In the present paper we study expansion in eigenfunctions of a non-self-adjoint system of differential equations of arbitrary even order on the half-axis \([0,\infty)\).
Consider a system of differential expressions of order \(2n\), which we write in the form:
\[ l(y)=y^{(2n)}+P_2(x)y^{(2n-2)}+P_3(x)y^{(2n-3)}+\cdots+P_{2n}y, \tag{1} \]
where \(y(x)=(y_1(x),\ldots,y_k(x))\) is a vector function; \(P_\nu(x)\), \(\nu=2,\ldots,2n\), are complex-valued matrix functions of order \(k\), summable on the interval \([0,\infty)\).
Denote by \(D\) the set of all vector functions \(y(x)\in L_k^2(0,\infty)\) such that: 1) the derivatives \(y^{(\nu)}(x)\), \(\nu=1,2,\ldots,2n-1\), exist and are absolutely continuous on every finite interval \([0,b]\), \(b>0\); 2) \(l(y)\in L_k^2(0,\infty)\).
Denote by \(D_A\) the set of all vector functions \(y(x)\in D\) satisfying the boundary conditions
\[ u_\nu(y)=A_{\nu,2n-1}y^{(2n-1)}(0)+A_{\nu,2n-2}y^{(2n-2)}(0)+\cdots+A_{\nu,0}y(0)=0, \]
\[ \nu=1,2,\ldots,n, \tag{2} \]
where \(A_{\nu,j}\) are complex matrices of order \(k\). We define the operator \(L_A\) as follows: its domain of definition is \(D_A\), and for \(y\in D_A\)
\[ L_Ay=l(y). \tag{3} \]
The operator \(L_A^*\), adjoint to \(L_A\), is constructed in an analogous way for the differential expression adjoint to (1),
\[ l^*(z)=z^{(2n)}+(P_2^*(x)z)^{(2n-2)}-(P_3^*(x)z)^{(2n-3)}+\cdots+P_{2n}^*z \tag{4} \]
and for the boundary conditions adjoint to (2),
\[ v_\nu(z)=B_{\nu,2n-1}z^{(2n-1)}(0)+B_{\nu,2n-2}z^{(2n-2)}(0)+\cdots+B_{\nu,0}z(0)=0, \]
\[ \nu=1,\ldots,n. \tag{5} \]
Put \(\rho^{2n}=-\lambda\). Let \(\omega_1,\ldots,\omega_{2n}\) be the roots of degree \(2n\) of \(-1\); divide the complex \(\rho\)-plane into \(2n\) equal sectors \(S_k\), \(k=0,1,\ldots,2n-1\), defined by the inequality
\[ \frac{k\pi}{n}<\arg\rho<\frac{(k+1)\pi}{n}. \]
In each sector \(S_k\) one can choose an ordering of the numbers \(\omega_1,\ldots,\omega_{2n}\) such that, for \(\rho \in S_k\),
\[ \operatorname{Re}(\rho \omega_1) \leqslant \operatorname{Re}(\rho \omega_2) \leqslant \cdots \leqslant \operatorname{Re}(\rho \omega_{2n}). \]
Denote by \(T_k\) and \(T_{k-1}\) the boundaries of the sector \(S_k\). Let the matrix functions \(P_\nu(x)\) satisfy the additional condition
\[ e^{\varepsilon_2 x}\left|P_\nu(x)\right| \leqslant C_\nu . \tag{6} \]
Consider the matrix equation
\[ Y^{(2n)} + P_2(x)Y^{(2n-2)}+\cdots+P_{2n}(x)Y=\lambda Y. \tag{7} \]
It can be shown that equation (7) has linearly independent solutions \(Y_j(x,\rho)\), \(j=1,2,\ldots,2n\), holomorphic with respect to \(\rho\) for \(\rho\in S_j\) and having the asymptotics:
\[ \text{as } x\to\infty \qquad Y_j^{(\nu)}(x,\rho)=\rho^\nu e^{\rho\omega_j x}\left[\omega_j^\nu\cdot 1+o(1)\right] \tag{8} \]
uniformly with respect to \(\rho\in S_j\),
\[ \text{as } \rho\to\infty \qquad Y_j^{(\nu)}(x,\rho)=\rho^\nu e^{\rho\omega_j x}\left[\omega_j^\nu\cdot 1+O\!\left(\frac{1}{\rho}\right)\right] \tag{9} \]
uniformly with respect to \(x\in[0,\infty)\).
Similarly, we construct solutions \(Z_j(x,\rho)\), \(j=1,\ldots,2n\), for the matrix equation
\[ Z^{(2n)}+(ZP_2(x))^{(2n-2)}-(ZP_3(x))^{(2n-3)}+\cdots+ZP_{2n}(x)=\lambda Z, \tag{10} \]
normalized in a definite manner.
Denote
\[ A(\rho)= \left| \begin{array}{ccc} u_1(Y_1)&\cdots u_1(Y_{n-1})&u_1(Y_n)\\ \cdots&\cdots&\cdots\\ u_n(Y_1)&\cdots u_n(Y_{n-1})&u_n(Y_n) \end{array} \right|, \]
\[ \widetilde A(\rho)= \left| \begin{array}{ccc} u_1(Y_1)&\cdots u_1(Y_{n-1})&u_1(Y_{n+1})\\ \cdots&\cdots&\cdots\\ u_n(Y_1)&\cdots u_n(Y_{n-1})&u_n(Y_{n+1}) \end{array} \right|. \tag{11} \]
We shall assume, for simplicity, that \(A(\rho)\ne0\), \(\widetilde A(\rho)\ne0\) for \(\rho\in T_k\), and that the eigenvalues of the operator \(L_A\) are simple.
Theorem 1. The spectrum of the operator \(L_A\) is continuous, for even \(n\), on the positive semiaxis (for odd \(n\), respectively, on the negative semiaxis) and is discrete in the entire complex \(\lambda\)-plane. The eigenvalues form a finite set. For values of \(\lambda\) not belonging to the spectrum, the resolvent \((L_A-\lambda 1)^{-1}\) of the operator \(L_A\) is a bounded integral operator with kernel \(K(x,t,\lambda)\) satisfying the conditions
\[ \int_0^\infty |K(x,t,\lambda)|^2\,dt<\infty,\qquad \int_0^\infty |K(x,t,\lambda)|^2\,dx<\infty. \]
Consider the auxiliary boundary-value problem on the interval \([0,b]\):
\[ l(y)=\lambda y;\qquad u_\nu(y)=0,\quad \nu=1,2,\ldots,n; \]
\[ u_{\mu b}(y)=y^{(\mu-1)}(b)=0,\quad \mu=1,2,\ldots,n. \tag{12} \]
For sufficiently large \(b\), to each eigenvalue \(\lambda_1,\ldots,\lambda_r\) of the operator \(L_A\) there corresponds exactly one eigenvalue \(\lambda_1(b),\ldots,\lambda_r(b)\) of the boundary-value problem (12), such that \(\lambda_k(b)\to\lambda_k\) as \(b\to\infty\). All
the remaining eigenvalues of the auxiliary boundary-value problem as \(b \to \infty\) satisfy the following asymptotic relations:
\[ \lambda=-\left(\rho_m^{(j)}\right)^{2n}, \qquad \rho_m^{(j)}\omega_n=\frac{m\pi i}{b}+\frac{1}{2b}\ln \xi_j\left(\frac{m\pi i}{\omega_n b}\right)+\frac{1}{b}O(1), \]
\[ j=1,2,\ldots,k, \tag{13} \]
uniformly with respect to \(\rho\) in the domain \(\left|\operatorname{Re}(\rho\omega_n)\right|\leqslant \varepsilon_1,\ \varepsilon_1<\varepsilon_2,\ 0\leqslant |\rho|\leqslant N\), \(\xi_j(\rho)\) are the roots of the algebraic equation of order \(k\) with respect to \(\xi\),
\[ \theta_k\xi^k+\theta_{k-1}\xi^{k-1}+\cdots+\theta_0=0, \tag{14} \]
where
\[ \theta_k= \left| \begin{array}{cccc} u_1(Y_1)&\cdots&u_1(Y_{n-1})&u_1(Y_n)\\ \cdot&\cdots&\cdot&\cdot\\ u_n(Y_1)&\cdots&u_n(Y_{n-1})&u_n(Y_n) \end{array} \right| \cdot \left| \begin{array}{cccc} 1&1&\cdots&1\\ \cdot&\cdots&\cdot&\cdot\\ \omega_n^{\,n-1}\cdot 1&\omega_{n+2}^{\,n-1}\cdot 1&\cdots&\omega_{2n}^{\,n-1}\cdot 1 \end{array} \right|, \]
\[ \theta_0= \left| \begin{array}{cccc} u_1(Y_1)&\cdots&u_1(Y_{n-1})&u_1(Y_{n+1})\\ \cdot&\cdots&\cdot&\cdot\\ u_n(Y_1)&\cdots&u_n(Y_{n-1})&u_n(Y_{n+1}) \end{array} \right| \cdot \left| \begin{array}{cccc} 1&\cdots&1\\ \cdot&\cdots&\cdot\\ \omega_{n+1}^{\,n-1}\cdot 1&\cdots&\omega_{2n}^{\,n-1}\cdot 1 \end{array} \right|. \]
Let \(y_j(x)\) be an eigenfunction of the operator \(L_A\) corresponding to the eigenvalue \(\lambda_j,\ j=1,2,\ldots,r\); then
\[ y_j(x)=\left[-\sum Y_i(x,\rho)\,T_{i\nu}u_\nu(Y_n)+Y_n(x)\right]c_j, \tag{15} \]
where \(c_j\) is a \(k\)-dimensional vector and
\[ \left( \begin{array}{ccc} T_{11}&\cdots&T_{1,n-1}\\ \cdot&\cdots&\cdot\\ T_{n-1,1}&\cdots&T_{n-1,n-1} \end{array} \right) \left( \begin{array}{ccc} u_1(Y_1)&\cdots&u_1(Y_{n-1})\\ \cdot&\cdots&\cdot\\ u_{n-1}(Y_1)&\cdots&u_{n-1}(Y_{n-1}) \end{array} \right)=1. \]
Let \(z_j(t),\ j=1,2,\ldots,r\), be the eigenfunctions of the adjoint boundary-value problem; then as \(b\to\infty\)
\[ \frac{y_j(x,b)z_j^*(t,b)} {\displaystyle\int_0^b (y_j,z_j)\,dx} = \frac{y_j(x)z_j(t)} {\displaystyle\int_0^\infty (y_j,z_j)\,dx} +o(1) \tag{16} \]
uniformly with respect to \(x,t\) in the square \(0\leqslant x,t\leqslant c,\ c>0\). The eigenfunctions of the boundary-value problem (12), corresponding to the eigenvalues (13), have the following asymptotics as \(b\to\infty\):
\[ y\left(x,\rho_m^{(j)}\right) = \left\{ -\sum_{i,k=1}^{n-1} Y_i\left(x,\rho_m^{(j)}\right)T_{ik} \left[u_k(Y_n)-a\xi_j\left(\rho_m^{(j)}\right)u_k(Y_{n+1})\right] +\right. \]
\[ \left. +Y_n\left(x,\rho_m^{(j)}\right) -a\xi_j\left(\rho_m^{(j)}\right)Y_{n+1}\rho\left(x,\rho_m^{(j)}\right) \right\} r_j\left(\rho_m^{(j)}\right)+o(1), \tag{17} \]
where \(a\ne 0\) is a constant depending only on \(\omega_n,\omega_{n+1},\ldots,\omega_{2n}\); \(r_j(\rho)\) is a uniquely determined \(k\)-dimensional vector.
The eigenfunctions of the boundary-value problem adjoint to problem (12) are constructed with the aid of solutions of equation (10) and have the form
\[ z\left(x,\rho_m^{(j)}\right) = \left\{ -\sum_{i,k=1}^{n-1} z_i^*\left(x,\rho_m^{(j)}\right)T'_{ik} \left[v_k(z_n^*)-\overline{a}\,\overline{\xi}'_j\,v_k(z_{n+1}^*)\right] +\right. \]
\[ \left. +z_n^*\left(x,\rho_m^{(j)}\right) -\overline{a}\,\overline{\xi}'_j\left(\rho_m^{(j)}\right) z_{n+1}^*\left(x,\rho_m^{(j)}\right) \right\} r'_j\left(\rho_m^{(j)}\right)+o(1), \tag{18} \]
where \(T'_{ik},\ \xi'_j,\ r'_j\) are constructed in the same way as \(T_{ik},\ \xi_j,\ r_j\).
Further,
\[ \frac{1}{b}\int_{0}^{b}\bigl(y(x,\rho_m^{(j)}),\,z(x,\rho_m^{(j)})\bigr)\,dx = -a[\xi_j+\xi'_j](r_j,r'_j)+o(1) \tag{19} \]
as \(b\to\infty\).
Let \(K_b(x,t,\lambda)\) be the kernel of the resolvent of the auxiliary boundary-value problem (12); then, as \(b\to\infty\),
\[ K_b(x,t,\lambda)=K(x,t,\lambda)+o(1) \tag{20} \]
uniformly with respect to \(x,t\) in every finite square \(0\le x,t\le c\), \(c>0\).
Theorem 2. Suppose that conditions (6) and (11) are satisfied, and suppose that the operator \(L_A\) has only simple eigenvalues \(\lambda_1=-\rho_1^{2n},\ldots,\lambda_r=-\rho_r^{2n}\); let \(y_1,\ldots,y_r\) be the corresponding eigenfunctions. Let \(K(x,t,\lambda)\) be the kernel of the resolvent of the operator \(L_A\). Then, for any point \(\lambda\) not belonging to the spectrum of the operator \(L_A\),
\[ K(x,t,\lambda) = \sum_{j=1}^{r} \frac{y_j(x)z_j^{*}(t)} {\displaystyle\int_{0}^{\infty}(y_j,z_j)\,dx} + \frac{\omega_n}{\pi i} \int_{T_k} \sum_{j=1}^{k} \frac{\widetilde y_j(x,\rho)\,\widetilde z_j^{*}(t,\rho)} {(\rho^{2n}+\lambda)a(\xi_j+\xi'_j)(r_j,r'_j)} \,d\rho, \tag{21} \]
where
\[ \widetilde y_j(x,\rho) = \left\{ -\sum_{i,k=1}^{n-1}Y_i(x,\rho)\,T_{ik} \bigl[u_k(Y_n)-a\xi_j u_k(Y_{n+1})\bigr] +Y_n(x,\rho) -a\xi_jY_{n+1}(x,\rho) \right\}r_j, \]
\[ \widetilde z_j^{*}(t,\rho) = r_j^{\prime *} \left\{ -\sum_{l,\mu=1}^{n-1} \bigl[v_\mu(z_n)-a\xi'_j v_\mu(z_{n+1})\bigr] T'_{\mu,l}z_l(t,\rho) +z_n(t,\rho)-a\xi'_jz_{n+1}(t,\rho) \right\}, \]
where the integral on the right converges absolutely and uniformly with respect to \(x,t\) in the domain \(0\le x,t<\infty\).
Denote by \(\widehat G_A\) the totality of all vector-functions \(g(x)\) satisfying the following conditions: 1) \(g(x)\), \(l(g)\) are summable on the interval \([0,\infty)\); 2) \(g^{(\nu)}(x)\), \(\nu=1,\ldots,2n-1\), exist and are absolutely continuous on every finite interval \([0,b]\); 3) \(u_\nu(g)=0\), \(\nu=1,2,\ldots,n\). Then, with the aid of formula (21), one can obtain an expansion of the function \(g(x)\) in the eigenfunctions of the operator \(L_A\) and an analogue of Parseval’s equality.
In conclusion the author expresses gratitude to M. A. Naimark for advice and comments.
Moscow State University
named after M. V. Lomonosov
Received
27 X 1961
REFERENCES
- M. A. Naimark, Tr. Mosk. matem. obshch., 3, 181 (1954).
- M. A. Naimark, Linear Differential Operators, 1954.
- V. M. Funtakov, DAN, 132, No. 4 (1960).
- M. G. Gimadislamov, DAN, 140, No. 1 (1961).