Abstract Generated abstract
This paper studies expansions in eigenfunctions for a non-self-adjoint system of differential equations of arbitrary even order on the half-axis in the space of vector functions. Under exponential decay assumptions on the matrix coefficients and regularity conditions on the boundary forms, it constructs asymptotic fundamental solutions, analyzes the spectrum and resolvent of the associated operator, and compares it with auxiliary boundary-value problems on finite intervals. The main result gives a representation of the resolvent kernel as a sum over the finite discrete eigenvalues plus an integral over the continuous spectrum, with absolute and uniform convergence. This representation is then used to indicate an eigenfunction expansion for suitable vector functions and an analogue of Parseval’s equality.
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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).