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UDC 517.941.92
EXPANSIONS OF MATRIX FUNCTIONS IN SOLUTIONS OF A SPECTRAL PROBLEM
N. A. ALIEV
As is known, the application of the residue method to the study of one-dimensional mixed problems^1) makes it possible to establish the necessary and sufficient conditions for its correctness. The justification of the residue method is very closely connected with the question of expanding functions in a series in the residues of the solution of the corresponding spectral problem [5]. The question of expanding functions in a series in the residues of solutions of spectral problems has been well studied in a number of works in the case of simple roots of the characteristic equation in the sense of Birkhoff [2, 3, 5].
In the case of multiple roots this question still remains open. Consequently, the question of justifying the residue method for solving mixed problems for multiple roots has likewise not been solved.
In this connection, in the present paper a theorem is proved on the validity of the formula mentioned above, connected with a system of ordinary linear differential equations of first order, in the case of multiple roots^2) of the characteristic equation in the sense of Birkhoff [5]. In particular, if the coefficients of system (1) do not contain negative powers of the parameter \(\lambda\), then from formula (23) there follows a formula for expanding an arbitrary vector-function in a series in the residues of the solution of the spectral problem (1), (2).
Consider a system of ordinary linear differential equations of first order of the form
\[ \frac{dy}{dx}=a(x,\lambda)y+f(x),\quad x\in[a,b] \tag{1} \]
with boundary conditions^3)
\[ \alpha y(a,\lambda)+\beta y(b,\lambda)=0, \tag{2} \]
where \(a(x,\lambda)\), \(f(x)\), \(\alpha\), \(\beta\), and \(y(x,\lambda)\) are square matrices of order \(n\), and, except for the last, all the matrices are known, while the first of them has the form
\[ a(x,\lambda)=\lambda a(x)+\sum_{\nu=1}^{l}\lambda^{-\nu}a^{(\nu)}(x); \tag{3} \]
\(l\geq 1\) is a fixed integer; \(\lambda\) is a complex parameter.
^1) Partial differential equations together with initial and boundary conditions will be called a mixed problem.
^2) The case of simple roots of Birkhoff’s equation was studied by M. L. Rasulov [5].
^3) The matrix coefficients \(\alpha\) and \(\beta\) in the boundary condition (2) are constant (i.e., the elements of these matrices do not depend on \(\lambda\)) matrices of order \(n\).
By \(n_i\) \((i=\overline{1,k})\) we shall denote the multiplicity of the root \(\theta_i(x)\) \((i=\overline{1,k})\) of the characteristic equation in the sense of Birkhoff [5], composed for equation (1). It is easy to see that
\[ n_1+n_2+\cdots+n_k=n . \tag{4} \]
For simplicity of exposition, suppose that the matrix \(a(x)\) is given in canonical form, i.e.
\[ a(x)=\bigl(a_{ij}(x)\bigr)_{i,j=1}^k,\quad a_{ij}(x)\equiv 0\quad (i\ne j), \tag{5} \]
\[ a_{jj}(x)= \begin{pmatrix} \theta_j(x) & 1 & 0 & \cdots & 0\\ 0 & \theta_j(x) & 1 & \cdots & 0\\ \cdot & \cdot & \cdot & \cdot & \cdot\\ \cdot & \cdot & \cdot & \cdot & \cdot\\ 0 & 0 & 0 & \theta_j(x) \end{pmatrix}, \tag{6} \]
where \(a_{jj}(x)\) is a square matrix of order \(n_j\). Taking into account the decomposition (5), (6), all the other coefficients can likewise be decomposed into matrix blocks of the form
\[ a^{(\nu)}(x)=\bigl(a_{ij}^{(\nu)}(x)\bigr)_{i,j=1}^k\quad (\nu=\overline{1,l}), \]
\[ a_{ij}^{(\nu)}(x)=\bigl(a_{ijrs}^{(\nu)}(x)\bigr)_{r,s=1}^{n_i,n_j}\quad (ij=\overline{1,k}). \tag{7} \]
For the further investigation we shall assume that the following restrictions hold.
\(1^\circ\). The matrices \(a(x)\) and \(a^{(\nu)}(x)\) \((\nu=\overline{1,l})\), which are the coefficients of the polynomial (3), have all continuous derivatives of all orders for \(x\in[a,b]\).
\(2^\circ\). Any two roots of the characteristic equation in the sense of Birkhoff are either distinct\(^1\) (for all values \(x\in[a,b]\)) or identically coincide.
\(3^\circ\). The canonical form (5)—(6) for the matrix \(a(x)\) does not change when \(x\in[a,b]\)\(^2\), and the number indicating the multiplicity of the root \(\theta_i(x)\) satisfies the condition
\[ n_i\le 2\quad (i=\overline{1,k})\quad k>1, \tag{8} \]
where equality is attained for at least one value of \(i\) (otherwise the roots would be distinct, which was studied by M. L. Rasulov [5]).
\(4^\circ\). The roots \(\vartheta_i(x)\) \((i=\overline{1,k})\) of the characteristic equation are different from zero; their arguments and the arguments of their differences do not depend on \(x\in[a,b]\).
\(5^\circ\). The boundary conditions (2) are linearly independent.
It is easy to see that from conditions \(1^\circ\)—\(3^\circ\) there follows the fulfillment of conditions \((i)\)—\((vi)\) of [4]. By obtaining inequality (3.1.12) from [5], and taking condition \(4^\circ\) into account, it is easy to show that conditions \((vii)\) [4] are also fulfilled in the entire complex \(\lambda\)-plane. Then the homogeneous equation corresponding to equation (1) has a fundamental solution whose asymptotic representation\(^3\) is obtained—
\(^1\) Distinct roots are denoted by \(\Theta_i(x)\) \((i=\overline{1,k})\).
\(^2\) This condition is equivalent to saying that the number of distinct roots of the Birkhoff equation [5] coincides with the number of distinct elementary divisors of the matrix \(a(x)\), and this number remains constant for all values \(x\in[a,b]\).
\(^3\) When the roots of the characteristic equation are simple, the asymptotics for a single higher-order equation was obtained by Birkhoff [2], and for systems of first-order equations by Tamarkin [3].
using the method of [4]\(^1\). Denoting this fundamental solution by \(\tilde y(x,\lambda)\), we can write for it the following representation\(^2\):
\[ \tilde y(x,\lambda)= \left( \sum_{\nu=0}^{m-1}\lambda^{-\nu} g^{(\nu)}(x)+ \frac{E(x,\lambda)}{\lambda^m} \right) e^{\lambda\int_a^x \Theta(\xi)\,d\xi}, \tag{9} \]
where \(\Theta(x)\) is a diagonal matrix whose diagonal elements are \(\theta_i(x)\) \((i=\overline{1,k})\), and each of them is repeated as many times as its multiplicity. All the coefficients are infinitely differentiable on \(x\in[a,b]\), and the last function \(E(x,\lambda)\), moreover, is bounded for large values of \(|\lambda|\). Taking condition \(4^\circ\) into account, the complex \(\lambda\)-plane can be divided into a finite number of such sectors in each of which, under a suitable enumeration of the roots \(\theta_i(x)\) \((i=\overline{1,k})\), condition (vii) of [4] would hold. This ensures the asymptotic representation for the fundamental solution (9) in the whole complex \(\lambda\)-plane\(^3\). Taking condition \(5^\circ\) into account, by the method of variation of constants for the solution of problem (1), (2) we obtain the following representation:
\[ y(x,\lambda)=\int_a^b G(x,\xi,\lambda)f(\xi)\,d\xi, \tag{10} \]
where
\[ \begin{aligned} G(x,\xi,\lambda) &=g(x,\xi,\lambda)-\tilde y(x,\lambda)(\alpha \tilde y(a,\lambda)+{}\\ &\quad{}+\beta \tilde y(b,\lambda))^{-1}(\alpha g(a,\xi,\lambda)+\beta g(b,\xi,\lambda)) \end{aligned} \tag{11} \]
is the Green matrix for problem (1), (2), and its principal part \(g(x,\xi,\lambda)\) has the form
\[ g(x,\xi,\lambda)=\pm \frac12\,\tilde y(x,\lambda)\tilde y^{-1}(\xi,\lambda) \begin{cases} +\ \text{for } a\le \xi \le x \le b,\\ -\ \text{for } a\le x \le \xi \le b. \end{cases} \tag{12} \]
Taking into account that some elements in the first coefficients of expression (9) are identically zero, for the determinant of this matrix we have
\[ \det \tilde y(x,\lambda) = \lambda^{-\frac12\sum_{i=1}^k n_i(n_i-1)} e^{\lambda\sum_{j=1}^k\int_a^x \vartheta_j(\xi)\,d\xi} \left\{ c(x)+\frac{E(x,\lambda)}{\lambda} \right\}, \tag{13} \]
where \(c(x)\ne0\) is a continuous function for \(x\in[a,b]\), and \(E(x,\lambda)\) is a continuous and bounded function for large values of \(|\lambda|\). Further, note that the negative integer
\[ -\frac12\sum_{i=1}^k n_i(n_i-1) \]
replaces the number \(p\) in work\(^4\) [4]. As is seen from (13), taking condition \(3^\circ\) into account, the matrix inverse to (9) can be represented in the form
\(^1\) Everywhere asymptotic representations are understood in the sense of Poincaré [1].
\(^2\) Since all coefficients of the formal solution are infinitely differentiable, the number \(m\) can be taken sufficiently large; we shall assume that the number \(m\) satisfies the condition given in the second theorem of [4], i.e. \(m>3p\).
\(^3\) We note that, when passing from one sector to another, the columns of the fundamental solution \(\tilde y(x,\lambda)\) may change places.
\(^4\) See Theorem 1 of [4]; in that work \(p\) is a positive integer, because the case of a small parameter \(\varepsilon\) is studied.
\[ \tilde y^{-1}(x,\lambda)=\lambda e^{-\lambda\int_a^x \Theta(\xi)\,d\xi} \left(\sum_{\nu=0}^{m-1}\lambda^{-\nu}\eta^{(\nu)}(x)+\frac{E(x,\lambda)}{\lambda^m}\right), \tag{14} \]
where \(\eta^{(\nu)}(x)\) are infinitely differentiable functions. Taking into account the identity
\[ \tilde y(x,\lambda)\tilde y^{-1}(x,\lambda)=I, \tag{15} \]
it is easy to establish a relation between \(\eta^{(\nu)}(x)\) and the matrices \(g^{(\nu)}(x)\). We give some of these expressions, which will be needed later:
\[ g^{(0)}(x)\eta^{(0)}(x)=0,\qquad g^{(0)}(x)\eta^{(1)}(x)+g^{(1)}(x)\eta^{(0)}(x)=I. \tag{16} \]
Using the representation (9) to find the spectrum of problem (1), (2), we arrive at the equation
\[ \Delta(\lambda)=\operatorname{opr}\bigl(\alpha \tilde y(a,\lambda)+\beta \tilde y(b,\lambda)\bigr)= \]
\[ =\operatorname{opr}\left( \alpha\left\{\sum_{\nu=0}^{m-1}\lambda^{-\nu}g^{(\nu)}(a)+\frac{E(a,\lambda)}{\lambda^m}\right\} +\beta\left\{\sum_{\nu=0}^{m-1}\lambda^{-\nu}g^{(\nu)}(b)+ \right.\right. \]
\[ \left.\left. +\frac{E(b,\lambda)}{\lambda^m}\right\} e^{\lambda\int_a^b \Theta(\xi)\,d\xi} \right)=0. \tag{17} \]
It is easy to see that equation (17) has the form (3.1.32) of [5].
Further, it is evident that from conditions \(1^\circ\)—\(5^\circ\) there follow conditions \(1^\circ,3^\circ\) of Rasulov [5], introduced in the third chapter. As for condition \(2^\circ\), it is violated in the case considered by us. To conditions \(1^\circ\)—\(5^\circ\) we add the following.
\(6^\circ\). Suppose that, after expanding the left-hand side of equation (17), similarly to (3.1.37) in [5], the resulting extreme coefficients \(M_1\) and \(M_\sigma\) are nonzero, and the intermediate coefficients are asymptotically constant.
Taking into account the condition \(6^\circ\) introduced by us and noting that, in the proof of Theorem 7 of [5], the simplicity of the roots of the characteristic equation was not used, it becomes clear that the assertion of Theorem 7 remains valid also for our case, i.e., for equation (17).
We now impose the following restriction on the right-hand side of equation (1).
\(7^\circ\). Let \(f(x)\) have continuous derivatives up to order two (inclusive), and let the function itself vanish at the ends of the interval, i.e., \(f(a)=f(b)=0\). After substituting (11) into (10), denote the principal part (corresponding to (12)) by \(y_0(x,\lambda)\). Then, taking into account condition \(7^\circ\) and integrating by parts the first term of \(y_0(x,\lambda)\) twice and the second once, we obtain\(^1\)
\[ y_0(x,\lambda)=g^{(0)}(x)\Theta^{-1}(x)\eta^{(0)}(x)f(x)- \]
\(^1\) The boundedness of \(E(x,\lambda)\) for large values of \(|\lambda|\) is obtained with the aid of the linear transformation that we made in the numerator of the Green’s function for each of its elements, which have been studied in detail by M. L. Rasulov [5].
EXPANSION OF MATRIX-FUNCTIONS IN SOLUTIONS OF A SPECTRAL PROBLEM
\[
-\lambda^{-1}\left\{g^{(0)}(x)\left[\Theta^{-1}(x)\left(\Theta^{-1}(x)\eta^{(0)}(x)f(x)\right)' + \Theta^{-1}(x)\eta^{(1)}(x)f(x)\right]+\right.
\]
\[
\left.+\,g^{(1)}(x)\Theta^{-1}(x)\eta^{(0)}(x)f(x)\right\}+\frac{E(x,\lambda)}{\lambda^2}.
\tag{18}
\]
In exactly the same way, taking condition \(7^\circ\) into account, by integration by parts for the remaining terms entering into (10), we obtain expressions of the form\(^1\) \(E(x,\lambda)/\lambda^2\). Then the asymptotic representation (18) may be adopted for the solution of the spectral problem (1), (2).
The next aim will be to compute the coefficients in formula (18). Substituting the fundamental solution (9) into the homogeneous equation corresponding to (1), in order to determine the unknown coefficients \(g^{(\nu)}(x)\), we arrive at an infinite system of equations. For our purposes it is sufficient to consider the following of them:
\[
g^{(0)}(x)\Theta(x)=a(x)g^{(0)}(x),
\]
\[
g^{(1)}(x)\Theta(x)+g^{(0)'}(x)=a(x)g^{(1)}(x).
\tag{19}
\]
Since \(\Theta(x)\) and \(a(x)\) are invertible matrices, from (19) we obtain
\[
g^{(0)}(x)\Theta^{-1}(x)=a^{-1}(x)g^{(0)}(x),
\]
\[
g^{(1)}(x)\Theta^{-1}(x)=a^{-1}(x)g^{(1)}(x)+a^{-1}(x)g^{(0)'}(x)\Theta^{-1}(x).
\tag{20}
\]
As is seen from the first expressions (16) and (20), the first coefficient in (18) vanishes identically.
Finally, taking (16) and (20) into account, we proceed to compute the last coefficient in (18):
\[
g^{(0)}(x)\left[\Theta^{-1}(x)\left(\Theta^{-1}(x)\eta^{(0)}(x)f(x)\right)' +\right.
\]
\[
\left.+\,\Theta^{-1}(x)\eta^{(1)}(x)f(x)\right]+
\]
\[
+\,g^{(1)}(x)\left(\Theta^{-1}(x)\eta^{(0)}(x)f(x)\right)=
\]
\[
= a^{-1}(x)g^{(0)}(x)\left(\left(\Theta^{-1}(x)\eta^{(0)}(x)f(x)\right)' +\right.
\]
\[
\left.+\,\eta^{(1)}(x)f(x)\right)+a^{-1}(x)g^{(1)}(x)\eta^{(0)}(x)f(x)+
\]
\[
+\,a^{-1}(x)g^{(0)'}(x)\Theta^{-1}(x)\eta^{(0)}(x)f(x)=
\]
\[
= a^{-1}(x)f(x)+a^{-1}(x)\left(g^{(0)}(x)\Theta^{-1}(x)\eta^{(0)}(x)f(x)\right)'=a^{-1}(x)f(x)+
\]
\[
+\,a^{-1}(x)\left(a^{-1}(x)g^{(0)}(x)\eta^{(0)}(x)f(x)\right)'=a^{-1}(x)f(x).
\tag{21}
\]
If we substitute the expressions obtained for the coefficients into (18), then the asymptotic representation of the solution of the spectral problem (1), (2) assumes the form
\[ y(x,\lambda)=-\frac{a^{-1}(x)}{\lambda}f(x)+\frac{E(x,\lambda)}{\lambda^2}. \tag{22} \]
The asymptotic formula (22) obtained by us suggests that in this case, too, the assertion of Theorem 8 from [5] holds.
Theorem. If conditions \(1^\circ\)—\(7^\circ\) are fulfilled, the following expansion formula holds:
\[ -\frac{1}{2\pi\sqrt{-1}}\lim_{\gamma\to\infty}\int_{\Gamma_\gamma} y(x,\lambda)\,d\lambda = \]
\(^1\) If for the remaining terms in (18) it is not possible to obtain the above-mentioned representation, then we strengthen condition \(7^\circ\).
\[ = -\frac{1}{2\pi\sqrt{-1}}\lim_{\nu\to\infty}\int_{\Gamma_\nu} d\lambda \int_a^b G(x,\xi,\lambda)f(\xi)\,d\xi = a^{-1}(x)f(x), \tag{23} \]
where \(\Gamma_\nu\) is a closed contour (\(\Gamma_\nu\) is constructed in the same way as in [5]) which, as \(\nu\to\infty\), encloses all the poles of \(G(x,\xi,\lambda)\) and the origin of coordinates in the complex \(\lambda\)-plane.
Remark 1. If the matrix \(a(x)\) is not in canonical form, then one can find a nonsingular matrix \(m(x)\) such that, by means of the substitution \(y(x,\lambda)=m(x)z(x,\lambda)\), the first coefficient of the resulting equation is brought to the form (5), (6).
Remark 2. The above scheme also applies in the case when the expression (3) contains a term corresponding to the zero degree of the parameter \(\lambda\), but under the additional condition\(^{1)}\)
\[ a^{(0)}_{j i n_j,1}(x)\equiv 0,\quad x\in[a,b]. \tag{24} \]
Furthermore, it is easily proved that if \(n_j\leq 2\), then, in order that the fundamental solution have an asymptotic form of type (9), condition (24)\(^{2)}\) is not only sufficient but also necessary. When using the second remark, one must take into account that the inverse matrix (14) again begins with the first power of \(\lambda\).
References
- Poincaré H. Acta Math., 8, 295—344, 1886.
- Birkhoff G. D. Trans. Am. Math. Soc., 9, 219—231, 1908.
- Tamarkin Ya. D. On Some General Problems in the Theory of Ordinary Differential Equations and the Expansion of Arbitrary Functions in Series. Petrograd, 1917.
- Turrittin H. L. Asymptotic expansions of solutions of systems of ordinary linear differential equations containing a parameter, Contributions to the theory of non linear oscillations, sol. 2. Princeton, 1952, p. 81—116.
- Rasulov M. L. The Method of the Contour Integral and Its Application to the Study of Problems for Differential Equations. Moscow, 1964.
Received by the editors
January 10, 1966
Azerbaijan State
University
\(^{1)}\) Condition (24) has meaning if \(\theta_j(x)\) is a double root, i.e. \(n_j\leq 2\).
\(^{2)}\) This condition is both sufficient and necessary in order that the formal solution contain only one term in the exponent, namely the term that corresponds to \(\lambda\).