On Eigenvalues and Eigenfunctions of Integro-Differential Operators1

Yu. K. Lando

The paper gives the construction of M. V. Keldysh series2 [1] in eigenfunctions and associated functions for a boundary-value problem for the integro-differential (i.-d.) equation:

\[ A_\lambda u=f,\qquad R_i u=0,\quad i=1,2,\ldots,n. \tag{1} \]

In problem (1), the i.-d. expression \(A_\lambda u\) and the forms \(R_i u\) are polynomials with respect to the complex parameter \(\lambda\):

\[ A_\lambda u=Lu+B_\lambda u, \]

\[ Lu=u^{(n)}(x)+\sum_{i=0}^{n-1}p_{n-i}(x)u^{(i)}(x)+\int_a^b K_{n-i}(x,y)u^{(i)}(y)\,dy, \]

\[ B_\lambda u=\sum_{k=1}^{m}\lambda^k B_k u, \]

\[ B_k u=\sum_{i=0}^{n-1}a_{n-i,k}(x)u^{(i)}(x)+\int_a^b q_{n-i,k}(x,y)u^{(i)}(y)\,dy, \]

\[ R_i u=\sum_{k=0}^{n-1}d_{ik}(\lambda)u^k(a)+\beta_{ik}(\lambda)u^k(b). \]

In particular, for \(m=1\), \(B_1u=u\), when the forms \(R_i u\) do not depend on the parameter \(\lambda\), one obtains the problem considered in [2].

The ideas of M. V. Keldysh (see the survey report of M. V. Keldysh and V. B. Lidskii [3]) were developed by many authors, especially in the direction of establishing the completeness of the system of eigen- and associated elements of various operators. Interesting results were obtained by D. E. Allakhverdiev [4], who established the \(m\)-fold completeness of the system of eigen- and associated elements of non-self-adjoint operators close to normal ones. As far as is known, Keldysh series for differential and i.-d. operators have not yet been constructed.

The paper also gives a simple proof of the \(m\)-fold completeness of the system of eigenfunctions and associated functions for one class of boundary-value problems for i.-d. equations, in particular, problems with boundary conditions regular in the sense of Birkhoff (see, for example, [5], p. 286).

If the coefficients \(p_i(x)\), \(a_i(x)\) are continuous on the segment \([a,b]\), the kernels are regular in the square \(a\le x,y\le b\), and the value \(\lambda=0\) is

nonself-adjoint for problem (1), which is also assumed in what follows, the boundary conditions have the form

\[ R_i u=\sum_{k=0}^{n-1}\alpha_{ik}(\lambda)u^{(i)}(a)+\beta_{ik}(\lambda)u^{(i)}(b)+\int_a^b \varphi_i(x,\lambda)u(x)\,dx=0, \tag{2} \]

where \(\varphi_i(x,\lambda)\) are polynomials with respect to \(\lambda\); then, as in [2], one can prove that the solution of problem (1), in the case of non-eigenvalues \(\lambda\), has the form

\[ u(x)=\int_a^b T(x,s,\lambda)f(s)\,ds. \]

Here \(T(x,s,\lambda)\) is a meromorphic function of the parameter \(\lambda\), the Green’s function of problem (1), possessing, as a function of \(x\) and \(s\), properties analogous to those known for the Green’s function of a boundary-value problem for a differential equation.

1. Adjoint problems

In what follows we shall assume that all coefficient derivatives and kernels that occur exist and are continuous. By \(\dfrac{\partial^\nu A}{\partial c^\nu}\) we shall denote the value of the derivative \(\dfrac{\partial^\nu A}{\partial \lambda^\nu}\) at \(\lambda=c\), \(\nu=0,1,\ldots\)

The integro-differential expression

\[ M^*z=\sum_{i=0}^{n}(-1)^i\frac{\partial^i}{\partial x^i} \left[ \overline{p_{n-i}}(x)z(x)+\int_a^b \overline{K_{n-i}}(t,x)z(t)\,dt \right] \]

will be called the adjoint integro-differential expression to

\[ Mu=\sum_{i=0}^{n}p_{n-i}(x)u^{(i)}(x)+\int_a^b K_{n-i}(x,y)u(y)\,dy. \]

The boundary-value problem

\[ A_\lambda^*z=g,\qquad P_i z=0,\qquad i=1,2,\ldots,n \tag{1*} \]

will be called adjoint to problem (1) if, for all functions \(u\) and \(z\) from the space \(C^n_{[a,b]}\) satisfying respectively the boundary conditions (1) and \((1^*)\), the identity

\[ \int_a^b (A_\lambda u)\,\overline{z}\,dx = \int_a^b u\,(\overline{A_\lambda^* z})\,dx \tag{3} \]

holds.

Let us prove the existence of the adjoint problem and find the form of the adjoint boundary conditions. Integrating by parts, we obtain

\[ \int_a^b \overline{z}\,(A_\lambda u)\,dx = P(\eta,\xi) + \sum_{i=0}^{n-1}u^{(i)}(a)\int_a^b \psi_i(x,\lambda)\overline{z}(x)\,dx + \]

\[ + \sum_{i=0}^{n-1}u^{(i)}(b)\int_a^b \varphi_i(x,\lambda)\overline{z}(x)\,dx + \int_a^b u\,(\overline{A_\lambda^*z})\,dx, \tag{4} \]

where \(P(\eta,\xi)\) is a bilinear form in the variables

\[ \eta=[u(a),\,u'(a),\,\ldots,\,u^{(n-1)}(a),\,u(b),\,u'(b),\,\ldots,\,u^{(n-1)}(b)], \]

\[ \xi=[\overline{z}(a),\ \overline{z}'(a),\ \ldots,\ \overline{z}^{(n-1)}(a),\ \overline{z}(b),\overline{z}'(b),\ \ldots,\ \overline{z}^{(n-1)}(b)], \]

\(P(\eta,\xi)\), \(\psi_i(x,\lambda)\), \(\varphi_i(x,\lambda)\), \(A_\lambda^{*}z\) are polynomials in \(\lambda\), \(A_\lambda^{*}z\) is the expression conjugate to the expression \(A_\lambda u\). Let us supplement the linear forms \(R_i\) by another \(n\) linear forms so that \(R_1, R_2,\ldots, R_n,\ldots, R_{2n}\) are linearly independent forms, and express \(u^{(k)}(a)\) and \(u^{(k)}(b)\) in terms of \(R_i u\) from the system of equations

\[ \sum_{k=0}^{n}\alpha_{ik}(\lambda)u^{(k)}(a)+\beta_{ik}(\lambda)u^{(k)}(b)=D(\lambda)R_i u,\qquad i=1,2,\ldots,2n, \]

where \(D(\lambda)\) is the determinant of the last system; substituting these expressions for \(u^{(k)}(a)\) and \(u^{(k)}(b)\) into (4), we obtain Lagrange’s identity

\[ \int_a^b \overline{z}(A_\lambda u)\,dx = \sum_{i=1}^{2n}\overline{P_i z}\,R_{2n-i}u + \int_a^b u(A_{\bar\lambda}^{*}\overline{z})\,dx, \tag{5} \]

where

\[ P_i z= \sum_{k=0}^{n-1}\gamma_{ik}(\bar\lambda)z^k(a) + \delta_{ik}(\bar\lambda)z^k(b) + \int_a^b \Phi_i(x,\bar\lambda)z(x)\,dx, \]

the coefficients \(\gamma_{ik}(\lambda)\), \(\delta_{ik}(\lambda)\), and the function \(\Phi_i(x,\lambda)\) are polynomials in \(\lambda\). If we put \(R_i u=0\), \(P_i z=0\), \(i=1,2,\ldots,n\), then we obtain identity (3). Thus the existence of the adjoint problem has been proved. From identity (3) it follows immediately that the eigenvalues of adjoint problems are conjugate, and also the well-known solvability condition for the nonhomogeneous problem in the case of an eigenvalue. Moreover, it follows from identity (3) that in the case of non-eigenvalues \(\lambda\) the Green’s functions of adjoint problems \(T(x,s,\lambda)\) and \(T^{*}(x,s,\bar\lambda)\) are conjugate (see [2, 6]).

Differentiate Lagrange’s identity with respect to \(\lambda\) \(\nu\) times. In each of the identities \((\nu=0,1,\ldots,k)\)

\[ \int_a^b \overline{z}\left(\frac{d^\nu A}{d\lambda^\nu}u\right)\,dx = \int_a^b u\left(\frac{d^\nu A^{*}}{d\lambda^\nu}\overline{z}\right)\,dx + \frac{d^\nu}{d\lambda^\nu}\sum_{i=1}^{2n}\overline{P_i z}\,R_{2n-i}u \]

we substitute, in place of the pair of functions \(u\) and \(z\), successively the pairs of functions

\[ \left(\frac{u_0}{\nu!},\, z_{k-\nu}\right),\quad \left(\frac{u_1}{\nu!},\, z_{k-\nu-1}\right),\quad \ldots,\quad \left(\frac{u_{k-\nu}}{\nu!},\, z_0\right). \]

Then we add all the identities obtained \((\nu=0,1,\ldots,k)\), and if, in addition, we require that the functions \(u_0,u_1,\ldots,u_k\) satisfy the boundary conditions

\[ \sum_{\nu=0}^{\mu}\frac{1}{\nu!}\frac{d^\nu R_i}{d\lambda^\nu}u_{\mu-\nu}=0,\qquad \mu=0,1,\ldots,k;\quad i=1,2,\ldots,n, \]

and the functions \(z_0,z_1,\ldots,z_k\) satisfy the boundary conditions

\[ \sum_{\nu=0}^{\mu}\frac{1}{\nu!}\frac{d^\nu P_i}{d\lambda^\nu}z_{\mu-\nu}=0,\qquad \mu=0,1,\ldots,k;\quad i=1,2,\ldots,n, \]

then we obtain the fundamental identity:

\[ \begin{aligned} \int_a^b \Bigg[ &(Au_0)\overline{z_k} +\left(Au_1+\frac{1}{1!}\frac{\partial A}{\partial\lambda}u_0\right)\overline{z_{k-1}} +\ldots +\left(Au_k+\frac{1}{1!}\frac{\partial A}{\partial\lambda}u_{k-1} +\right.\\ &\left.\left. +\ldots+\frac{1}{k!}\frac{\partial^k A}{\partial\lambda^k}u_0\right)\overline{z_0} \right]dx = \int_a^b \Bigg[ u_0\left(\overline{A^*z_k} +\frac{1}{1!}\overline{\frac{\partial A^*}{\partial\lambda}}\,z_{k-1} +\right.\\ &\left. +\ldots+\frac{1}{k!}\overline{\frac{\partial^k A^*}{\partial\lambda^k}}\,z_0\right) +\ldots +u_{k-1}\left(\overline{Az_1} +\frac{1}{1!}\overline{\frac{\partial A^*}{\partial\lambda}}\,z_0\right) +u_k(\overline{A^*z_0}) \Bigg]dx, \end{aligned} \tag{6} \]

since the nonintegral terms vanish by virtue of the boundary conditions. The boundary-value problems

\[ A_\lambda u_0=f_0,\qquad A_\lambda u_1+\frac{1}{1!}\frac{\partial A}{\partial\lambda}u_0=f_1, \]

\[ \ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots \tag{7} \]

\[ A_\lambda u_k+\frac{1}{1!}\frac{\partial A}{\partial\lambda}u_{k-1} +\ldots+\frac{1}{k!}\frac{\partial^k A}{\partial\lambda^k}u_0=f_k, \]

\[ \sum_{\nu=0}^{\mu}\frac{1}{\nu!}\frac{\partial^\nu R_i}{\partial\lambda^\nu} u_{\mu-\nu}=0,\qquad \mu=0,1,\ldots,k;\quad i=1,2,\ldots,n; \]

\[ A_\lambda^*z_0=g_0,\qquad A_\lambda^*z_1+\frac{1}{1!}\frac{\partial A^*}{\partial\lambda}z_0=g_1, \]

\[ \ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots \tag{7*} \]

\[ A_\lambda^*z_k+\frac{1}{1!}\frac{\partial A^*}{\partial\lambda}z_{k-1} +\ldots+\frac{1}{k!}\frac{\partial^k A^*}{\partial\lambda^k}z_0=g_k, \]

\[ \sum_{\nu=0}^{\mu}\frac{1}{\nu!}\frac{\partial^\nu P_i}{\partial\lambda^\nu} z_{\mu-\nu}=0,\qquad \mu=0,1,\ldots,k;\quad i=1,2,\ldots,n, \]

will be called adjoint. From the fundamental identity (6) it follows that if \(\lambda=c\) is an eigenvalue of problem (7), then \(\lambda=c\) is an eigenvalue of the adjoint problem \((7^*)\). In the case of the eigenvalue \(\lambda=c\), for the solvability of problem (7) it is necessary that the equalities

\[ \int_a^b\left(f_0\overline{z_r}+f_1\overline{z_{r-1}}+\ldots+f_r\overline{z_0}\right)dx=0, \qquad r=0,1,\ldots,k, \tag{8} \]

hold for any eigenvectors \((z_0,z_1,\ldots,z_k)\) of the adjoint problem \((7^*)\). It can be proved that conditions (8) are sufficient for the solvability of problem (7), and that the ranks of the eigenvalues of problems (7) and \((7^*)\) are equal.

2. Principal part of the Green function

Let \(u_i(x)\) \((i=1,2,\ldots,p)\) be the eigenfunctions of the boundary-value problem (1) corresponding to the eigenvalue \(\lambda=c\).

The functions \(u_{i1}, u_{i2}, \ldots, u_{ik}\) are called associated with the eigenfunction \(u_i=u_{i0}\) if the vector-function \((u_i, u_{i1}, \ldots, u_{ik})\) is an eigenfunction for problem (7), i.e.

\[ A_c u_i=0,\qquad A_c u_{i1}+\frac{1}{1!}\frac{\partial A}{\partial c}u_i=0, \]

\[ \cdots\cdots\cdots\cdots\cdots \tag{9} \]

\[ A_c u_{ik}+\frac{1}{1!}\frac{\partial A}{\partial c}u_{ik-1} +\cdots+\frac{1}{k!}\frac{\partial^k A}{\partial c^k}u_i=0, \]

\[ \sum_{\nu=0}^{\mu}\frac{1}{\nu!}\frac{d^\nu R_j}{dc^\nu}\,u_{i\mu-\nu}=0, \qquad \mu=0,1,\ldots,k;\quad j=1,2,\ldots,n. \]

The number \(m_i\) is called the multiplicity of the eigenfunction \(u_i\) if there exists a chain of functions associated with \(u_i\) whose length is equal to \(m_i-1\), but no such chain of length \(m_i\) exists.

If the forms \(R_i u\) do not depend on \(\lambda\), then the eigenfunctions and associated functions satisfy the boundary conditions \(R_i u=0\), but in the adjoint problem the expressions \(P_i z\) will be polynomials in \(\lambda\).

In a neighborhood of the point \(\lambda=c\) we have

\[ T(x,s,\lambda)= \frac{g_0(x,s)}{(\lambda-c)^{k+1}} +\frac{g_1(x,s)}{(\lambda-c)^k} +\cdots+ \frac{g_k(x,s)}{\lambda-c} +g_{k+1}(x,s,\lambda), \]

where \(g_{k+1}(x,s,\lambda)\) is an analytic function at the point \(c\). From the properties of the Green function \(T(x,s,\lambda)\) it follows that

\[ A_\lambda\bigl[g_0(x,s)+(\lambda-c)g_1(x,s)+\cdots+(\lambda-c)^k g_k(x,s)+ \]

\[ +(\lambda-c)^{k+1}g_{k+1}(x,s,\lambda)\bigr]=0. \]

Differentiating the last equality with respect to the parameter \(\lambda\) \(i\) times \((i=0,1,\ldots,k)\), and then putting \(\lambda=c\), we find that, for a fixed value of \(s\), the kernels \(g_0(x,s), g_1(x,s), \ldots, g_k(x,s)\), as functions of \(x\), satisfy the system of equations (9). We shall also prove that these kernels satisfy the boundary conditions (9). That is, the vector-function

\[ [g_0(x,s),\, g_1(x,s),\, \ldots,\, g_k(x,s)] \]

for a fixed value of \(s\) is an eigenfunction for problem (9), and therefore will be a linear combination of the eigenvector-functions \((u_i,u_{i1},\ldots,u_{ik})\). Since the Green functions of adjoint problems are conjugate, by analogy with the preceding we conclude that, for a fixed value of \(x\), the vector-function

\[ [\overline{g_0}(x,s),\, \overline{g_1}(x,s),\, \ldots,\, \overline{g_k}(x,s)], \]

considered as a function of \(s\), is an eigenfunction for the problem adjoint to (9), and will be a linear combination of the eigenvectors \((z_i,z_{i1},\ldots,z_{im_i-1})\) of the adjoint problem. Therefore the Green function \(T(x,s,\lambda)\) in a neighborhood of the pole is representable in the form

\[ T(x,s,\lambda)= \sum_{j=1}^{p} \frac{u_j(x)\overline{z_j}(s)}{(\lambda-c)^{m_j}} + \frac{u_{j1}(x)\overline{z_j}(s)+u_j(x)\overline{z_{j1}}(s)} {(\lambda-c)^{m_j-1}} + \]

\[ +\ldots+\frac{u_{j m_j-1}(x)\overline{z_j(s)}+u_{j m_j-2}(x)\overline{z_{j1}(s)}+\ldots+u_j(x)\overline{z_{j m_j-1}(s)}}{\lambda-c} +g_{k+1}^{*}(x,s,\lambda), \]

where \(\max m_j=k+1\), \(g_{k+1}(x,s,\lambda)\) is an analytic function at the point \(\lambda=c\), \(u_j,u_{j1},\ldots,u_{j m_j-1}\) are the eigenfunctions and associated functions of problem (1), and \(z_j,z_{j1},\ldots,z_{j m_j-1}\) are the eigenfunctions and associated functions of the adjoint problem \((1^*)\).

Remark. The expression for the Green function \(T(x,s,\lambda)\) in a neighborhood of its pole can be obtained in another way, if one uses M. V. Keldysh’s representation of the resolvent of a completely continuous operator and applies this representation to the integral equation to which problem (1) is reduced by the substitution

\[ u(x)=\int_a^b T(x,s,0)z(s)\,ds, \]

when the boundary conditions do not depend on the parameter \(\lambda\). The kernel of the integral equation will be a polynomial of degree \(m\) with respect to \(\lambda\).

3. M. V. Keldysh series

It is easy to see that the eigenvalue \(\lambda=c\) is a root of the polynomial

\[ \Phi_k^i(x,\lambda)=A_\lambda\left[u_{i0}+(\lambda-c)u_{i1}+\ldots+(\lambda-c)^k u_{ik}\right] \]

of multiplicity \(k+1\), since

\[ \Phi_k^i(x,c)=A_cu_{i0}=0; \]

\[ \frac{\partial \Phi_k}{\partial c} =A_cu_{i1}+\frac{1}{1!}\frac{\partial A}{\partial c}u_{i0}=0;\ \ldots\ \frac{\partial^k\Phi_k^i(x,c)}{\partial c^k} = \]

\[ =k!\left(Au_{ik}+\frac{1}{1!}\frac{\partial A}{\partial c}u_{ik-1} +\ldots+\frac{1}{k!}\frac{\partial^k A}{\partial c^k}u_{i0}\right)=0. \]

Therefore the expression \(\Phi_k^i/(\lambda-c)^{k+1}\) is a polynomial in \(\lambda\) of degree \(m-1\), i.e.

\[ A_\lambda\left[ \frac{u_{i0}}{(\lambda-c)^{k+1}} +\frac{u_{i1}}{(\lambda-c)^k} +\ldots+ \frac{u_{ik}}{\lambda-c} \right] = \sum_{\nu=0}^{m-1}Y_{\nu k}^i(x)\lambda^\nu, \tag{10} \]

where \(Y_{\nu k}^i(x)\) are the coefficients of this polynomial.

Denote

\[ \varphi_k^i(x,\lambda)= \frac{u_{i0}(x)}{(\lambda-c)^{k+1}} +\frac{u_{i1}(x)}{(\lambda-c)^k} +\ldots+ \frac{u_{ik}(x)}{\lambda-c}. \]

Then for \(\lambda\ne c\)

\[ A(\varphi_k^i)=\sum_{\nu=0}^{m-1}Y_{\nu k}^i(x)\lambda^\nu, \]

\[ A\left(\frac{\partial\varphi_k^i}{\partial\lambda}\right) +\frac{1}{1!}\frac{\partial A}{\partial\lambda}(\varphi_k^i) = \sum_{\nu=1}^{m-1}\binom{\nu}{1}Y_{\nu k}^i(x)\lambda^{\nu-1}, \]

\[ \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot \tag{11} \]

\[ A\left(\frac{1}{k!}\frac{\partial^k \varphi_k^i}{\partial \lambda^k}\right) + \frac{1}{1!}\frac{\partial A}{\partial \lambda} \left(\frac{1}{(k-1)!}\frac{\partial^{k-1}\varphi_k^i}{\partial \lambda^{k-1}}\right) + \]

\[ +\frac{1}{2!}\frac{\partial^2 A}{\partial \lambda^2} \left(\frac{1}{(k-2)!}\frac{\partial^{k-2}\varphi_k^i}{\partial \lambda^{k-2}}\right) +\cdots+ \frac{1}{(k-1)!}\frac{\partial^{k-1}A}{\partial \lambda^{k-1}} \left(\frac{\partial \varphi_k^i}{\partial \lambda}\right) + \]

\[ +\frac{1}{k!}\frac{\partial^k A}{\partial \lambda^k}(\varphi_k^i) = \sum_{\nu=k}^{m-1} \binom{\nu}{k} Y_{\nu k}^i(x)\lambda^{\nu-k}. \]

Therefore the vector-function

\[ \left( \varphi_k^i(x,\lambda),\ \frac{1}{1!}\frac{\partial \varphi_k^i(x,\lambda)}{\partial \lambda},\ \ldots,\ \frac{1}{k!}\frac{\partial^k \varphi_k^i(x,\lambda)}{\partial \lambda^k} \right) \tag{12} \]

for any values \(\lambda \ne c\) is a solution of the system of equations (11). In addition, if the boundary conditions do not depend on the parameter \(\lambda\), then the functions \(\dfrac{\partial^\nu \varphi_k^i}{\partial \lambda^\nu}\) satisfy them, since then the functions \(u_{i0}, u_{i1}, \ldots, u_{ik}\) satisfy the boundary conditions \(R_i u_{i\nu}=0\). Consequently, if the boundary conditions (1) do not depend on the parameter \(\lambda\), then the vector-function (12) is a solution of the system of equations (11), satisfying the boundary conditions

\[ R_j\left(\frac{\partial^\nu \varphi_k^i}{\partial \lambda^\nu}\right)=0,\quad j=1,2,\ldots,n;\quad \nu=0,1,\ldots,k, \]

for any value of the parameter \(\lambda\ne c\). Let \(\lambda_j\ne c\) be an eigenvalue of the boundary-value problem (1); then \(\lambda_j\) is an eigenvalue of the boundary-value problem for the system (7) with boundary conditions

\[ R_i(u_j)=0,\quad i=1,2,\ldots,n;\quad j=1,2,\ldots,k. \]

Therefore, from the solvability condition (8), we obtain:

\[ \left(\sum_{\nu=0}^{m-1}Y_{\nu k}^i\lambda_j^\nu,\ z_{jl}\right) + \left(\sum_{\nu=1}^{m-1}\binom{\nu}{1}Y_{\nu k}^i\lambda_j^{\nu-1},\ z_{j\,l-1}\right) +\cdots+ \]

\[ + \left(\sum_{\nu=l}^{m-1}\binom{\nu}{l}Y_{\nu k}^i\lambda_j^{\nu-l},\ z_j\right)=0,\quad \begin{array}{l} l=0,1,\ldots,m_j-1,\\ k=0,1,\ldots,m_i-1, \end{array} \tag{13} \]

where \(z_{j0}, z_{j1}, \ldots, z_{j\,m_j-1}\) are the eigenfunctions and associated functions of the adjoint problem corresponding to the eigenvalue \(\lambda_j\), \((f,g)=\displaystyle\int_a^b f\overline{g}\,dx\).

Let \(\lambda_j=\lambda_i=c\) be an eigenvalue of problem (1). We take the identity

\[ A[u_{i0}+u_{i1}(\lambda-c)+\cdots+u_{ik}(\lambda-c)^k] = (\lambda-c)^{k+1}\sum_{\nu=0}^{m-1}Y_{\nu k}^i\lambda^\nu \]

and differentiate it successively with respect to \(\lambda\) \(k+1, k+2 \ldots m_i\) times, and then set \(\lambda=c\); after simplification we obtain:

\[ \frac{1}{1!}\frac{\partial A}{\partial c}u_{ik} +\frac{1}{2!}\frac{\partial^2 A}{\partial c^2}u_{ik-1} +\cdots+ \frac{1}{(k+1)!}\frac{\partial^{k+1}A}{\partial c^{k+1}}u_{i0} = \sum_{\nu=0}^{m-1}Y_{\nu k}^{i}c^\nu, \]

\[ \frac{1}{2!}\frac{\partial^2 A}{\partial c^2}u_{ik} +\frac{1}{3!}\frac{\partial^3 A}{\partial c^3}u_{ik-1} +\cdots+ \frac{1}{(k+2)!}\frac{\partial^{k+2}A}{\partial c^{k+2}}u_{i0} = \sum_{\nu=1}^{m-1}\binom{\nu}{1}Y_{\nu k}^{i}c^{\nu-1}, \tag{14} \]

\[ \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot \]

\[ \frac{1}{(m_i-k-1)!}\frac{\partial^{m_i-k-1}A}{\partial c^{m_i-k-1}}u_{ik} + \frac{1}{(m_i-k)!}\frac{\partial^{m_i-k}A}{\partial c^{m_i-k}}u_{ik-1} +\cdots+ \]

\[ +\frac{1}{(m_i-1)!}\frac{\partial^{m_i-1}A}{\partial c^{m_i-1}}u_{i0} = \sum_{\nu=m_i-k-2}^{m-1} \binom{\nu}{m_i-k-2} Y_{\nu k}^{i}c^{\nu-(m_i-k-2)}, \]

\[ \frac{1}{(m_i-k)!}\frac{\partial^{m_i-k}A}{\partial c^{m_i-k}}u_{ik} + \frac{1}{(m_i-k+1)!}\frac{\partial^{m_i-k+1}A}{\partial c^{m_i-k+1}}u_{ik-1} +\cdots+ \]

\[ +\frac{1}{m_i!}\frac{\partial^{m_i}A}{\partial c^{m_i}}u_{i0} = \sum_{\nu=m_i-k-1}^{m-1} \binom{\nu}{m_i-k-1} Y_{\nu k}^{i}c^{\nu-(m_i-k-1)}. \]

Since the functions \(u_{i0}, u_{i1}, \ldots, u_{i m_i-1}\) satisfy the homogeneous system of equations (9) \((k=m_i-1)\), the first \(m_i-k-1\) identities (14) can be rewritten as follows:

\[ Au_{ik+1}=-\sum_{\nu=0}^{m-1}Y_{\nu k}^{i}c^\nu, \]

\[ Au_{ik+2}+\frac{1}{1!}\frac{\partial A}{\partial c}u_{ik+1} = -\sum_{\nu=1}^{m-1}\binom{\nu}{1}Y_{\nu k}^{i}c^{\nu-1}, \]

\[ \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot \]

\[ Au_{i m_i-1} + \frac{1}{1!}\frac{\partial A}{\partial c}u_{i m_i-2} +\cdots+ \frac{1}{(m_i-k-2)!} \frac{\partial^{m_i-k-2}}{\partial c^{m_i-k-2}}u_{ik+1} = \]

\[ = -\sum_{\nu=m_i-k-2}^{m-1} \binom{\nu}{m_i-k-2} Y_{\nu k}^{i}c^{\nu-(m_i-k-2)}. \]

From the solvability condition (8) follow the equalities (13) for \(k+l\le m_i-2\), \(\lambda_i=c\). Suppose that equality (13) is valid for all eigenfunctions \(z_j\) of the adjoint problem \((1^*)\) corresponding to the eigenvalue \(\lambda_j=\lambda_i=c\) in the case \(k+l=m_i-1\). Therefore the boundary-value problem

\[ Av_1=-\sum_{\nu=0}^{m-1}Y_{\nu k}^{i}c^\nu, \]

$$ A v_2 + \frac{1}{1!}\frac{\partial A}{\partial c}v_1 = -\sum_{\nu=0}^{m-1}\binom{\nu}{1}Y_{\nu k}^i c^{\nu-1}, $$

$$ \cdots \tag{15} $$

$$ A v_{m_i-k}+\frac{1}{1!}\frac{\partial A}{\partial c}v_{m_i-k-1} +\cdots+ \frac{1}{(m_i-k-1)!}\frac{\partial^{m_i-k-1}A}{\partial c^{m_i-k-1}}v_1 $$

$$ = -\sum_{\nu=m_i-k-1}^{m-1}\binom{\nu}{m_i-k-1}Y_{\nu k}^i c^{\nu-(m_i-k-1)}, $$

$$ R_\nu(v_j)=0,\qquad \nu=0,1,\ldots,n,\quad j=1,2,\ldots,m_i-k $$

is solvable and has a solution \((v_1,v_2,\ldots,v_{m_i-k})\). Adding the \(j\)-th rows of the equalities (14) and (15), we shall see that the multiplicity of the eigenfunction \(u_i\) will be greater than \(m_i\), which is impossible. Consequently, there will be at least one eigenfunction \(z_i\) for which the left-hand side of expression (13) will be equal to 1 in the case \(k+l=m_i-1\). The remaining eigenfunctions \(z_j\) may be regarded as chosen so that for them equality (13) will also be valid in the case \(k+l=m_i-1\). Therefore one may assume that the eigenfunctions and associated functions are chosen so that the equalities

$$ \left(\sum_{\nu=0}^{m-1}Y_{\nu k}^i\lambda_j^\nu,\ z_{jl}\right) + \left(\sum_{\nu=1}^{m-1}\binom{\nu}{1}Y_{\nu k}^i\lambda_j^{\nu-1},\ z_{jl-1}\right) +\cdots+ $$

$$ + \left(\sum_{\nu=l}^{m-1}\binom{\nu}{l}Y_{\nu k}^i\lambda_j^{\nu-l},\ z_j\right) = \delta_{ij}\delta^0_{k\,m_j-l-1}, $$

$$ k=0,1,\ldots,m_i-1,\qquad l=0,1,\ldots,m_j-1. $$

Suppose that the boundary conditions in the adjoint problem \((1^*)\) do not depend on the parameter \(\lambda\), but in problem (1) may be polynomials with respect to the parameter \(\lambda\) and have the form (2). We shall indicate two simple cases in which these assumptions are realized.

1. If the boundary-value problem (1), whose boundary conditions do not depend on the parameter \(\lambda\), is regarded as the adjoint problem \((1^*)\), then in problem \((1^*)\) the boundary conditions will be polynomials with respect to the parameter \(\lambda\).

2. If

$$ B_k u=a_k(x)u(x)+\int_a^b q_k(x,y)u(y)\,dy $$

and the boundary conditions \(R_i u=0\) do not depend on the parameter \(\lambda\).

Put

$$ A_\lambda^*\left[ \frac{z_{i0}}{(\lambda-c)^{k+1}} + \frac{z_{i1}}{(\lambda-c)^k} +\cdots+ \frac{z_{ik}}{\lambda-c} \right] = \sum_{\nu=0}^{m-1}Z_{\nu k}^i\lambda^\nu . $$

Then, analogously to the preceding, we obtain the equalities

\[ \left(\sum_{\nu=0}^{m-1} Z_{\nu k}^{i}\lambda_j^\nu,\, u_{jl}\right) + \left(\sum_{\nu=1}^{m-1} \binom{\nu}{1} Z_{\nu k}^{i}\lambda_j^{\nu-1},\, u_{jl-1}\right) +\cdots+ \]

\[ + \left(\sum_{\nu=l}^{m-1} \binom{\nu}{l} Z_{\nu k}^{i}\lambda_j^{\nu-l},\, u_j\right) = \delta_{ij}\delta_{k m_j-l-1}, \qquad \begin{aligned} &k=0,1,\ldots,m_i-1,\\ &l=0,1,\ldots,m_j-1. \end{aligned} \]

Or

\[ \sum_{\nu=0}^{m-1} (Z_{\nu k}^{i}, u_{jl}^{\nu}) = \delta_{ij}\delta_{k m_j-l-1}; \qquad \begin{aligned} &k=0,1,\ldots,m_i-1,\\ &l=0,1,\ldots,m_j-1, \end{aligned} \tag{16} \]

where

\[ u_{jl}^{\nu} = \binom{\nu}{l}\lambda_j^{\nu-l}u_j + \binom{\nu}{l-1}\lambda_j^{\nu-l+1}u_{j1} +\cdots+ \binom{\nu}{1}\lambda_j^{\nu-1}u_{jl-1} + \lambda_j^\nu u_{jl} = \]

\[ = \left. \frac{d^\nu}{dt^\nu} \left\{ e^{\lambda_j t} \left( u_{jl}+\frac{t}{1!}u_{jl-1}+\cdots+u_j\frac{t^l}{l!} \right) \right\} \right|_{t=0}. \]

Assigning the values \(\nu=0,1,\ldots,m-1\), one can construct \(m\) systems of derived chains of eigenfunctions and associated functions \(\{u_{jl}^{\nu}\}\).

The system of functions

\[ \{u_{jl}^{\nu}\}, \qquad \nu=0,1,\ldots,m-1 \]

is called \(m\)-fold complete if any system of \(m\) functions from the domain of definition of the operator \(A\) can be represented as the limit of linear combinations of the functions \(u_{jl}^{\nu}\):

\[ f_{\nu+1}=\sum_{i=1}^{\infty}\sum_{k=0}^{m_i-1} a_{ik}u_{ik}^{\nu}, \qquad \nu=0,1,\ldots,m-1, \tag{17} \]

where the coefficients \(a_{ik}\) must not depend on the number of the function \(\nu\). If the expansions (17) hold and the series (17) may be integrated term by term, then from the equalities (16) we find that

\[ a_{ik}=\sum_{\nu=0}^{m-1}(f_{\nu+1}, Z_{\nu m_i-k-1}^{i}). \tag{18} \]

The coefficients \(a_{ik}\), determined by formula (18), will be called the M. V. Keldysh coefficients, and series with such coefficients will be called M. V. Keldysh series of the system of functions \(f_1,f_2,\ldots,f_m\).

Hence it follows

Theorem 1. If the system of eigenfunctions and associated functions of problem (1) is \(m\)-fold complete and the series (17) can be integrated term by term, then the series (17) are M. V. Keldysh series and the expansion (17) is unique.

4. The case of \(m\)-fold completeness

Suppose that the integro-differential expression \(A_\lambda u\) has the form

\[ A_\lambda u=Lu+\lambda^m u, \]

the boundary conditions do not depend on the parameter \(\lambda\). It is obvious that if \(\lambda_j\) is an eigenvalue of problem (1), then

\[ \lambda_{jk}=\lambda_j \exp 2\pi ik/m,\qquad k=0,\,1,\ldots,m-1 \]

will also be eigenvalues of problem (1). We shall agree to denote by \(\lambda_j\) the eigenvalues in the sector
\[ -\frac{\pi}{m}<\arg\lambda\leqslant\frac{\pi}{m}. \]
It is easy to verify directly that the following is true.

Lemma. If \(\lambda_j\) is an eigenvalue of problem (1), in which
\[ A_\lambda u=Lu+\lambda^m u, \]
and the boundary conditions do not depend on \(\lambda\), and if
\[ u_{j0},\,u_{j1},\,\ldots,\,u_{jm_j-1} \]
are the eigenfunction and associated functions corresponding to this eigenvalue,

\[ u_{j0}^{\nu},\,u_{j1}^{\nu},\,\ldots,\,u_{jm_j-1}^{\nu},\qquad \nu=0,\,1,\ldots,m-1 \]

are the derivative chains of these functions, then the chains of functions

\[ u_{j0}^{k\nu}=u_{j0}^{\nu}\exp 2\pi ik\nu/m,\qquad u_{j1}^{k\nu}=u_{j1}^{\nu}\exp 2\pi ik(\nu-1)/m,\ \ldots, \]

\[ u_{jm_j-1}^{k\nu}=u_{jm_j-1}^{\nu}\exp 2\pi ik(\nu-m_j+1)/m \]

are the derivatives corresponding to the eigenvalue

\[ \lambda_{jk}=\lambda_j\exp 2\pi ik/m,\qquad k=0,\,1,\ldots,m-1. \]

From the lemma it follows:

Theorem 2. If the system of eigenfunctions and associated functions of problem (1) is complete,
\[ A_\lambda u=Lu+\lambda^m u, \]
and the boundary conditions do not depend on \(\lambda\), then this system of functions is \(m\)-fold complete.

Indeed, if the conditions of Theorem 2 are satisfied, then each system of functions

\[ \{u_{j0}^{\nu},\,u_{j1}^{\nu},\,\ldots,\,u_{jm_j-1}^{\nu}\},\qquad \nu=0,\,1,\ldots,m-1 \]

is complete. If there are \(m\) functions \(f_1,f_2,\ldots,f_m\) from the domain of definition of the operator \(A\), then each of them can be expanded in the \(\nu\)-th derivative chains corresponding to the eigenvalues \(\lambda_{j0}=\lambda_j\):

\[ f_{\nu+1}(x)=\sum_{j=1}^{\infty} \left(b_{j0}^{\nu}u_{j0}^{\nu}+b_{j1}^{\nu}u_{j1}^{\nu}+\ldots+ b_{jm_j-1}^{\nu}u_{jm_j-1}^{\nu}\right),\qquad \nu=0,\,1,\ldots,m-1. \]

On the other hand, if we use the lemma and take all eigenvalues \(\lambda_{jk}\) \((k=0,\,1,\ldots,m-1)\), then these expansions can be written as follows:

\[ f_{\nu+1}=\sum_{j=1}^{\infty} \left(a_{j0}u_{j0}^{\nu}+a_{j1}u_{j1}^{\nu}+\ldots+ a_{jm_j-1}u_{jm_j-1}^{\nu}\right)+ \]

\[ +\sum_{j=1}^{\infty} \left(a_{j0}^{1}u_{j0}^{\nu 1}+a_{j1}^{1}u_{j1}^{\nu 1}+\ldots+ a_{jm_j-1}^{1}u_{jm_j-1}^{\nu 1}\right)+\ldots+ \]

\[ +\sum_{j=1}^{\infty} \left(a_{j0}^{m-1}u_{j0}^{\nu\,m-1}+a_{j1}^{m-1}u_{j1}^{\nu\,m-1}+\ldots+ a_{jm_j-1}^{m-1}u_{jm_j-1}^{\nu\,m-1}\right), \]

where the coefficients \(a_{jk}^\nu\) are determined from the system of equations

\[ a_{jk}+a'_{jk}\exp 2\pi i(\nu-k)/m+\ldots+a_{jk}^{m-1}\exp 2\pi i(m-1)(\nu-k)/m=b_{jk}^{\nu}, \]

\[ \nu=0,\ 1,\ \ldots,\ m-1 \]

and do not depend on the number \(\nu+1\) of the function. If the convergence of the series is uniform, then on the basis of Theorem 1 the coefficients \(a_{ik}\) are M. V. Keldysh coefficients and the expansion is unique.

Suppose that the boundary conditions \(R_i u=0\) are regular. The order \(i_k\), etc., of the expression \(B_k u\) is less than

\[ \frac{n(m-k)}{m}-1,\qquad k<m,\qquad p_1(x)=0,\qquad K_1(x,y)\equiv0,\qquad B_m u=u. \]

Introduce the substitution

\[ u^{(n)}(x)+\lambda^m u(x)=z(x),\qquad R_i u=0,\qquad i=1,\ 2,\ \ldots,n, \tag{19} \]

where \(z(x)\) is a new unknown function. Let \(g(x,s,\lambda^m)\) be the Green’s function of problem (19). Then one can choose a sequence of circles \(\Gamma_k\) such that their radius \(R_k\) tends to infinity, and on them the following estimates will hold for the Green’s function \(g(x,s,\lambda^m)\):

\[ |g|<\frac{c}{|\lambda|^{\frac{n-1}{n}m}},\qquad \left|\frac{\partial g}{\partial x}\right|<\frac{c}{|\lambda|^{\frac{n-2}{n}m}},\ \ldots,\ \left|\frac{\partial^{n-1}g}{\partial x^{n-1}}\right|<c \tag{20}. \]

(see [6], p. 68). In what follows we shall assume that \(\lambda\in\Gamma_k\); obviously, such values are not eigenvalues for problem (19). Replacing \(u(x)\) in equation (1) by the expression

\[ u(x)=\int_a^b g(x,s,\lambda^m)z(s)\,ds, \]

we obtain, with respect to \(z(x)\), the integral equation

\[ z(x)+\int_a^b k(x,s,\lambda)z(s)\,ds=f(x), \tag{21} \]

for whose kernel \(k(x,y,\lambda)\) on the circles \(\Gamma_k\) the estimate will hold:

\[ |k(x,y,\lambda)|<\beta:|\lambda|^\gamma, \]

where \(\beta\) and \(\gamma\) are certain positive constants. It follows from this that the resolvent \(R(x,s,\lambda)\) of the integral equation (21) exists and tends to zero when \(\lambda\to\infty\) along the circles \(\Gamma_k\). Therefore

\[ u(x)=\int_a^b\left[g(x,s,\lambda^m)+\int_a^b R(t,s,\lambda^m)\,dt\right]f(s)\,ds. \]

For the Green’s function

\[ T(x,s,\lambda)=g(x,s,\lambda^m)+\int_a^b R(t,s,\lambda)g(x,t,\lambda^m)\,dt \]

on the circles \(\Gamma_k\) the estimates (20) are valid.

Consider the sequence of integrals

\[ I_k=\frac{1}{2\pi i}\int_{\Gamma_k}\frac{T(x,s,\lambda)}{\lambda}\,d\lambda =T(x,s,0)+\sum_{\nu=1}^{N_k}H_\nu(x,s), \]

where \(N_k+1\) is the number of poles of the function \(T(x,s,\lambda)/\lambda\) inside the circle \(\Gamma_k\), and \(H_\nu(x,s)\) are the residues of the function \(T(x,s,\lambda)/\lambda\) at these poles. Since \(I_k\to 0\) as \(\lambda\to\infty\) along \(\Gamma_k\), the series

\[ -\sum_{\nu=1}^{\infty}H_\nu(x,s)=T(x,s,0) \]

converges uniformly with respect to \(x\) and \(s\). From the representation of the Green function \(T(x,s,\lambda)\) we find that the residues \(H_\nu(x,s)\) are sums of terms of the form

\[ a_j(s)u_j(x)+a_{j1}(s)u_{j1}(x)+\cdots+a_{im_j-1}(s)u_{im_j-1}(x). \]

Therefore every function \(F(x)\) having continuous derivatives up to order \(n\) inclusive and satisfying the boundary conditions is expanded in a series in the eigenfunctions and associated functions of the boundary-value problem, which converges to \(F(x)\) uniformly under a certain grouping of its terms. Hence, and from Theorem 2, it follows that

Theorem 3. The system of eigenfunctions and associated functions of the boundary-value problem

\[ u^{(n)}(x)+\sum_{i=0}^{n-2}\left[p_{n-i}(x)u^{(i)}(x)+\int_a^b K_{n-i}(x,y)u^{(i)}(y)\,dy\right]+\lambda^m u(x)=0, \]

\[ R_i u=0,\qquad i=1,2,\ldots,n, \]

is \(m\)-fold complete if the boundary conditions are regular and do not depend on the parameter \(\lambda\).

References

1. Keldysh M. V. DAN SSSR, 77, No. 1, 1951, pp. 11–14.
2. Lando Yu. K. Izv. AN BSSR, ser. phys.-techn. sci., No. 4, 1960, pp. 11–21.
3. Keldysh M. V. and Lidskii V. B. Proceedings of the Fourth All-Union Mathematical Congress, 1, 1963, pp. 101–120.
4. Allakhverdiev D. E. DAN SSSR, 115, No. 2, 1957, pp. 207–210.
5. Kamke E. Handbook of Ordinary Differential Equations. IL, 1950.
6. Naimark M. A. Linear Differential Operators. GITTL, Moscow, 1954.

Minsk Pedagogical Institute
named after A. M. Gorky

1. Report read at the First Belarusian Mathematical Conference, January 25–28, 1964. ↩

2. The definition of such series will be given later. ↩