MATHEMATICS

Yu. L. RODIN

ON THE EIGENFUNCTIONS OF A CERTAIN INTEGRAL EQUATION

(Presented by Academician I. N. Vekua, 19 VIII 1959)

I. N. Vekua \((^1)\) constructed a theory of solutions of the elliptic system of differential equations

\[ U_{\bar z}=|A U_*+\overline{B\bar U}+C_*. \tag{1} \]

An important place in the construction of this theory was occupied by the proof of the fact that the equation

\[ U(z)+\frac{1}{\pi}\iint_T \frac{B(t)\overline{U(t)}}{t-z}\,dT=0 \tag{2} \]

has no nontrivial solutions.

In constructing the theory of generalized analytic functions on Riemann surfaces, consideration of a certain integral equation analogous to (2) is of great significance. The present note contains an example of such an equation having eigenfunctions.

Let the contour \(\Gamma\) bound, on a closed Riemann surface \(R\) of genus \(p\), a domain \(T\). Let \(A(\zeta,z)\) be an elementary function of the first kind of the domain \(T\), covariant in \(\zeta\), invariant in \(z\), analytic in both variables and having in \(T\) a single pole of the first order with residue \(+1\) at the point \(P[\zeta]=P[z]\). One of the possible constructions of such a function is given in \((^2)\).

For what follows, the behavior of \(A(\zeta,z)\) in the cylindrical domain \((R-T)_\zeta\times(R-T)_z\) is important. In this domain \(A(\zeta,z)\) has a pole of the first order with residue \(-1\) at the point \(P[\zeta]=P_0\), provided that \(P[z]\ne P_0\). At the point \(P[\zeta]=P[z]=P_0\), the function \(A(\zeta,z)\) is regular. There are poles of the first order at the points \(P[z]=P_\mu\) \((\mu=1,2,\ldots,p)\), provided that \(P[\zeta]\ne P_\mu\) \((\mu=1,2,\ldots,p)\).* In the case \(P[z]=P[\zeta]=P_\mu\), the function \(A(\zeta,z)\) is regular. There is a pole of the first order with residue \(+1\) at the point \(P[\zeta]=P[z]\ne P_\nu\), \((\nu=0,1,\ldots,p)\). In addition, we shall assume that, for \(P[z_0]=P_0\), \(A(\zeta,z_0)\equiv0\). For this purpose it is enough to subtract from the function constructed in \((^2)\) the covariant of the first kind \(A(\zeta,z_0)\). The point \(P_0\) is chosen arbitrarily in \(R-T\). The points \(P_\mu\) \((\mu=1,2,\ldots,p)\) are also chosen with a large degree of arbitrariness, more precisely:

Theorem 1. In order that a system of points be able to be the set of singularities (in \(z\)) of the function \(A(\zeta,z)\), it is necessary and sufficient that there not exist a covariant of the first kind \(Z'(z)\) vanishing at all these points.

* In \((^2)\) it is proved that the number of these points does not exceed \(p\). The fact that there are exactly \(p\) of them was proved by S. Ya. Gusman. This will, in particular, follow from Theorem 1.

Proof. Necessity of the condition*. Let there exist a covariant \(Z'(z)\) having zeros at the indicated points. Then the product \(A(\zeta,z)Z'(z)\) is a covariant in \(z\), having a single pole at \(z=\zeta\). Hence it follows that \(Z'(z)\equiv 0\).

In particular, there cannot be fewer than \(p\) such points, since for any \(p-1\) points there exists a covariant possessing the indicated property \((^4)\).

Sufficiency of the condition follows from the properties of this set of points \((^2)\). The indicated points form a divisor \(\Delta\) \((^3)\). Obviously, \(\operatorname{ord}\Delta=p\), \(\dim(W/\Delta)=0\). Therefore, by the Riemann–Roch theorem, \(\dim\Delta=1\), i.e. on \(R\) there exists no rational function having poles only at the points of \(\Delta\).

Consider the system of differential equations in \(T\)

\[ U_{\bar z}=B(z)\overline{U}, \tag{3} \]

where the coefficient depends on the local parameter as follows:

\[ B^*(z^*)=B(z)\frac{d\bar z}{d\bar z^*}, \]

where \(z\) and \(z^*\) are local parameters. For constructing the theory of this equation, the basic role is played by the equation

\[ U(z)+\frac{1}{\pi}\iint_T B(t)\overline{U(t)}\,A(t,z)\,dT=0. \tag{4} \]

Theorem 2. If there exists a solution \(U_0(z)\), regular in \(T\), of equation (3), analytically continuable into \(R-T\) in such a way that in this domain it is a multiple of the divisor \(1/\Delta\) and has a zero at the point \(P_0\), then equation (4) is solvable.

Consider the function

\[ H(z)\equiv U_0(z)+\frac{1}{\pi}\iint_T B(t)\overline{U_0(t)}\,A(t,z)\,dT. \tag{5} \]

Obviously, this is a function analytic on \(R\), a multiple of the divisor \(1/\Delta\). In view of the fact that \(\dim\Delta=1\), \(H(z)\equiv\mathrm{const}\). At the point \(P_0\) the right-hand side of (5) vanishes; therefore \(H(z)\equiv 0\). Consequently, \(U_0(z)\) is an eigenfunction of equation (4), \(\lambda=1/\pi\). It remains to construct a function satisfying the requirements of Theorem 2.

In the domain \(T\) consider the system

\[ W_{\bar z}=D(z)w, \tag{6} \]

where \(D^*(z^*)=D(z)\,d\bar z/d\bar z^*\). We shall construct a solution \(W_0\), regular in \(T\), of this system, analytically continuable into \(R-T\) as in Theorem 2. The function \(W_0(z)\) satisfies the system

\[ W_{0\bar z}=\tilde B(z)\overline{W_0}, \tag{6'} \]

where

\[ \tilde B(z)=D(z)W_0(z)/\overline{W_0(z)}. \]

Thus, the construction of the required example is reduced to the construction of a function \(W_0(z)\), for which we use the theory of the Riemann problem \((^{5,6})\). Put \(T^+=T\) and \(T^-=R-T-\Gamma\). The function \(W_0(z)\) has in \(T^+\) the representation

\[ W_0(z)=\varphi^+(z)\exp\omega(z), \]

where \(\varphi^+(z)\) is regular analytic in \(T^+\), and

\[ \omega(z)=-\frac{1}{\pi}\iint_T D(t)\,A(t,z)\,dT. \]

In \(T^-\) the desired function has the representation \(W_0(z)=\varphi^-(z)/\Delta(z)\), where \(\varphi^-(z)\) is a regular analytic function in \(T^-\) and \(\Delta(z)\) is a regular analytic

* The proof of necessity given here is due to S. Ya. Gusman.

into a \(T^{-}\)-function whose zeros are determined by the divisor \(\Delta\). We arrive on \(\Gamma\) at the problem

\[ \varphi^{+}(t)=\frac{e^{-\omega(t)}}{\Delta(t)}\varphi^{-}(t). \tag{7} \]

The index of this problem is equal to \(p\); consequently, by (6), this problem is solvable.

In order to normalize the solution at the point \(P_{0}\), it is sufficient to have at least two linearly independent solutions. Problem (7) has no more than \(p+1\) solutions\({}^{(5)}\). Let \(G(t)\) be a function of index \(p\) such that the problem \(\psi^{+}(t)=G(t)\psi^{-}(t)\) has exactly \(p+1\) solutions. Its existence follows from the results of work \({}^{(5)}\).

In order that problem (7) have the same number of solutions, it is sufficient\({}^{(5)}\) that

\[ \int_{\Gamma}\ln\!\left[G(t)\Delta(t)e^{\omega(t)}\right]\,dZ_{k}(t)=0 \qquad (k=1,2,\ldots,p), \tag{8} \]

where \(dZ_{k}\ (k=1,2,\ldots,p)\) is a basis of differentials of the first kind on the surface \(R\). We obtain

\[ \int_{\Gamma}\omega(t)\,dZ_{k} = -\int_{\Gamma}\ln G(t)\Delta(t)\,dZ_{k}. \tag{9} \]

By a simple calculation we verify that

\[ \int_{\Gamma}\omega(t)\,dZ_{k} = 2i\iint_{T^{+}}D(t)Z'_{k}(t)\,dT \qquad (k=1,\ldots,p). \tag{10} \]

Choosing \(D(z)\) so that condition (8) is satisfied, we obtain that problem (7) has \(p+1\) solutions, which makes it possible, for \(p>1\), to obtain the required normalization.

The author expresses his deep gratitude to his scientific adviser, Prof. L. I. Volkovyskii.

Perm State University
named after A. M. Gorky

Received
1 VIII 1959

CITED LITERATURE

\({}^{1}\) I. N. Vekua, Matem. sborn., 31 (73), No. 2, 217 (1957).
\({}^{2}\) H. Behnke, K. Stein, Math. Ann., 120, 430 (1949).
\({}^{3}\) M. Schiffer, D. C. Spencer, Functionals on Finite Riemann Surfaces, IL, 1957.
\({}^{4}\) N. G. Chebotarev, Theory of Algebraic Functions, 1948.
\({}^{5}\) L. Yu. Rodin, Proceedings of the IV All-Union Conference on the Theory of Functions of a Complex Variable, in press.
\({}^{6}\) Yu. L. Rodin, DAN, 129, No. 6 (1959).