UDC 517.917

MATHEMATICS

L. S. Yakupov

ON SOLUTIONS OF FINITE FORM OF CERTAIN HILL EQUATIONS

(Presented by Academician I. G. Petrovskii on 2 XI 1967)

It is known that the radial wave equation reduces to the form

\[ d^2y/d\tau^2+(\delta+\varepsilon e^\tau+\mu e^{2\tau})y=0 \tag{1} \]

(the Hill equation with imaginary period), for which, under certain relations among \(\delta,\varepsilon,\mu\), solutions in closed form have been found \({}^{1}\).

Solutions of finite form can also be found for the Hill equation with real period, analogous to (1):

\[ d^2y/d\tau^2+(\delta+\varepsilon_c\cos\tau+\varepsilon_s\sin\tau+\mu\cos 2\tau)y=0. \tag{2} \]

Equation (2), under certain relations among \(\delta,\varepsilon_c,\varepsilon_s,\mu\), has solutions in the form of a segment of a Fourier series multiplied by

\[ E(\tau)=\exp\left[\int(\lambda_0+\lambda\sin\tau)d\tau\right]. \tag{3} \]

For \(y=E(\tau)f(\tau)\), equation (2) is satisfied if

\[ f''+2(\lambda_0+\lambda\sin\tau)f'=(\alpha+2\beta\cos\tau)f, \tag{4} \]

\[ \delta=-(\alpha+\lambda_0^2+\mu),\qquad \varepsilon_c=-(2\beta+\lambda),\qquad \varepsilon_s=-2\lambda_0\lambda,\qquad \mu=\lambda^2/2. \]

From (4), after the substitution

\[ f(\tau)=\sum_{n=0}^{N} \frac{a_n\cos n\tau+b_n\sin n\tau} {c_n\cos(n+\tfrac12)\tau+d_n\sin(n+\tfrac12)\tau} \]

and elementary transformations, one obtains a system of algebraic equations, the study of which leads to Theorems 1 and 2.

Theorem 1. Let \(\lambda_0\) be a constant, \(\lambda=\sqrt{2\mu}\), \(N=0,1,\ldots\), \(L_1=\operatorname{diag}\{2\lambda_0,4\lambda_0,\ldots,2N\lambda_0\}\) be a diagonal matrix; \(B,A\) be codiagonal matrices:

\[ B=\operatorname{codiag}\left\{ \begin{array}{cccccc} 1,&\dfrac{(N+2)\lambda}{2^2},&\dfrac{(N+3)\lambda}{3^2},&\ldots&\ldots,&\dfrac{2N\lambda}{N^2}\\ (N-1)\lambda,&(N-2)\lambda,&\ldots&\ldots,&\lambda& \end{array} \right\}, \]

\[ A=\operatorname{codiag}\left\{ \begin{array}{cccccc} 0,&\dfrac{(N+1)\lambda}{1},&\dfrac{(N+2)\lambda}{2^2},&\ldots&\ldots,&\dfrac{2N\lambda}{N^2}\\ 2N\lambda,&(N-1)\lambda,&(N-2)\lambda,&\ldots&\lambda& \end{array} \right\}. \]

If \(\varepsilon_c=-(2N+1)\lambda\), \(\varepsilon_s=-2\lambda_0\lambda\), and the sum \(\delta+\mu+\lambda_0^2\) is equal to an eigenvalue of the matrix

\[ A^*= \left[ \begin{array}{c:c} A& \begin{array}{cc} 0&\ldots 0\\ &-L_1 \end{array} \\ \hdashline \begin{array}{c} 0\\ \vdots\\ 0 \end{array} \quad L_1 & B \end{array} \right] \]

and to this value there corresponds the eigenvector \((a_0^*, \ldots, a_N^*, b_1^*, \ldots, b_N^*)\), then equation (2) has the solution

\[ y=E(t)\sum_{n=0}^{N}(a_n^*\cos nt+b_n^*\sin nt). \]

Corollary 1.1. If \(\varepsilon=-(2N+1)\lambda\), the sum \(\delta+\mu\) is equal to an eigenvalue of the matrix \(A\), and to this value there corresponds the eigenvector \((a_0,\ldots,a_N)\), then the equation

\[ d^2y/d\tau^2+(\delta+\varepsilon\cos\tau+\mu\cos2\tau)y=0 \tag{5} \]

has a \(2\pi\)-periodic even solution

\[ y=e^{-\lambda\cos\tau}\sum_{n=0}^{N}a_n\cos n\tau; \tag{6} \]

if, however, \(\delta+\mu\) is equal to an eigenvalue of the matrix \(B\), to which the vector \((b_1,\ldots,b_N)\) corresponds, then equation (5) has a \(2\pi\)-periodic odd solution

\[ y=e^{-\lambda\cos\tau}\sum_{n=1}^{N}b_n\sin n\tau. \tag{7} \]

Corollary 1.2. The solutions (6) and (7), as \(N\to\infty\), tend to \(2\pi\)-periodic Mathieu functions of integral order.

Theorem 2. Let \(L_2=\operatorname{diag}\{\lambda_0,3\lambda_0,\ldots,(2N+1)\lambda\}\),

\[ C=\operatorname{codiag} \left\{ \begin{array}{cccccc} (N+2)\lambda, & (N+3)\lambda, & \ldots, & \ldots, & \ldots, & (2N+1)\lambda \\ (1/2)^2+(N+1)\lambda, & (3/2)^2, & (5/2)^2, & \ldots, & (N-1/2)^2, & (N+1/2)^2 \\ N\lambda, & (N-1)\lambda, & \ldots, & \ldots, & \lambda \end{array} \right\}; \]

\(D\) is the codiagonal matrix obtained from \(C\) by changing the sign of \(\lambda\) in the element \(c_{11}\).

If \(\varepsilon_c=-2(N+1)\lambda\), \(\varepsilon_s=-2\lambda_0\lambda\), and the sum \(\delta+\mu+\lambda_0^2\) is equal to an eigenvalue of the matrix

\[ C^*= \begin{bmatrix} C & -L_2\\ L_2 & D \end{bmatrix}, \]

to which there corresponds the eigenvector \((c_0^*,\ldots,c_N^*,d_0^*,\ldots,d_N^*)\), then equation (2) has the solution

\[ y=E(\tau)\sum_{n=0}^{N}\left[c_n^*\cos(n+1/2)\tau+d_n^*\sin(n+1/2)\tau\right]. \]

Corollary 2.1. If \(\varepsilon=-2(N+1)\lambda\), \(\delta+\mu\) is equal to an eigenvalue of the matrix \(C\), and to this value there corresponds the eigenvector \((c_0,\ldots,c_N)\), then equation (5) has a \(4\pi\)-periodic even solution

\[ y=e^{-\lambda\cos\tau}\sum_{n=0}^{N}c_n\cos(n+1/2)\tau; \tag{8} \]

if, however, \(\delta+\mu\) is equal to an eigenvalue of the matrix \(D\), to which there corresponds the eigenvector \((d_0,\ldots,d_N)\), then equation (5) has a \(4\pi\)-periodic odd solution

\[ y=e^{-\lambda\cos\tau}\sum_{n=0}^{N}d_n\sin(n+1/2)\tau. \tag{9} \]

Corollary 2.2. The solutions (8) and (9), as \(N\to\infty\), tend to \(4\pi\)-periodic Mathieu functions of integral order.

Received
27 X 1967

REFERENCES

1. J. Lbornik, Sitzungsber. Osterreichische Akad. d. Wissensch., Math.-Naturwissensch. Kl., Abt IIa, 166 (1957).