Abstract Generated abstract
The paper studies finite form solutions of a Hill equation with real period containing first and second harmonics. By applying an exponential substitution and seeking the remaining factor as a finite trigonometric sum with integral or half integral frequencies, the problem is reduced to algebraic eigenvalue conditions for explicitly defined matrices. The resulting theorems give parameter relations under which the equation has closed finite Fourier type solutions, and the corollaries specialize these results to even and odd periodic solutions related, in the limiting case, to Mathieu functions.
Full Text
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
- J. Lbornik, Sitzungsber. Osterreichische Akad. d. Wissensch., Math.-Naturwissensch. Kl., Abt IIa, 166 (1957).